Impact-parameter dependence of the relativistic photoeffect and other high-energy photoprocesses

Abstract

The resolution in space of processes involving high-energy photons incident on atoms or bare atomic nuclei is investigated. A simple analysis, based on momentum transfer, gives first indications of the length scale being defined by the Compton wavelength of the electron for both the photoeffect and electron-positron pair creation with the electron bound to the atomic nucleus. Since the simple method of converting a momentum transfer q to a distance of $\ensuremath{\Elzxh}/q$ has potential pitfalls, we continue with a detailed wave-packet study. This study, which is undertaken for the case of the photoeffect, involves the incidence of a photon localized in space and time on a hydrogenlike atom. The wave-packet approach confirms the Compton wavelength, and not the extent of the atomic state, to be the decisive measure for photon energies in excess of the electron rest energy ${\mathrm{mc}}^{2}.$ In addition, it provides a direct and detailed picture of the impact-parameter dependence of the process. As an introduction to the wave-packet study, we compare calculations based on a plane-wave representation of the unbound lepton to lowest-order perturbative calculations.