Calculations of the electron-energy-loss probability in thin uniaxial crystals at oblique incidence

The complete differential electron-energy-loss probability is calculated for scattering of a fast electron by a thin uniaxial crystal foil at oblique incidence. Retardation effects are considered in the present calculation. In the long-wavelength limit the local dielectric tensor ${\lim }_{q\to 0}\overleftrightarrow{\epsilon }(\overrightarrow{\mathrm{q}},\omega )$ is expected to give a good description of the response of the electron gas in solids to the fields of the incident electron. Using this dielectric approach, the scattering cross section of both volume and surface excitations are obtained. Plasmon anisotropy and the excitation of both extraordinary and ordinary Čerenkov radiation are discussed. The dispersion relations for both extraordinary and ordinary surface waves are also obtained and studied in detail. As an example, we have calculated the scattering probability distribution function for a 500-Å-thick graphite foil under normal incidence of 75-keV electrons.