Effects of incomplete phonon relaxation on x-ray emission edges in simple metals

Abstract

A theory for phonon broadening in x-ray emission is presented. The excitation and subsequent emission are treated as one quantum-mechanical process, within the Born-Oppenheimer approximation. The phonon part of the problem is exactly solvable for harmonic phonons with a linear coupling to the core hole. Explicit expressions for lifetime-dependent phonon broadening, line shape, and Stokes shift are given and calculated numerically for Li, Na, Al, and K. For long core-hole lifetimes the phonon broadening in x-ray absorption and in x-ray emission is the same, and there is a Stokes shift of twice the relaxation energy between the edges. When the lifetime becomes shorter the Stokes shift gradually disappears, and when the lifetime width $\ensuremath{\Gamma}$ is comparable to the Debye energy there is an additional temperature-independent contribution to the phonon broadening, which may be called incomplete phonon relaxation. This effect is small for Al, but it should be present in the emission spectra from the alkalis provided $\ensuremath{\Gamma}$ is in the right range. For Li, we obtain good agreement with both the observed Stokes shift and the emission edge shape reported recently by Callcott and Arakawa, and by Crisp, and the effect of incomplete phonon relaxation is likely to be the explanation of the longstanding problem of the broad Li emission edge. We also find indication that the effect is present in the emission spectra of Na and K.