Radiative Tail for Inelastic Electron Scattering

The cross-section differential in the energies and angles of all final particles for the bremsstrahlung which accompanies nuclear excitation is derived, in first Born approximation and neglecting nuclear recoil, in terms of the form factors corresponding to the Coulomb and current interactions, which are left arbitrary through-out the calculation. This cross section is integrated over all photon angles, without approximation, and its limit for large energies ($E\gg m{c}^2$) of the initial and final electron is discussed. As in the case of elastic electron scattering considered previously, photons emitted in the direction of either the incoming or outgoing electron contribute to this integrated cross section terms of order $\ln \left(\frac{E}{m{c}^2}\right)$, while all other photon directions contribute terms of relative order 1. We give explicit expressions for these terms, as well as a numerical evaluation for some typical cases; the formulas are also valid for electron scattering angles $\vartheta$ equal or very close to 180°. It is shown explicitely that the logarithmically divergent terms which appear in the integral of our high-energy cross section over the energy of the final electron are cancelled exactly if to this integral one adds the radiative corrections to the inelastic electron scattering cross section.