## Introduction

A quantitative understanding of any spectroscopy technique based on the analysis of reflected, transmitted, or emitted charged projectiles from solid surfaces relies on an accurate description of the inelastic interaction of the projectiles with the target. In the bulk of the solid, the stopping of the projectile can be accurately described within the semiclassical dielectric formalism for infinite media: the presence of the charged projectile perturbs the equilibrium charge distribution of the solid, which becomes polarized and induces an electromagnetic (e.m.) field that acts as a stopping force on the projectile. Within the semiclassical approximation, the Fourier spectrum of the field set up by the projectile is interpreted as a distribution of energy losses and momentum transfers at individual, discrete inelastic interactions. Thus, the semiclassical approximation serves as the link between the classical dielectric description and the quantal nature of inelastic interactions. The presence of an interface, e.g. solid-vacuum or between two solids, imposes strict boundary conditions on the total (and hence also on the induced) e.m. field. Thus, the stopping of the projectile close to the interface is substantially different than in the bulk of the solid: an additional dependency is found on the distance to the interface (at either side) and on the surface crossing direction.

A collective response of the electrons in the surface region of the solid due to the passage of charged projectiles through the interface was predicted in the 1950s by Ritchie[1] and confirmed experimentally by Powell and Swan.[2, 3] A number of models have been developed in the last decades to understand the collective (and single-particle) response.[4-23] Different approaches and approximations are adopted: some models use a classical electrodynamics framework whereas others use many-body quantum theory, some assume a simplified dielectric response of the solid, some are valid for particular trajectories of the projectile (e.g. perpendicular or parallel to the surface). Simplifying mathematical assumptions are often made in order to highlight the relevant physics, to obtain more treatable expressions, and to keep the computation time within reasonable limits. Unfortunately, the effect of the different approximations on the resulting physical description of the stopping of the projectile has not been scrutinized.

It is the purpose of the present work to present a self-contained description of the dynamics of charged projectiles near a planar interface within the semiclassical dielectric formalism, taking special care to avoid further approximations such as those addressed above. A common framework is provided which allows one to compare a subset of models in the literature and to expose the effects of their implied approximations. The calculation procedure presented here has been implemented in a Monte Carlo algorithm for the simulation of reflection-electron-energy-loss spectra. Finally, the quality of the model will be assessed by means of comparisons of simulated reflection electron energy loss spectra (REELS) with experimental REELS on an absolute scale.

This work is structured as follows: In section “Theory”, the dielectric formalism for the stopping of charged projectiles in an infinite medium is briefly reviewed and extended to take into account the presence of a planar interface to another medium. An expression for the differential inelastic inverse mean free path (DIIMFP) of a charged projectile in the vicinity of a planar interface is derived. In section “Results”, the dependence of the derived DIIMFP on the velocity of the projectile, on the distance at each side of the interface, and on the surface crossing direction is examined. Calculated DIIMFPs are compared with those given by another model from the literature, exposing the effects of its implied approximations. Next, a Monte Carlo algorithm for the simulation of electron reflection experiments is described. Simulated spectra are compared in absolute units with experimental spectra. Finally, in section “Conclusions”, the presented results are summarized, and conclusions are drawn.