Banerji, G

Sil, N C

English

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B\r G. B aniaeji an d  N. C SilDepartment of Theoretical Physics,Indian Association for the Cultivation of Science, Jadavput\ Calcutta-^2 {Received 4 Amfust 1970)(Jalculatioua uf dilToionlial ami total oi-(j,sh Hoctiotis for tho clastic Hcattcriiig ol oloctioiis and posilrojis by neutral liolmm atom have boon jjorfomied ovm- tho raTJgo ol impact cnoi;4ics n0“700cV oiiiploying the iSchwiiigor vaiiaticmal pnuciplo for scattering anijili- turle. The Hartioo-L^ock static potential for holimn atom has been used in tho calcula­tions. Tho results are compared with the Bmn approximation calculations and also with experimental findings.I ntroduction111 the piosent paper calculations are carried out Ibr the differential and total cross sections of tho elastically scattered electrons by helium atom for various (Miei’gies between 50 and 700 eV, the effects of exchange and polarization are considered to be not so important. The corresponding results for positron scat­tering arc also reported.Experimental measurements on the clastic eolliaion between electrons and lielium atiuns have been carried out by several woikors (Hughes et al 1932, Werner 1933, Vrieiis et al 1968, Bromberg 1969) over a Avido range of electron impact energies. A number of theoretical investigations have also been made on the c—He elastic scattering. The calculation of scattering cross sections of high energy electrons by helium atom has been performed by Mnkherjee (1961) in Born approxi­mation A\'here the use of a refined wave function which includes tho correlation lunction depending on tho mutual distance of the atomic eloctrons has been made. JCiin & Inokuti (1968) have used the twenty-term Hylleraas wave function of Hart & Herzberg (1957) in their Born-approximation calculations. Here we apply tlio Schwinger variational principle for tho scattering amplitude (Lippman & Schwinger 1950) to the same collision problem. The static field of helium atom IS represented by a bnear combination of several Yukawa potentials (Tietz 1966). The form of the trial wave function taken by us has been previously used by Mowoi- (1955) for tho calculation of differential cross section in tho elastic scattering of electrons from neon atom whore it has yielded results very close to the numerical solution.Unlike other variational principles, Sehwingei’ principle does not require that tile trial functions involved should have a particular asymptotic form. This principle lias been puhin two forms. In one, by maldng the usual expansions in spherical harmonics an infinite set of independent integral equations and hence[ndian J. Phys. 44, 431-437 (1070)E lastic sca tterin g  o f  e le ctro n s  and  p o s itro n s  b y  h eliu m  a tom431432 G . B a n e r ji a n d  N . C. Sila corresponding set of variational expressions for the phase shifts lias been obtained, while in the other, the entire scattering amplitude has been expressed in a sta­tionary form. The calculation of scattering cross section by summing over the individual phase shifts, though accurate, involves a lot o f numerical computations, whereas the calculation by the direct estimation of scattering amplitude is intriii- Hically much simpler.The choice of tJie potential as a linear combination of several Yukawa poten­tials is motivated by the fact that for this potential the integrals occuring in the variational iirincijile can be evaluated in closed form for some suitable trial func­tions This evaluation is possible lor the relatively simple form of the Fourier transform of tlui Yukawa potential. For other potentials the variational formula­tion may not yield closed form expressions for the scattering amplitudes Unless the integrals can be evaluated in closed foiin, computations using the Schwinger variational formulation become very tedious and have practically no advantage over exact numerical integration of the differential equation.M a t h e m a t i c a l , f o u m u l a t t o n—►The scattering of a particle of mass m by a potential F(r) is dcseiabed by an exact solution to the iritegral eqiiation(M(>tt & Massy 1905)^  e i -1- (1)vherc Tk I r-r> Iis the free space Green function for the Helmholtz equation and E ™ isthe energy of the incident particle. The vector =  k ni, where the unit vectorA. --- ► A111 specifics the direction of incidence and the vector r =  rn is the radius vectoi which specifies the position of the particle The amplitude for the scatteringfrom direction to direction — ^ 2 defined by/(n j.