Theory of electron energy loss in a random system of spheres

We derive an expression for the inverse longitudinal dielectric function ε^-1(k,{ω}) of a random system of identical spherical particles with dielectric function ε₁({ω}) in a host with dielectric function ε₂({ω}). A spectral representation allows us to separate geometrical and material effects by writing ε^-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[-ε^-1(k,{ω})] we calculate the energy loss probability per unit path length for fast electrons passing through a system of colloidal aluminum particles.