  The interaction of a fast electron with a photonic crystal is investigated by
solving the Maxwell equations exactly for the external field provided by the
electron in the presence of the crystal. The energy loss is obtained from the
retarding force exerted on the electron by the induced electric field. The
features of the energy loss spectra are shown to be related to the photonic
band structure of the crystal. Two different regimes are discussed: for small
lattice constants $a$ relative to the wavelength of the associated electron
excitations $\lambda$, an effective medium theory can be used to describe the
material; however, for $a\sim\lambda$ the photonic band structure plays an
important role. Special attention is paid to the frequency gap regions in the
latter case.
