Probability and shape of the spectral line of a single bulk characteristic energy loss of a fast electron in a medium with electron absorption and strong spatial dispersion

The probability of single characteristic energy loss of a fast electron in a reflection experiment has been calculated. Unlike many works concerning this subject, the bremsstrahlung of bulk plasmons in the non- Cherenkov ranges of frequencies and wavevectors of a plasmon has been taken into account. The contributions to the probability of single loss and to the shape of the spectral line from a quantum correction that is due to the interference of elastic and inelastic electron scattering events have been determined. The probability has been calculated in the kinetic approximation for the relative permittivity, where the short-wavelength range of the plasmon spectrum is correctly taken into account. In view of these circumstances, the expression for the mean free path of the electron with respect to the emission of a bulk plasmon that was obtained by Pines [D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963)] has been refined. The coherence length of the fast electron in the medium-energy range under consideration has been estimated. The shape of the spectral line of energy losses in the non-Cherenkov frequency range has been determined. It has been shown that the probability of the single emission of the bulk plasmon incompletely corresponds to the Poisson statistics.