Analytical inversion of the dielectric matrix of a metallic superlattice of varying charge density: The angular dependence in the long-wave limit

The analytical inversion of the dielectric matrix of a superlattice (SL) of varying electron density is performed, giving the macroscopic dielectric function and the loss function of the system with local field effects included. The charge density is assumed to be arbitrary periodic function of one coordinate. The local hydrodynamic approximation is used. The long-wave limit is treated. It is proved, that, if the wave propagation is normal to the axis of inhomogeneity, then the system behavior is the same as for a uniform electron gas with a charge density equal to that of the superlattice, averaged over one period. The general formula obtained is illustrated by example of a model SL, which demonstrates the behavior of the macroscopic dielectric function and the loss function at different angles of wave propagation.