Inelastic Scattering of Electrons with a Definite Momentum Transfer in a Crystal

A formulation is presented which describes the inelastic scattering of electrons by a crystal with a definite momentum transfer and a small energy loss. The Bragg reflection of both the elastic and inelastic waves is treated in the frame of the dynamical theory. This formulation is especially suitable for the discussion of the inelastic scattering in the plasmon-excitation process, where the Bragg condition for the inelastic wave is not so much altered from that for the elastic one because of the small change of momentum in this process. A numerical example for the intensity of the inelastic waves is demonstrated to describe the equi-thickness interference fringes formed by plasma loss electrons.