General properties of small-amplitude fluctuations in magnetized and unmagnetized collision poor plasmas. I. The dielectric tensor

Abstract

The dielectric tensor of a collision poor plasma determines all the physical properties of small-amplitude fluctuations for given initial plasma particle distribution functions, as it enters the Maxwell operator as the only plasma-specific quantity. For the important class of gyrotropic initial particle distribution functions fa(0)(p∥,p⊥) in a uniform magnetic field, we rigorously investigate the general properties of the plasma fluctuations without specifying the explicit momentum dependence of the gyrotropic distribution function. Two alternative forms of the relativistically correct dielectric tensor are derived which differ from nonrelativistic expressions in the literature. The first standard form is expressed in terms of infinite series of Bessel functions, whereas in the second form these infinite series are calculated with the Lerche–Newberger sum rules, yielding products of Bessel functions with complex indices for the individual elements of the Maxwell operator. The second form of the dielectric tensor is well suited to simplify the tensor in the special cases of parallel wave vectors and unmagnetized plasmas. For unmagnetized plasmas it is shown that aperiodic electrostatic and transverse fluctuations can only exist in symmetric distribution functions with f(−p∥,p⊥)=f(p∥,p⊥). Because this includes isotropic distribution functions, the more thorough investigation of this special case reveals that no electrostatic and fluctuations with positive growth rates γ=kcS>0 exist in an isotropic unmagnetized plasma, excluding both aperiodic (with R=0) and wave-like (with R≠0) instabilities, where R=ωR/(kc) denotes the real part of the phase speed. The second form of the dielectric tensor is also most appropriate to investigate fluctuations in magnetized equal mass plasmas, such as electron-positron-pair and/or proton-antiproton plasmas. Here for arbitrary wave vector orientation the dispersion relation factorizes into three separate modes. For fluctuations with parallel wave vectors in isotropic plasmas of arbitrary composition, the electromagnetic stability of such isotropic plasma populations is proven, relativistically generalizing the known corresponding nonrelativistic theorem.