Unstable longitudinal plasma oscillations in a magnetic field: Nonrelativistic and relativistic considerations

The nonrelativistic and relativistic stability properties are investigated of longitudinal waves propagating in a plasma embedded in an ambient magnetic field, when the wave propagation direction is not necessarily either parallel or perpendicular to the ambient magnetic field. The analysis is based on the concept introduced by Harris [Phys. Rev. Lett. 2, 34 (1959)] of neutral points in wavenumber space to determine plasma instability to one side or the other of such neutral points. The critical need is to determine whether a particular plasma distribution function permits the existence of a neutral point. Relativistic considerations, although necessary to include for many astrophysical plasmas, complicate significantly the determination of instability conditions. In this paper it is shown how one can provide a general argument for such neutral point determination and for determining instability rates in the neighborhood of such neutral points. Only waves independent of resonant wave-particle effects are considered.