FACTA UNIVERSITATIS Seres: Physcs, Chemstry and Technology Vol., N o 4, 00, pp. 183-188 OPTICAL POTENTIAL APPROACH TO THE SLOW POSITRON SCATTERING FROM HELIUM ATOM UDC 530.145; 539.1 A. R. Tančć 1, M. R. Nkolć 1 Insttute for Nuclear Scence Vnča, lab 00, Belgrade, Yugoslava Unversty of Nš, Faculty of Scence and Mathematcs, Department of Physcs Abstract. We have extended our prevous calculatons to nclude the contrbuton of the random-phase approxmaton (RPA) (mproved) optcal potental to the low energy elastc scatterng n the electron+he system. Our mproved RPA calculatons are shown to be n better agreement wth the expermental values than other theoretcal results. Key words: postron-helum elastc scatterng, random phase approxmaton INTRODUCTION The exstence of postron was predcted theoretcally by Drac [1], and [] and Oppenhemer [3], and confrmed expermentally by Anderson [4] Blackett and Occhaln [5]. Postron has been extensvely used as probes n dfferent areas of physcs: n the hgh energy physcs, where the nteracton of postrons wth electrons gves very nterestng nformaton; n the condensed matter physcs the beams of postrons are used to obtan unque nformaton about crystal defects, ther formaton and confguraton, and, to determnate the Ferm (F) surface, too. The study of atomc collson phenomena has great mportance n the theory of thermonuclear research, chemcal reactons, gaseous electroncs and lasers. The electron-atom approxmate theores may be used as a senstve self-test by applyng them n the case of postron-atom scatterng. Durng the last 0 years postron-atom collson problems have been studed extensvely. The development of new expermental technques to obtan low energy monoenergetc beam has stmulated ths surge of actvty [6]. A comprehensve revew of progress n the postron atom (molecule) scatterng has been gven by many researchers [7]. In spte of mportant advances n recent years, both theoretcally and expermentally, our knowledge of postron-atom scatterng s stll ncomplete. The change of the sgn of the charge n e + -atom scatterng as opposed to the e -atom scatterng, has several mportant consequences. The exchange effects between the projectle and the target electrons are absent. In contrast to the electron, the postron s Receved November 1, 001
184 A. R.TANČIĆ, M. R. NIKOLIĆ attracted by the target electrons and repelled by the nucle. Ths attracton must be taken nto account adequately f accurate results are to be obtaned. In the case of postron scatterng by atoms the statc potental energy s postve, whereas the lowest order term n the polarzaton potental s negatve. Thus two major components n the e + -target nteracton tend to cancel each other. The slow postron scatterng has the specfcty that the ncdent e + combnes wth one of the electrons n the target to produce a bound postron-electron system (postronum). That s to say, a new nelastc threshold appears where postronum formaton s energetcally allowed. Ths nelastc threshold always les below the frst nelastc threshold of exctaton. The theoretcal analyss of postron scatterng from atomc systems represents a very dffcult task of scatterng theory, requrng that a double perturbatve expanson, wth respect to both e + - e and e - e Coulomb nteractons, respectvely, be carred out. In the case of e + - He scatterng accurate results have been obtaned by the varatonal methods [1,13,14], and the optcal potental method [14,15,16,17]. In ths paper we have mproved a method, whch recently was suggested n ref [15,16]. Ths method s based on the many-body theory, namely, the random phase approxmaton (RPA) [19], and a smple "energy shft'' technque was suggested whereby the nfluence of the vrtual postronum formaton (VPF) channel (.e. the spatal correlaton between the projectle and target electrons durng the scatterng) could be taken nto account. All notatons used n the paper follow those n ref. [15]. THEORY N +1 As an ntal approxmaton for the postron, the wave functons PE (radal part) are used, whch were obtaned from the "frozen" Hartree-Fock () atomc state. The excted contnuum state radal functons are normalzed to the δ-functons of the energy. The scattered postron wave functon can be wrtten as a soluton of the followng equaton [7]: Σ N + 1 1 Σ PE = PE Σ ( r, r'; E) PE ( r') dr' (1) H E N + 1 where H s the radal part of the sngle-partcle Hamltonan ( PE are ts egenfunctons) and Σ (r,r';e) s the optcal potental (.e. the rreducble part of the Σ sngle-partcle Green functon n the coordnate representaton). P have the asymptotc E form Σ E P 1 π {sn[ kr + δ πk ] + tan δ π cos[ kr + δ ]} where k E s the postron momentum, and δ s the correlaton correcton to the frozen-core value ( δ ) of the phase shft, δ = δ + δ. ()
