Threshold Behavior of the X-Ray Spectra of Light Metals

Abstract

A simple model to describe the x-ray process in light metals is used to study the transition rate near the threshold. For the absorption case, the problem consists of calculating the response of a band of free electrons to the sudden creation of a core hole potential and to the injection of the core electron into the band. Recently, Nozières et al. showed that the transition rate becomes singular at the threshold. We offer an explanation for how this comes about, and we give a rederivation of Nozières' result by a different approach, related to Zubarev's method of Green's functions. We set up and solve, for large times, integral equations for quantities directly connected to the response function. The model is studied explicitly in the case of absorption for spinless particles and $s$-wave scattering. The results can be extended to include emission processes, electrons with spin, and higher angular momenta.