Model for X-ray Cherenkov radiation by MeV electrons at grazing incidence

A possible source of radiation in the X-ray range can be obtained by penetration of a dielectric slab by electrons with sufficient kinetic energy. We present a near-exact solution of Maxwell equations to obtain the intensity of Cherenkov (and transition) radiation from slabs thicker than the absorption length, the only approximation being that the observation angle of the radiation with respect to the electron axis is taken to be small. Contrary to earlier models our model accounts for all relativistic electron energies, dielectric constants and grazing incidence angles. Also scattering of the electrons is incorporated very accurately, including the effect of scattering on the electro-magnetic fields within the slab. We report significant differences with earlier calculation for radiation yields from e.g. Si. A new effect has been found for intermediate angles of incidence (around 45 deg.). At these angles, the cone of Cherenkov radiation exhibits pronounced minima near the direction perpendicular to the plane of incidence. We present arguments based on the special characteristics of Cherenkov radiation why in that direction internal reflection of the radiation occurs at the medium-vacuum interface