Electron spill-out effects in plasmon excitations by fast electrons

A new non-retarded hydrodynamic approach to the interaction between a fast electron and a diffuse metal-vacuum interface is presented. The metal is characterized by the parameters of a dispersive bulk dielectric function which slowly fade at the interface. The response of the medium is described by the induced charge density, which is self-consistently calculated. This formalism is applied to the study of the energy loss spectrum (EELS) experienced by a fast electron passing by a metal-vacuum interface. In the case of a sharp interface analytical expressions for the loss probability, fully equivalent to that of the Specular Reflection Model (SRM), are found. In an Al interface the effects of the electron density spill-out (modeled according to Lang-Kohn density) on both the longitudinal (EELS) and transverse components of the momentum transfer are studied. The influence of the interface profile on the surface plasmon dispersion in EELS is also discussed, showing that in agreement with previous theoretical and experimental works the dispersion of surface plasmon turns out to be much weaker than the one calculated in the SRM. A possible extension of the theory to study interfaces between transition metals and insulators is also discussed.