The Elastic Scattering of Electrons from Cadmium Atom with the use of Model-Potential Approach

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1 Basah Jounal of Scienec (A) Vol.6(),77-84, 008 The Elastic Scatteing of Electons fom Cadmium Atom with the use of Model-Potential Appoach A. A. Khalf Physics Depatment, College of Science, Univesity of Basah, Basah, Iaq. Abstact Calculations of low-intemediate enegies of electons scatteing fom the gound state of cadmium atom ae pefomed using the optical model appoach. The diffeential coss sections (DCS s) ae calculated with the semi-empiical polaization potential added to the Hatee-fock potential of the fixed coe. The ageement between ou calculated (DCS s) and the available theoetical esults and expeimental data was good fo the electon scatteing. الاستطارة المرنة للا لكترونات من ذرة الكادميوم باستخدام نموذج الجهد البصري د.علاء عبد الحسن خلف قسم الفيزياء - كلية العلوم - جامعة البصرة البصرة - العراق الخلاصة استخدم الجهد البصري لا جراء حسابات استطارة الا لكترونات عند الطاقات الواطي ة- المتوسطة من ذرة الكادميوم في حالتها الا رضية. تم حساب المقاطع العرضية التفاضلية للا ستطارة باستخدام صيغة جهد تجريبية لجهد الاستقطاب مضاف ا الى جهد هارتري-فوك الا ستاتيكي. كان التوافق بين حساباتنا المستحصلة للمقاطع العرضية وما تم المقارنة مع ما متوفر من حسابات الباحثين وقياساتهم العملية جيدا لحالة استطارة الا لكترونات..Intoduction In the pesent investigation we have detemined the diffeential coss sections fo the elastic electon scatteing fom cadmium atom in the enegy egion (-75)eV. The method of patial wave along with optical model potential fomalism has been employed. Fo electons scatteing, the spheical symmetic optical potential compises of the static field, polaization and exchange inteactions. The only othe investigations on elastic ( e Cd) scatteing is that of Pangantiwa and Sivastava (989), to ou knowledge. Since the diffeential coss section povide moe igoous testing fo theoies and expeiments, as compaed to 77

2 A. A. Khalf The Elastic Scatteing... the total coss section (TCS), it maybe of geat inteest to pesent the calculated (DCS) of the electon fom cadmium. The optical model potential appoach (OMPA) has poved to be quite successful and attacted consideable inteest in the past few yeas to study electonatom elastic scatteing [K.Batschat et.al (988,990), A.Jain 990, B.H.Bansdent et al.985, and P.Khan et.al.984]. In geneal, fo shot-ange inteactions between pojectile and taget only a finite numbe of patial waves make contibutions to the scatteing amplitude, and the numbe of contibuting patial waves inceases with the incease in impact enegy. If the pojectile-taget inteaction has a long-ange tail, then in pinciple, an infinite numbe of patial waves will contibute to the scatteing amplitude.. Theoy In the optical potential appoach the basic idea is to analyze the elastic scatteing of a paticle fom a complex taget by eplacing complicated inteactions between the pojectile and the taget paticles by an optical potential o a pseudopotential in which the incident paticle moves. Thus once the optical potential is detemine, the many-body poblem is educed to an equivalent one-body poblem. The diffeential equation fo the scatteing electon, in atomic units ( e = m = =.0) is given by. ( + k Vopt ) Ψ( ) = 0 () Whee (k) is the incident wave vecto of the pojectile. Howeve, the equivalent optical potential, in geneal, is a complex, non-local, non-spheically symmetic and enegy-dependent potential and it is exact detemination is non-possible at pesent. One is equied to take V opt () in an appoximation fom. In this investigation () is epesented by a localized, eal, and enegy-dependent potential which is spheically symmetic, so that equation () can be solved using the patial wave method. We have pefomed V opt () in equation () by static field-polaizationexchange appoximation, and expessed as V opt 78

3 Basah Jounal of Scienec (A) Vol.6(),77-84, 008 V = V00 + V V () opt ex + pol Whee the static field V00 = Φ / V (,,..., Z, ) / Φ (3) Whee Φ,,...,, ) is the gound state wave function of the taget atom having ( Z (Z) electons (Z=54 fo Cd) and V,,..., Z, ) is the inteaction potential due to the taget and the incident paticle given as ( Whee (Q=-) fo electon. N NQ V (,,..., Z, ) = Q (4) / j= / j V pol () is the polaization potential and V ex () is the electon exchange tem. The non-local exchange tem is conveted into an equivalent local exchange potential following Tuhla (98), expessed as V ex 3 3 π 3 = [ ρ( )] (5) Whee ρ () is the spheical one electon chage of the cadmium atom. Hatee-Fock wavefunction fo the gound state of (Cd) atom as given by Clementi and Roetti (974), have been employed to evaluate V ( ) and ρ (). The semi-empiical fomula of polaization potential poposed by Nocoss and Seaton (976), given as 00 Whee V g pol α d g = (6) 4 = (7) 6 exp( ρ ) 79

