Incomplete Relaxation in a model problem for the Auger process

Excitation of a core electron and the subsequent emission of an Auger electron is described in a simplified model, where the core electrons couple only to the plasma oscillations of the conduction electrons and not to individual electron-hole excitations. Results are evaluated to lowest nontrivial order, namely when only terms linear in the interaction with the exciting probe and linear in the Auger coupling are kept. This leads to a four-particle Green’s function which can be evaluated exactly within the present simple model. It is shown explicitly that for long core-hole lifetimes the Auger spectrum is the same as in the two-step approximation (complete relaxation). The possibilities of high-energy satellites in the case of short core-hole lifetimes are discussed.