Ion Wave Instabilities

The dispersion relation for electrostatic oscillations in a magnetic field is derived on the basis of the Boltzmann equation for arbitrary velocity distributions and for propagation in an arbitrary direction under the following restrictions: (1) The thermal velocities of the particles and the phase velocity of the wave are small compared to those of light; (2) the component of the wave vector k perpendicular to the magnetic field is small compared to the reciprocal of the gyration radius of an ion at the larger of the mean ion, and electron thermal, energies; (3) the magnetic field B is uniform. The dispersion relation is formally identical with that for electrostatic oscillations in the absence of a magnetic field. The dispersion relation is examined for stability under the further restrictions that: (4) The mean thermal energy of the ions is small compared to that of the electrons; (5) the electron distribution function for the component of the velocity v along B has a single maximum. It is found that a very small shift along B of this maximum relative to the ions leads to an unstable ion oscillation. The growth rates and frequencies of these oscillations are determined; their possible applications are discussed. Some further results on the ``two stream'' instability are given.