Excitation of Dielectric Spheres by Electron Beams

Abstract

Recent new developments and applications of the scanning transmission microscope(STEM) have stimulated renewed interest in the interaction of high-energy electron beams with small particles and surfaces[1–7]. Usually the classical theory of energy loss has been employed to analyze the experimental energy-loss spectra for planar, spherical or cylindrical geometries [8–14]. A new general approach, based on the self-energy formalism of the many body problem has been presented recently[15]. In this approach, the mean potential energy Σ_o of the incoming electron, in a state defined by a wave function ψ_o(r) and energy E_o is written as the average of an effective local potential V_eff(r), representing the complex interaction of the incoming electron with the many body target

$${\Sigma_o}=\int{\psi_o^*} (\bar r){V_{eff}}(r){\psi_o}(\bar r)dr$$

((1))

We use atomic units (e² = h = m) throughout this paper. The real part of Σ_o gives us the lowering of the energy of the particle due to virtual excitations of the medium, and the imaginary part is directly related to the probability of energy loss due to creation of real excitations. The effective potential is written as a spatial integral of the non local self-energy[16], Σ (r, r’, E_o), which in turn can be expressed in terms of the Green’s function G(r, r’, ω+E₀) and the causal screened interaction W(r, r’, ω)[16].