Comments on core-hole lifetime effects in deep-level spectroscopies

The effects of a finite core-hole lifetime on deep-level spectra in solids have recently attracted much interest. We here present a solvable model which enables a detailed analysis of lifetime effects. For x-ray photoemission and x-ray absorption, our analysis gives support for the usual convolution approach where the infinite lifetime spectrum is convoluted with a Lorentzian lifetime broadening. In particular, we demonstrate the inadequacy of a recent intuitive approach which predicts a suppression of the phonon broadening for short core-hole lifetimes. This is done by testing the different theories on our model. We also show that a straightforward diagrammatic analysis leads to the same conclusion. Finally, we discuss the phonon response to the total process of core-hole creation and subsequent annihilation in an x-ray or Auger emission event. It is explained in some detail that the lifetime independence of the shake-up connected with the first step in this process is in no contradiction with the fact that the phonon system is uneffected by the total deep-level process in the limit of short core-hole lifetimes. We also analyze the process of $1s$ hole creation and subsequent Auger $K-LL$ emission and give explicit expressions for this process.