Energy losses of fast electrons in multi-layer systems

Starting from Maxwell's equations a formula for the energy loss probability of fast electrons in systems of a given number of thin layers is derived for normal incidence. Retardation effects are neglected. For three layers of arbitrary thickness the equations are fully solved. As examples, a two-metallic-layer system with energy-dependent dielectric functions is discussed. Furthermore, the tarnish of metals e.g. Al–Al₂O₃, Be–BeO, Ag–Ag₂S is considered as three-layer systems. The energy loss spectrum of the system Al₂O₃–Al–Al₂O₃ is calculated in agreement with the experiment. It turns out that the bulk longitudinal optical mode of the covering layer is excited in the infra-red. In the case of the oxidation of a dielectric film the transversal mode of the oxide is expected to prevail in the energy loss spectrum.