Theory of electron energy loss in a random system of spheres

We derive an expression for the inverse longitudinal dielectric function ${\mathrm{\varepsilon }}^{\mathrm{-}1}$(k,ω) of a random system of identical spherical particles with dielectric function ${\mathrm{\varepsilon }}_1$(ω) in a host with dielectric function ${\mathrm{\varepsilon }}_2$(ω). A spectral representation allows us to separate geometrical and material effects by writing ${\mathrm{\varepsilon }}^{\mathrm{-}1}$(k,ω) in terms of a spectral function, which depends only on the wave vector k and the geometry of the system. Multipoles of arbitrary order are included. Using a mean-field theory and introducing the two-particle correlation function, we carry out a configuration average and find a simple result for the spectral function. From the loss function Im[-${\mathrm{\varepsilon }}^{\mathrm{-}1}$(k,ω)] we calculate the energy loss probability per unit path length for fast electrons passing through a system of colloidal aluminum particles.