Single-integral expressions for electron cyclotron damping, relativistically and with an arbitrary distribution function

Abstract

The relativistic expressions for the dielectric-tensor elements can be expressed as a double integral over both parallel and perpendicular momenta. The anti-Hermitian (damping) parts can be integrated over a delta function for either one of these two momenta, leaving a single integral over the other momentum variable. We review these expressions for either integration option, and compare them with statements in the literature. We give the limits on the remaining integral explicitly, for all positive or negative values of the parallel refractive index, whose magnitude can be less or greater than one. This single-integral form can be applied to electron cyclotron harmonic damping for any distorted electron momentum distribution function. As an example, we recover the results for a Maxwellian distribution. We also provide approximate results when the distribution function can be taken as a Maxwellian only in the perpendicular-momentum direction. A computer program will be written for a distribution function subjected to distortion by both electron cyclotron and lower-hybrid waves.