4 A. A. Khalf The Elastic Scatteing... The facto ( α d ) is the static dipole polaizability, and it s value is (43.7 a 3 ) fo (Cd), whee ( a ) is Boh adius. ( g ) is the cut-off function designed to make the polaization potential finite at the oigin, (ρ) is the cut-off paamete, and equal (.5805) fo (Cd). Afte defining suitable choice fo V opt (), as given by equation (), we substitute it in to equation () and using the standad patial wave expansion fo Ψ (), then we get ( + ) ( + k V opt ) u = 0 (8) Whee u () is the adial scatteing asymptotic wave function given as π u sin( k + δ ) k (9) Whee ( δ ) is the phase shift coesponding to the th patial waves. The scatteing amplitude is obtained using the following expession: iδ f ( θ ) = ( + )[ e ] p (cosθ ) ik (0) The diffeential and total coss sections ae obtained in the conventional manne fom f (θ ). 3.Results and Discussion We pesent in Table() and figues (,) ou vaious calculated esults, whee figues pesent a compaison with the available theoetical and expeimental esults. Table(): The elastic scatteing phase shifts in (ad) fo thee patial waves (s,p,d) diffeent values of (E). 80

5 Basah Jounal of Scienec (A) Vol.6(),77-84, 008 E(eV) Electon Phase Shift δ δ δ In figue() we pesent the diffeential coss sections fo the scatteing of electons fom cadmium atom at the impact enegies (,5,0 and 5)eV. So fa, thee is no theoetical calculations o expeimental measuements ae available to compae it with ou esults at these enegies to ou knowledge. 8

6 A. A. Khalf The Elastic Scatteing... Diffeential Coss Section(a 0 /s) E= ev E=5 ev E=0 ev E=5 ev Scatteing Angle (deg) Figue(): The Diffeential Coss Section fo electon at incident Enegies (,5,0 & 5eV). In figue (a) ou (DCS) fo elastic scatteing of electons fom cadmium atom at (40)eV ae compaed with the expeimental data of Mainkovic et al.(987) and the calculations of Pangantiwa and Sivastava (989). In figues (b&c) ou (DCS) fo ( e Cd) elastic scatteing at the incident enegies (60 & 75)eV espectively, ae compaed with the coesponding absolute (DCS) measuements of Nogueia et al.(987), which ae available only up to scatteing angles of (70 o ) and the calculations of Pangantiwa and Sivastava (989). Ou esults at these enegies also show simila stuctues as shown by the expeimental and theoetical and ou esults at (40)eV impact enegy. Also ou esults show minima at a nealy the same scatteing angles as pedicted by expeiment. In the angula egion (60 o -76 o ) and (07 o -43 o ) ou esults ae found to be lowe than the expeimental data in Fig.(a). 8

7 Basah Jounal of Scienec (A) Vol.6(),77-84, Diffeential Coss Section (a o /S) 00 0 Pesent Wok Pangantiwa and Sivastava Mainkovic (a) Scatteing Angle (deg) 000 Pesent Wok (b) Diffeential Coss Section (a o /S) Pangantiwa and Sivastava Nogueia Scatteing Angle (deg) 000 Pesent Wok (c) Diffeential Coss Section (a o /S) Pangantiwa and Sivastava Nogueia Scatteing Angle (deg) Figue():The diffeential coss section fo electon scatteing by Cadmium at incident enegies (a)40 ev (b)60 ev (c)75 ev. 4.Conclusions We apply the patial wave method along with the optical model potential to calculate phase shifts and diffeential coss sections fo the scatteing of electons fom cadmium atom. Fo electons scatteing, ou esults wee in a good ageement with the calculations and expeimental measuements that we had made a compaison with it. 83

8 A. A. Khalf The Elastic Scatteing... The diffeence between ou esults and the calculations of Pangantiwa and Sivastava (989), maybe attibute to the following easons. The fomula of polaization potential and the scatteing amplitude of them quite diffe of us. We think that the neglecting of the inelastic channels like ionization, absoption channels effect in the behavio of (DCS) cuves, whee we notice the inceasing of minima in (DCS) as the enegy inceases. In etun this has come in getting on esults coesponding with the eseaches calculations. Refeences [].A.Jain, Phys.Rev.A 4,437(990). [].A.W.Pangantiwa and R.Sivastava,Phys.Rev.A 40,346(898). [3].B.H.Bansdent, I.E.McCathy and A.T.Stelbovics, J.Phys.B: At.Mol. Opt. Phys.8,83 (985). [4].B.Mainkovic, V.Pejcev,Dfillipovic and L.Vuskouic, in Abstact of Contibuted, Poceedings of the Fifteenth Intenational Confeence on the Physics of Electonic and Atomic Collision, Bighton, 987, edited by J.Geddes, M.B.Gillbody, A.E.Kingston, and C.J.Latima(Queen s Univesity, Belfast,987),P86. [5].D.G.Tuhla, in Chemical Application of atomic and Molecula Electostatic Potentials,New Yok,P3,98). [6]D.R.Nocoss and M.J.Seaton, J.Phys.B:At.Mol.Opt.Phys.9,983(976). [7].E.Clementi and C.Roetti, At.Data.Nucl.Data Table 4,77(974). [8].J.C.Nogueia, W.R.Newell and W.M.Johnstone, J.Phys.B:At.Mol. Opt.Phys.0,L537(987). [9].K.Batschat, R.P.McEachan, and A.D.Stauffe,J.Phys.B:At.Mol. Opt.Phys.3,349(990). [0].K.Batschat, R.P.McEachan, and A.D.Stauffe,J.Phys.B:At.Mol. Opt.Phys.,349(988). [].P.Khan, S.K.Datta,D.Bhattachayya and A.S.Ghosh,Phys.Rev.A 9,39 (984). 84