Resonant particle instabilities in a uniform magnetic field

We present here a simple expression for the resonant particle growth rate of arbitrary plasma waves propagating at an arbitrary angle to a uniform magnetic field in a spatially homogeneous plasma. We then are able to see that cyclotron harmonic wave-particle resonances contribute to instability when the resonant particle pitch angle distributions are anisotropic, while the phase velocity resonances of the Landau form are unstable to beam type phenomena. A sufficient condition for resonant particle stability is that particle distributions be isotropic and monotonically decreasing with increasing energy everywhere.