RelativisticK-Shell Photoeffect

Expressions for the relativistic $K$-shell photoeffect cross sections, correct to first order in $\alpha Z$ (inclusive), are established. For this purpose a second order calculation must be carried out, that is, electronic spinors correct to second order in $\alpha Z$ must be employed in the matrix element. The final continuum state spinor of the electron, whose exact analytic form is not known, is described by means of the Born approximation. To avoid the divergences, peculiar to the application of this method to the pure Coulomb field, the case of the screened potential is considered at the beginning. The matrix element, which is evaluated in momentum space, remains singular in the limit of no screening. Nevertheless, it is shown that the differential cross section, as issuing from a very laborious trace evaluation, is to first order finite in this limit and has the behavior one would expect. Indeed, its zero-order approximation in $\alpha Z$ coincides with Sauter's formula, as it should. Further, in the nonrelativistic and extreme relativistic limits the cross section determined reduces to results established by other means.