Theory of Inelastic Processes in Low-Energy Electron-Loss Spectroscopy. I

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Abstract
The possibility of incorporating inelastic processes into a theory of low-energy electron diffraction from solids is considered. Particular emphasis is placed on the line shape and intensity of the bulk and surface plasmon characteristic loss peaks. The incident electron is regarded as an elementary particle capable of undergoing certain fundamental processes. A propagator is developed for the electron in terms of the dielectric properties of the crystal. Vertex amplitudes for emission of plasmons are derived. A distinction is made between bulk plasmons for a semi-infinite crystal and bulk plasmons for an infinite crystal. An equivalent optical potential is obtained for the semi-infinite dielectric and is compared to the infinite dielectric potential. In the course of discussing the surface plasmon vertex, a generalized Nozières-Pines sum rule is derived. The crystal scattering vertex is introduced and discussed in the light of previous work on elastic processes. The various diagrams contributing to the inelastic processes are then analyzed. With certain simplifying assumptions one obtains an analytic matrix element. Finally, we discuss how one extracts the desired line shape from the matrix element.