Plasma instabilities associated with anisotropic velocity distributions

The general dispersion relation is derived for small amplitude waves in a fully-ionized plasma in an external magnetic field. This derivation is based on the Vlasov equations (Boltzmann equations without collision terms for the electrons and ions plus Maxwell's equations). The dispersion relation involve integrals over the zero-th order velocity distributions. It is found that for sufficiently anisotropic velocity distributions waves exist which have exponentially growing amplitudes. A number of special cases are discussed. It is shown that streams of charged particles passing through a plasma may excite either longitudinal or transverse waves. Other instabilities exist when the distributions are such that the particles have their velocity vectors perpendicular to the magnetic field.