We present an analytic, Mie-theory based solution for the energy-loss and the
photon-emission probabilities in the interaction of spherical nanoparticles
with electrons passing nearby and through them, in both cathodoluminescence
(CL) and electron energy-loss spectroscopies (EELS). In particular, we focus on
the case of penetrating electron trajectories, for which the complete fully
electrodynamic and relativistic formalism has not been reported as yet. We
exhibit the efficiency of this method in describing collective excitations in
matter through calculations for a dispersive and lossy system, namely a sphere
described by a Drude permittivity, and discuss possible complications when
computing contributions from higher-order modes. Subsequently, we use the
analytic solution to corroborate the implementation of electron-beam sources in
a state-of-the-art numerical method methods. We show that the two approaches
produce spectra in good mutual agreement, and demonstrate the versatility of
DGTD via simulations of spherical nanoparticles characterized by surface
roughness. The possibility of simultaneously employing both kinds of
calculations (analytic and numerical) facilitates a better understanding of the
rich optical response of nanophotonic architectures excited by fast electron
beams.Comment: 10 pages, 6 figure