Parametric Cherenkov instability of a relativistic electron beam in a dissipative medium with periodic density variation

The parametric Cherenkov instability of a relativistic electron beam is studied in a stationary periodic medium, taking into account the damping of electromagnetic waves in this medium. The spatial growth rate is found as a function of the transverse wave vector and the angle between the direction of beam propagation and the unit vector of the reciprocal lattice of the medium. It is shown that for a broad parameter region the behavior of the instability is determined by the dissipation of the medium. It is also shown that the maximum of the spatial growth rate is essentially independent of the angle between the beam and the reciprocal lattice vector.