Radiative Corrections to the Scattering of Electrons and Positrons by Electrons

M. L. G. Redhead
1953 Proceedings of the Royal Society A  
219 accompanied by a negligibly small intensity of potassium X-rays. Allowing for the presence of the calcium Kfi component (relative intensity 0*10) the calcium K a line was found to have a Gaussian distribution with a standard deviation of 0-30 keV. The standard deviations of the calcium K fi and potassium K a and Kfi components were then calculated assuming that the standard deviation is pro portional to the square root of the energy (this was confirmed by measuring the width of the copper
more » ... dth of the copper calibration peak). I t is now possible to calculate, for a mixture of calcium and potassium X-rays, the ratio R of the number of counts above any selected energy, to the total counts in the peak as a function of n the ratio of calcium to total X-ray counts: in fact, R = A(E 0) + B(E0) n. From this equation n is calculated for the experimentally observed value of R. I t is clear th a t errors in this estimate of n can arise from several causes. The main ones are errors in energy calibration, statistical fluctuations of the counting rate, and errors in the estimation of the background. The effect of each one of these separately can be minimized for any given value of n by appropriate choice of E 0, but there is no unique optimum E 0. Taking into consideration the relative magnitude of each disturbing factor it wras concluded th at a value for E 0 of 3-69 keV (the calcium Ka energy) was about the best choice for the most important range of n, 0-4 to 0-1, involved in this work. Several runs were made on each source and the peaks analyzed. The mean values of the potassium and calcium X-ray counting rates thus obtained are given in tables 1, 2 and 3. The e6 corrections to the Moller formula for the scattering of electrons by electrons and the Bhabha formula for the scattering of positrons by electrons, arising from the interaction of the particles with virtual photons, are formulated using the Feynman-Dyson techniques. After removing ultra-violet divergences by mass and charge renormalization the crosssection still suffers from a logarithmic infra-red divergence. This is cancelled by adding on the cross-section for the production of a single real photon of low energy during the collision. The result is evaluated assuming that the maximum bremsstrahlung energy radiated is small compared with the rest energy of the electron, as viewed from the laboratory frame. Nonrelativistic and extreme relativistic approximations to the formulae are presented, together with the results of exact calculations for a laboratory energy of 20me2. I n t r o d u c t io n | A more detailed evaluation of radiative corrections to the Moller formula has also been made by G. Lomanitz in his 1950 Cornell thesis (unpublished), according to a footnote in the paper of Brown & Feynman referred to above. M = M1e + M2e2+ ...+ M i ei + (M ) | J f | a = j M% |2 e4 + 2MM2 e6 + 0(es), ( 1*2)
doi:10.1098/rspa.1953.0183 fatcat:zef4jtdbjvh4jm2a3cce6ujmiy