Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam

An analytical expression for the energy loss suffered by a fast electron passing near a homogeneous dielectric sphere is derived within a fully relativistic approach. The sphere is described by a frequency-dependent dielectric function. The electromagnetic field induced by the passage of the electron is then obtained by expressing the solution of Maxwell's equations for this geometry in terms of the scattering of the multipole expansion of the incoming electromagnetic field at the sphere. The energy loss is derived from the induced field acting back on the electron. The variation of the energy-loss spectra with both the radius of the sphere and the impact parameter of the electron trajectory is studied in detail. Part of the energy loss is transformed into radiation, which is also investigated. For spheres characterized by real dielectric functions, like those of ionic materials in the transparency-frequency region, it is shown that the entire energy loss is transformed into radiation. Examples of loss spectra and radiation emission spectra are given for a material described by a Drude-like dielectric function (e.g., Al) and for ${\mathrm{SiO}}_{2}.$