Radiative Tail in Elastic Electron Scattering

The Bethe-Heitler cross section with arbitrary form factor is integrated over all photon angles,without approximation. The limit of this integrated cross section for large energies $\epsilon$ of the initial and final electron is considered in detail. It is shown that the contribution to this cross section from photons emitted in the direction of either the incoming or the outgoing electron gives the terms of order $\ln \epsilon$ found by Schiff, but that the contribution from all other photon directions is of relative order 1, and we give explicit expressions for these terms, as well as a numerical evaluation for some typical cases, since for the energies of experimental interest, viz., 30-1000 MeV, the logarithm is not very large: $\ln \epsilon \approx 4-8$. Furthermore, keeping the terms of order 1 is of particular importance for electron scattering angles $\vartheta$ near 180°, since, as is shown, the terms in $\ln \epsilon$ all have the factor ${cos}^2\frac{1}{2}\vartheta$, which is not the case for the terms of order 1. It is shown that all the formulas given are also valid for $\vartheta$ equal or very close to 180°.