Energy Losses and Collective Excitations in a Crystal

The relation which exists between the Coulomb energy losses of an electron passing through a crystal and the absorption of light by the same crystal is examined in the Born approximation. It is found that at long wavelengths the validity of the Born approximation is the only necessary criterion for the equality of the corresponding averaged matrix elements.The coupling of an incident electron to the collective modes of a crystal is next considered. The modifications of previous results for the transition probability, required by the presence of a crystal structure, are discussed. When the possibilities for Bragg scattering are extracted, a single function is found to describe the energy losses of a fast incident particle. Although simple band models may be chosen, approximations equivalent to the random-phase approximation lead to a result which retains the possibility of describing observed complex absorption spectra. It is demonstrated how a collective mode of the system may appear when the imaginary part of the polarizability varies from zero sufficiently rapidly.