Optcal Potental Approach to the Slow Postron Scatterng from Helum Atom 185 Fg. 1. Some of the RPA dagrams coreespondng to (3) The RPA contrbuton to the scatterng process was determned by usng the relatonshp [7] δ RPA N + 1 RPA N + 1 π PE Σ PE (3) RPA δ (E) gves the contrbuton to the phase shft, due to the RPA optcal potental and double vertcal lne represents reduced matrx elements. The expresson (3) corresponds to a frst Born approxmaton to a "two-potental formula" [19]. The effect of the Hartree dstortng feld (the frst potental) s fully taken nto account, n terms of the P N+1 functons. Fg.. Phase shft for s (δ 0 ) and p (δ 1 ) waves,---- our RPA results, - - - our results For the process whch wll be consdered the momentum-space T-matrx elements can be expressed n terms of the followng Lppmann-Schwnger relatonshp [15]: 1 1 1 0 k k1 k T ( ε ) k = k Σ( ε ) k + dk π k Σ ( ε ) k k T ( ε ) k + δ (4) where the -th radal component of the optcal potental have been evaluated at the ε = k scatterng energy. The optcal potental Σ s gven by Hartree term augmented by the RPA contrbuton: Σ(ε ) = Σ (ε ) + Σ RPA (ε ), Ther matrx elements can be convenently gven accordng to fg. 1a (the postron s depcted by double lne, the electron and hole by a sngle lne and the Coulomb nteracton by a wavy lne):
186 A. R.TANČIĆ, M. R. NIKOLIĆ k Σ RPA ( ε ) k = β F, j> F L, 1 k, β VL k1 1, j k11, j dk1 RPA 0 ε ( ε j εβ ) ε1 [( + 1)(L + 1)] V L + δ k. β 1 (5) The summaton β F s performed over occuped states, whereas j > F means the summaton s carred out over all dscrete excted states and ntegraton performed over the contnuum. The hgher "tme-forward" dagrams of the RPA method (fgures 1b. and 1c.) N ( LS ) are taken nto account by usng the wave functons P( β for the states ε ) j (.e. k ) (fg. 1a), calculated n the "frozen'' on core wth the hole n the state β. We estmated the contrbuton of the "tme backward'' dagrams (for example, fg 1d) - ther contrbutons was contrbutons was about 5.3%. Dagram 1e represent the results of the summaton of the nfnte sequence of dagrams whch represents the VPF terms. Fg 3. Total cross secton: sold lne our result, dash [15], dot-dash [16], dots [19], black square [13] In order to ntroduce the effect of vrtual postronum formaton VPF (the results of the summaton of an nfnte sequence of the dagrams whch represents the VPF terms are presented n fgure 1g) t s suggested [15] the "energy shft'' technque by replacng the bndng energy hole ε β n the denomnator of Eq. (5) by ε β E (where p E p = 0.5Ryd). The bound state of the postron-electron par s known as postronum.the bndng between a postron and an electron s the attractve Coulomb nteracton. The energetc of postronum formaton s descrbed by the Ore gap model (A. Ore, Naturvdenskap Rekko No.9, Unv. Bergen Arbok. 1973). It states that the probablty of postronum formaton s maxmum when the postron energy durng thermalzaton les wthn a gap where no other electronc energy transfer s possble. To capture an electron from an atom or a molecule n a medum wth onzaton energy E, the knetc energy
Optcal Potental Approach to the Slow Postron Scatterng from Helum Atom 187 Ep of the postron must be greater than E E {ps}, E {ps} beng the bndng energy of the postronum. In vacuum, the bndng energy of a postronum atom s 6.8 ev but may be smaller n the medum. If the postron energy E p s greater than the onzng energy E of the medum, the postronum atom s formed wth a knetc energy greater than ts bndng energy and t wll rapdly break up n further collsons. Furthermore, above the lowest exctaton energy E {ex}, nelastc processes occur. These processes defntely slow down the rate probablty of postronum formaton. Thus the postronum formaton s most probable wth the energy n the range E E {ps} < E p < E {ex} whch s the Ore gap. As a suggested n ref. [17] such energy shft s equvalent to lowerng the thresholds of the 1 S and 1 P nelastc channels by the same amount ( 6.80eV). We have solved equaton (4) numercally by usng the "mproper" calculaton for contrbutons of the RPA matrx elements (some of the contrbutons of estmated RPA thrd order dagrams). Fg 4. Momentum transfer cross secton: : sold lne our result, dash [15], dot-dash [16], dots [18], black square [14]. RESULTS AND CONCLUSIONS The contrbuton of dagrams wth transferred angular momentum = 0,1,,3,4 was calculated, and when > 4 the contrbuton was estmated. The values of the s and p phases are plotted n fgure. Ths s a clear ndcaton of the fact that the approxmaton (3) (whle useful for electron atom scatterng), breaks down for processes nvolvng postrons, as ponted out n ref. [16,17]. In the lower energy range (below about 4 ev) the contrbuton of the vrtual postronum formaton (VPF) channel (mssng from RPA calculatons) s most mportant. 4π The elastc scatterng cross secton (both total, σt = ( + 1)sn δ, and k = 0
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