Retardation effect on energy losses of electrons moving parallel to solid surfaces

When a charged particle moves parallel and close to a solid surface, it suffers an energy loss arising from the induced potentials caused by the interactions between the charged particle and the surface. For the fast moving charged particle, the induced potentials could be affected by the electromagnetic retardation effect. In the present work, the retardation effect on the induced potentials was studied using a dielectric function with spatial dispersion for an electron of high energy moving parallel to the solid surface. Appropriate boundary conditions and the Lorentz gauge were employed to calculate the induced potentials by solving Maxwell equations in the Fourier space using the dielectric response theory. Analytical formulas of the differential inverse inelastic mean free path (DIIMFP), inelastic mean free path (IMFP), and stopping power (SP) were derived by considering the retardation effect using relativistic energy and momentum conservation relations and applying the extended Drude dielectric function with spatial dispersion. The DIIMFP, IMFP, and SP were calculated for electron moving parallel to the Cu surface with different electron energies. Results calculated with the retardation effect were compared to corresponding data without the retardation effect.