Inelastic scattering of fast electrons by crystals

Generalized fundamental equations for electron diffraction in crystals, which include the effect of inelastic scattering described by a nonlocal interaction, are derived. An expression is obtained for the cross section for any specific type of inelastic scattering (e.g., inner-shell ionization, Rutherford backscattering). This result takes into account all other (background) inelastic scattering in the crystal leading to absorption from the dynamical Bragg-reflected beams (in practice mainly due to thermal diffuse scattering). There is a contribution to the cross section from all absorbed electrons, which form a diffuse background, as well as from the dynamical electrons. The approximations involved in assuming that the interactions leading to inelastic scattering can be described by a local potential are discussed, together with the corresponding expression for the cross section. It is demonstrated by means of an example for K-shell electron energy loss spectroscopy that nonlocal effects can be significant.