Theory of High-Energy Bremsstrahlung and Pair Production in a Screened Field

The differential cross sections for high-energy bremsstrahlung and pair production in a screened Coulomb field are calculated without the use of the Born approximation. It is shown that for pair production the correction to the Born approximation occurs only for momentum transfers $q$ of order $\mathrm{mc}$, for any amount of screening. For bremsstrahlung, however, the correction is only important for $q\text{'}\mathrm{s}$ of order $\left(\frac{m{c}^2}{E}\right)mc$, where $E$ is the energy of the electron. As in the case of no screening, the correction to the differential cross section for bremsstrahlung is found to be given by a factor multiplying the Bethe-Heitler cross section. It is then shown that the bremsstrahlung cross section integrated over the angles of the final electron is additive, just as in the case of pair production: one part is the Bethe-Heitler cross section including screening; to this is then added the Coulomb correction which is independent of screening.