Calculation of Bremsstrahlung Cross Sections with Sommerfeld-Maue Eigenfunctions

Abstract

A formula for the differential cross section of bremsstrahlung is calculated with the aid of Sommerfeld-Maue eigenfunctions, i.e., under the assumption of a pure Coulomb field and low atomic numbers ($\ensuremath{\alpha}Z\ensuremath{\ll}1$). This expression is valid for all energies of both electrons and photons, and it is shown that the previously well-known formulas of Sommerfeld, Sauter, Bethe, Heitler, Scherzer, and Bethe and Maximon are contained as special cases. Hence this cross section is correct for high electron energies and, apart from a small spin correction, for nonrelativistic energies even if $\ensuremath{\alpha}Z\ensuremath{\ll}1$ is not satisfied. The formula has been programmed, and in addition total cross sections have been obtained by numerical integration over the angles of the emitted electrons and photons. Comparison with the Born approximation allows derivation of the Coulomb correction. Several cases of interest are compared with experiment, in particular the elementary bremsstrahlung process itself, for which experimental results are now available.