EFFECT OF VACUUM POLARIZATION ON THE EXCITATION OF HYDROGEN ATOM BY ELECTRON IMPACT

The vacuum polarization potential is included in the interaction Hamil- tonian for IS 2S excitation of the hydrogen atom by electron impact. The excitation amplitude calculated field theoretically is found to be lowered by

with close-coupling approximation showing good agreement with the measurements of Williams and Willis [4].Callaway, McDowell and Morgan [5] did the calculation in a hybrid pseudostate close-coupling distorted wave model.Here also agreement with the experimental result is good.Roy and Bhattacharyya [2] tried the problem in a field theoretic way and compared the result with the then available experi- mental result of Stebbing, Fite, Hummer, and Brackmann [I].Even now this simple scattering problem in atomic physics is not a closed chapter.There is room for theorization and approximation.
Although vacuum polarization gives a very small correction to the Coulomb potential, it is of interest to see its influence on the excitation amplitude.
In the muonic atom the effect of vacuum polarization has been extensively studied and found to affect the energy levels [6].In non-re,lativistic approximation vacuum polarization is known to affect the hydrogen S-states and in particular the 2S level is lowered by 27 MHz relative to the 2P level.As such it is not out of place to look for the effect of vacuum polarization on the excitation amplitude of the hydrogen atom.

MATHEMATICAL FORMALISM.
Quantum electrodynamics predicts deviation from Coulomb potential behavior near a point charge.The potential energy of a negative charge -e located at a distance r from another point charge is not given entirely by the Coulomb expression e 2 [ 0(r) 0(r') d 3 J ir_ r, r dBr There is a correction of order , the Uehling potential.It arises from the polarization of the vacuum of the virtual electrons of mass m and is given by [6] e 2 [ 0(r) o(r') { Ir_ r, Z0(r r )} dBr d3r The amplitude for IS 2S excitation of the hydrogen atom by electron impact due to static Coulomb interaction V I as computed by Roy and Bhattacharyya [2] is Mfi 4/ Z *rr' ss' 63(k_ + k,_,) gcd ('k3) gab(kk'k3) -ik,--,i ab,cd , , d3k d3, d3k 3 X r X s X r' X s' d3 d3k (2.5) which can be written as ab,cd *rr' ss' , , We can proceed in a similar way to compute contributions of the vacuum polarization x d3k d3 d3k d3 d3R d3k 3 (2.7) The total amplitude W for the process in the co-ordinate space is obtained by adding Mfi to M'fi, and integrating over k, k', , ' and k 3 we get as in the #ab(X, X+R) and cd(X' X+R) are initial and final state solutions respectively of the Schr6dinger equation for the combined system of hydrogen atom and a free electron.
Writing these in a slater determinant form, equation (2.8) becomes W 463 f I ab,cd (Pc Qc 2S (X) Xc 2s(X+R) Xd , , f (X) Xc f (X+R) Xd Is(X) Xa Is(X+R) Xb i (X) Xa i (X+R) Xb 1 (R)) d3X d3R where Is(X), 2s(X) are respectively the ground state and 2S excited state wave functions of the hydrogen atom.i(X) and f(X) are the plane wave solutions for the inci- dent and scattered electrons respectively.After integrating over X and R (all quantities are in atomic units) and taking for Pauli spin vector Putting Z01R) from (2.3) in A I and knowing that R varies from 0 to % (the Compton wave length for electrons), integral A I becomes %2 i- A I 4I D -(2n% i) (2.12)  being very small (45.778  transfer this reduction becomes -0.47 t2/729 while for high momentum transfer its value is -0.47/t4.Thus change in the amplitude for low momentum transfer is greater than that for high momentum transfer.These changes are very small and give a correction of the order of 0.4 to 0.5 per cent to the differential cross-section.So there is every possibility for these to escape detection during experiment.But with the increase in the precision of the experi- ment we hope that these small changes may be detected in the future.

ACKNOWLEDGFRMENT.
We would like to thank U.G.C. for financial support of this work under the project "Atomic and Molecular Collisions" at Jadavpur University.
is the Euler's constant and 0(r) is the charge density of the electron field [ *(r) ir) d3r p(r) J (2.4) C O + D O +-(A I + B I + C I + D I) (2.10)With P and Q as the momenta of the incident and scattered electrons, 0 0.375 ID(t + 9)/t I D 512t2/(t 2 + 9) 3,

3 exp
integrate B1, CI, and D I we have taken only the first term in the expansions of