Oblique electromagnetic instabilities for an ultra relativistic electron beam passing through a plasma

We present an investigation of the electromagnetic instabilities which are triggered when an ultra relativistic electron beam passes through a plasma. The linear growth rate is computed for every direction of propagation of the unstable modes, and temperatures are modelled using simple waterbag distribution functions. The ultra relativistic unstable spectrum is located around a very narrow band centered on a critical angle whose value is given analytically. The growth rate of modes propagating in this direction decreases like $k^{-1/3}$