Energy-Loss Spectrum of Fast Electrons in a Dielectric Slab. I. Nonretarded Losses and Cherenkov Bulk Loss

Abstract

The energy lost, in the phonon-energy range, by a fast electron beam passing through a polar dielectric slab is analyzed using classical electrodynamics. The medium is represented by a frequency-dependent dielectric constant $\epsilon \left(\omega \right)=\frac{{\epsilon }_{\infty }({{\omega }_L}^2-{\omega }^2)}{({{\omega }_T}^2-{\omega }^2)}$ having one single pole at the transverse optical-phonon frequency ${\omega }_T$ and one zero at the longitudinal-mode frequency ${\omega }_L$. When retardation effects are excluded ($c=\infty$), two kinds of losses occur: bulk losses resulting from the emission of the longitudinal optical phonon at ${\omega }_L$ and surface losses due to the excitation of "surface" vibrations whose frequencies lie within the gap (${\omega }_T, {\omega }_L$). Both effects can be obtained exactly in terms of closed analytic functions of $\omega$. When retardation is taken into account, radiation losses take place as well, either through Cherenkov radiation in the bulk of the slab or through the so-called "transition radiation" occuring at the surfaces of the slab. Here we will study the Cherenkov loss only. Surface radiative losses are treated in part II. Comparison with experimental results for alkali-halide crystals and with calculations due to Fujiwara and Ohtaka is presented.