-re ,)  =  J e’*” '!''' V(r)fi(r)dr,whei'eU{r) =  2mV{r)!W.For the approximate determination of the scattering amplitude by the Schwinger variational method we take, following Mower (1955), the functional for the scat­tering amplitude as— 5 ir^{r)Ur)f^{r)dr + / J ft(r)U{r)0{r, r')U(r')^^{r')drdr'\ (2)Avhei’e and ^2. solutions of (1). Using trial functions of the form, ^   ^ —>, „  ikn..r , ™ -ikn^.r  . ,  „i/ri =  U^ e * +<^ 26 * , a =  1, 2the expression (2) for the scattering amplitude is obtained asElastic scattering of electrons etc. 433[ /(» !, - n , ) ]  =  2(7i/ji(n i.-m 2)+20a/jiK , TOj)— —” 2)—y&2(%t—^2)]— ng) fbz('*h> ^2)]’ \\ hwo/n (» i. - % )  =  J =  ( - X 1 1/1^)(3)(i)_  4n {-h\U\v)(v\U\k,) J*(2;r)3 J " ~ V~jfc2 /are, j cspoctively, the first and second terms in the Born series of approximations to the scattered amijlitude. Substitutingfbiinp 5j2)//6i(%> —^h) == ^ and/ft2(^i, —th) == —ng)tli(5 expression (3) may be rewritten as~  fb ii'^ v  —W2)[2(7 4^-2C'2A— —Wg)) - 2 UiC'2A(1-/a(Wi, Wa))]Now, adjusting the parameters according to the conditions d[f]ldCt =  0, i =  1, 2 the scattering amplitude is given byl /(%  — ^ a)J =/i»i(^i> — “ ^z) )~2A2( l “ / (^” n ^b))]/[( l-^ a K , -Wa))2-A2(l-/<-(S.p Wa))** (5)The Hartree—Pock screening factor (Tietz 1966) for neutral atoms can be expressed anal;y'tically in the formf[r) =  S oLte-'^ i^  (6a)iyo that the electrostatic scattering potential may be written as(6b)43 4 G. Baiierji and N. C. SilWith this potential the first and second Born scattering ami)litudes are obtained from (4) as (Morse & Bosliback 1953, Lewis 1956)2mZ V. at0( 7 )^vllcrcUeM(yi, y,) -  =  0 *In \tan'_x Tand , ... .2TJ^kh^ yj) ’- a -0 -- 0 and i =  ;jothorwis(3sp 1 , T2+(is!+/?)2r  '27r2/c«Rwith R =  r/c^ (-K'“+ r « H r r ) “—S =  k\K-My%+yjYl T ^  yiyj{yi-\ryi)---> --- > --- ► --- > ---  ^ --- > A, AK =  P — kj^ —k^ , 0 =  arc cos ( /^^ .(“ -^a))-Using the equations (7) and the definitions of A, fi, we can determine the scattciing amplitude from the expression (5) and licncc the differential cross-sections for the seattei'ing of particles by aii atom. The total scattering cross-section 0  may be obtained by integration of the differential cross-section through solid angle or by employing the optical theoremQ — '%)■R e s u l t s  a n d  d i s c u s s i o nWe have calculated the differential and total cross-sections of elastically scattered electrons and positrons having incident energy between 60 and 700 eV by the Schwinger variational method and the Born approximation. Into- grations over angles yielding total cross-sections have been performed numerically by Simpson’s rule with suitable intervals. The parameters occurring in the ex­pression for the potential (c.f. equations 6a and 6b), which reproduces the Hartree* Fock field of helium atom, areai =  1 0000, aa =  - 0  6195 =- -0.1846, === 0.6195, =  0 1846y, =  2.4907 =  3.8530 yg =  6.1212, y  ^ =  2.8530, y  ^ =  5.1212Elastic scattering of electrons etc. 435Tlio reaulia of our calculation for tho differential cross sections are shown in figure 1. Wo compare our results for the clc^ctron aoattoring for 50 oV and 700SCATTERING ANGLE (D E G R E E )Figures 1.....................  ejtperimental findings for oloolron scattering.