Use of Furry-Sommerfeld-Maue Wave Functions in Pair Production and Bremsstrahlung

An analytic expression for the differential pair-production cross section is obtained using full Furry-Sommerfeld-Maue wave functions for the electron and positron. It is shown that this expression reduces to the Bethe-Maximon approximation and the Born approximation in the appropriate limits, and that it goes into the corresponding Elwert-Haug approximation for bremmstrahlung via the substitution rule. It is found that the cross sections of the Born approximation and the Bethe-Maximon approximation may be related to the cross section of the present approximation by a normalization theory similar to that used to relate point-Coulomb and screened "exact" calculations for pair production. No such theory works for bremsstrahlung because a larger-$r$ region is of importance in the bremsstrahlung process. The region of validity of each approximation is found by comparison with "exact" numerical calculations. There exists a region of low energies and small $Z$ where the present theory for pair production and the Elwert-Haug approximation for bremsstrahlung are valid and are significantly better than the Born and the Bethe-Maximon approximations. The failure of these calculations at low energies for intermediate and high $Z$ is attributed to the fact that the lowest partial waves for the electron and positron, which are dominant at low energies, are very poorly represented in the Furry-Sommerfeld-Maue approximation.