Banerjee, S N

Sil, N C

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EFFECT OF HIGHER ORDER PARTIAL WAVE PHASE- SHIFTS IN ELASTIC ELECTRON SCATTERING BYHELIUM ATOMS. N. BANERJEE a n d  N. C. SilD k pa rtm ej t^  o p  T iiboretioa I i P h y sio s ,India?! Association' for the Cultivation op Soibnoe,J adavpue, Caloutta-32.(Received September 29, 1966)ABSTRACT. In elastic e“He scattering in the energy range of 13.6 ev to 54.4 ev we have computed in the Bom approximation the higher order phase-shifts ( i> 3 ) ; making use of lower order phase-shifts (t.e. for 1, 2) from the calculation of LaBahn and Callaway (1966), we find that the inclusion of the higher order phase-shifts considerably improves the value of the total and differential cross-sections in bringing them to better agreement with the experimental results.44I N T R O D U C T I O NRecently LaBahn and Callaway (1966) have investigated the elastic scattering of electrons by helium atom in the energy range of 0 to 60 ev. Taking into account both the exchange and polarisation effects, they have calculated the 8 , P  and D- wave phase-shifts in throe different approximations viz. i) adiabatic—exchange(ii) Dynamic-exchange with all components and (iii) dynamic-exchange with only the dipole component. They have not, however, considered the phase-shifts for I >  3, which are expected to be of considerable importance specially at higher energies.For scattering in the energy of 13.6 ev to 54.4 ev, we have computed in the Born approximation those higher order phase-shifts which contribute appreciably to the scattering cross-sections. For such energies and for such higher order phase- shifts, it will bo enough to take only the asymptotic part of the potential which in effect will be a term of the form —a/r* coming from the polarisation potential, tlie screened coulomb potential and other short-range potentials will have no appreciable contribution to it. As the exchange effect decreases with increase in energy and increase in order of phase-shifts, it will be consistent to neglect ex­change terms. The fact that the higher order phase-shifts have small values gives us the reasonable justification for using the Bom’s approximation in calculating the phase-shifts.358T H E O R YThe Born approximation for the phase-shift is given by* V(T)[ji{r)rr^dr 0Effect of Higher order Partial Wave Phaseshifist etc. 369whereand the symbols have their usual significances, V(r) is the total potential.With potential F(r) =  —{ajr*), we may Mfrite (cf. Mott and Massey, 1966)(hy-^ r(3)T(Z-2+3/2)P  - \ 2 /  • [r(2)p .f ( l+ 2+ i )the constant a is the polarisability of the He atom and its value is 1.370 Oq®. The differential cross section 1(d) is given bym  =  I S(2i+ l)c*’«. sin viPi (cos 6) | ®K q4:7Tand total cross section Q === y ~ S(2?+l)) sin^^ .^f^ orR E S U L T S  A N D  D I S C U S S I O N SThe phase-shifts in radioiis for I 3,4:, 5 and 6 waves are given in Table I  for five different energies.TABLE I IEnergy in ev 1^3 f==4 l==5 i = 613.6 .0374 .0124 .0067 —21.25 .04285 .019493 .010496 .0062930.60 .066834 .030379 .016358 .0098141.65 .08439 .0383 .02065 .0123954.40 .109786 .049903 .02687 .01622LaBahn and Callaway (1966) have calculated 8  ^ P, D wave phase shifts in three different types of approximations, of which the dynamic exchange approxi-360 S. N. Banerjee and N. C. Silmation with only the dipole component gives the best agreement with experi­ment. Hcncf  ^ we have calculated the total and differential cross sections by using the aV, P, D wave phawe-shifts of Labahn and Callaway (1966) djuamic exchatigo approximatin with dipole component only and the higher order phase shifts computed here in the Born a2>proximation.Fig. 1. Energy (ov). Tho total cross soctiou Q is plotted against energy. Curve A_Prosoiii Calculatioii; Curve 15—Caleulotiou by LtiBolm and Callaway (10G6) ; .—oxporimontal results of'Normand (1930); dasliod and dotted curve is that of Bandel and Golden (1965).In fig. 1 , wo have plotted our calculated vahics of the total cross section Q against energy for the dynamic exchange case with tho dipolo component only.CoesS3tDifforoiiiial cross section 1($) is plotted against 0 for 21.25 ev energy. Curve A—Pre­sent Calculation ; Curve B—Calculation using only S, P. D wave phase-shifts of LaBahn andCallaway; experimentalresults at 20 ov.Fig. 3-Differential cross seotion J(0) is plotted against 0 for 30.6 ev energy. Curve A—Present Calculation ; Curve B— Calculation using tho 8, P, D wave phase-shifts of LaBahn and Callaway.Effect of Higher order Partial wave Phaseshifts, etc. 301The theoretical results of LaBahn and Callaway (1966) in the dynamic exchange approximation with dipole component only and the experimental values of Bandel and Golden (1965) and Normand (1930) are shown for comparison. To get accurate­ly Normand’s experimental data of elastic scattering excitation cross section data of Gabriel and Heddle (1960) are added to those of ionisation cross section of Smith (1930) and the sum is subtracted from the total cross section data of Normand (1930), after increasing its value by 25% accojrding to the suggestion of Gabriel and Heddle (1960). The discrepancy still left n|ay be partially due to over simpli­fication of the dynamics of the scattering elytron in describing the distortion interaction and due to the influence of inelastic processes on the elastic scattering. In figs. 2 and 3, we have shown calculated diff(?rential cross section I{0) for 21.25 ev and 30.6 ev electron energies respectively against scattering angles 0 together with the available experimental resufes. These calculations have been done only for the dynamic exchange case in dipole approximation.From the above figures, we see that the tendency of higher order phase-shifts is io give a sharp increase in the differential cross section for forward scattering angles. I t  should be mentioned that we have neglected exchange in the energy range of 13.6 ev to 54.4 ev in our comjiutation of the phase shifts. This is justi­fied because in the energy x'ango concerned, influence of exchange decreases parti­cularly for higher order partial waves.A K N O W J. 3: D U M E X TThe authors are thankful to Prof. D. Basu for his kind interest and valuable discussions throughout the progress of the work.K E F E R E X O E SBandel, H. W., and Golden, D. E., 1965, P h y s .  R e v .  138, A14.Gabriel, A. B. and Hoddlo, D.W.O., 1960, P r o c .  R o y .  S o c . ,  4258, 124.LaBahn, K. W. and Oallaway, J., 1966, P h y a .  R e v . ,  147, 28.Mott, N. F. and Massey, H. S. W., 1965, T h e  T h e o r y  o f  A t o m i c  C o l l i s i o n s ,  Oxford at the Clarendon Press, Third Edition,Normand, C. E., 1930, Phys. Rev. 85, 1217.Smith, P. T., 1930, P h y s .  R e v . ,  86, 1293.

Effect of Higher Order Partial Wave Phase- Shifts in Elastic Electron Scattering by Helium Atom

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Effect of Higher Order Partial Wave Phase- Shifts in Elastic Electron Scattering by Helium Atom

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