--------------Born results----------•—  variational calculation for electron scattering. —. . —  variational calculation for positron scattering.436 G . B a n er ji a n d  N . C. SilAt high energies the results for the differential cross-section for electron and positron scattering are nearly tho same. But with the decrease o f energy there is a marked difference between the two results, as expected. The results for' the differential cross-sections for electron scattering obtained by variationa] calculations are in slightly better agreement with experiment than those calculated by Born-approximation at all energies. In the high energy region {E ^  600 oA^ ) our theoretical results for tho differential cross-sections agree well with the ex­perimental findings for all scattering angles, which contribute appreciably to the total cross-section. For small angle of scattering the ratio of the experimental findings and the corresponding Born results for the differential cross-sectious is quite large, more so for low incident energy. In the incident energy region of 300 eV and above, for larger angles, this ratio decreases below unity with the in­crease of angle. Again with further increase of scattering angle the value of the ratio gradually increases and finally becomes greater than unity. This character is also maintained in the variational calculation.We obtain the total cross sections for the systems in two ways, by the integration of differential cross sections over the solid angle and by using tho optical theorem. Tn table 1, we have presented these two results of total cross-sectionseV with tlie experimental data of Hughes et al (1932) and that for 300 eV witlithe experimental data of Vriens ei al (1968).T a b l e  1 Total cross sections Q (in units of where «o Bohr radius)for elastic scattering of electrons and positrons by helium atomIncident Q* Q energy (Born) (eV)’Q** <2***electrons positrons electrons positrons50 0.711006 0.8043 0.3131 0.89298 0 35746100 0.762 0.410396 0 4063 0.2204 0.40267 0 2672015150 0 443 0 287944 0.2767 0.1742 0 31380 0 21207200 0..308 0.221654 0.2111 0.1463 0.23731 0 17607300 0.190 0.1516999 0.1440 ■ 0.1095 0.16929 0 12014400 0.142 0.1152831 0.1096 0.8823X 10-1 0.11976 0.102044500 0.929577X 10-1 0.8808X 10-1 0.7415X 10-1 0.95901X 10■-1 0 842013x10“^600 0.778729X 10-1 0.7454x10-1 0.6394X 10-1 0 .79957xl0 - 1 0.71729x10-1700 0 609988x10'! 0.0434x10-1 0.6627x10-1 0.68549x10--1 0.62427x10-1________________ ^*Experimental results of Vriens et al (1908)**Calculated from tho variationally obtained differential cross-sections by integration through the solid angle♦*"'Calculated from the variationally obtained amplitude by using tho optical theoremElastic scattering of electrons etc. 437and the corresponding Born results and compared them with the experimental findings of Vrions et al (J 968) It is seen from the table that at 100 cV and abovetJie cross sections obtained by integration are nearly equal to the Born cross-sec­tions, but the cross-sections obtained by employing the optical theorem are always greater than those two results and closer to experimental findings.A cknow ledgem entThe authors thank Dr. S. C. Mukherjee for fruitful discussions They also thank The Kuljian Corporation (India) Pvt. Ltd., Calcutta, for extending facilities for using their IBM 1130 electronic computer.R bfeuencesnronibtM'p i). P. 1969 ,7 Chev}. Phys. 50, No. 9, 3906.Hart J. r  Si- Herzborg, CJ 1957 Phya. liev. 106, 79.Hughos A. L , IVIcMillon, J. H. & Webb, G M. 1932 Phys Hev 41. 154 Kim Y. Iv. St, IiiokuU, M 1968 Phys. Rev. 165, 39.Lewis R  R. 1956 Phya. Rev. 102, 537.Lippman 13. _A St Schwinger, J 1950 Phya. Rpv. 79, 469.j\loi\so 3’ . M. & Feshback H 1953 Methods of Theoretical Physics, Motirraw-Hill Book Co.Inc , New York,Mott N F, & Mas«oy, H S. W. 1965 The Theory of Aton}ir Collisions, Clarendon ProHS, Oxford. Mower L. 1955 Phys. Rev. 99, 1005. lUukberjoo S C. 1961 Indian J Phys. 35, 333.Tiefcz T. 1965 Nuovo Cimento 36. 1305.ViienR L,, Kuyatl, C K. & Mielczarok, S. R. 1908 Phys Rev. 170, No. 1, 163,Wornor S 193;i Proc. Roy. Roc A139, T13.

Elastic Scattering of Electrons and Positrons by Helium Atom

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Elastic Scattering of Electrons and Positrons by Helium Atom

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