Many-Electron Theory of Nondirect Transitions in the Optical and Photoemission Spectra of Metals

The theory of the singular readjustment of a conduction band to a hole formed in an x-ray absorption event is extended to the case where the hole has finite mass, as in the $d$ band of Cu. Although the resulting recoil removes the singularity, the effect may still be quite large, and results in an electron-induced Debye-Waller-factor reduction of the intensity of direct ($k$-conserving) optical or photoemission events. This reduction depends on the mass of the $d$-band hole, and is accompanied by inelastic contributions in which the photon energy is shared between an interband transition and a number of low-energy electron-hole pairs.