Following our discussion of XAS and its polarization dependence in the last two chapters we here continue with the discussion of two other important first order processes, namely x-ray emission spectroscopy (XES) and Thomson scattering, outlined in Sect.  II-4.5 and illustrated schematically in Fig.  II-4.4 b and c. X-ray emission is the inverse of the x-ray absorption process and both can be described within KHD theory by transitions between two states. The interaction Hamiltonian for the two processes is the same, given by \( \mathcal{H}_{int}= e \, \textbf{r} \!\cdot \!\textbf{E}\) in ( II-4.22 ). In XES, one starts with an initial state that is an excited state with a core hole in an inner shell and considers its decay to another excited electronic state, where the core hole is now in an outer electronic shell, typically the valence shell.