In the fast ignition scenario for inertial fusion, a relativistic electron beam is supposed to travel from the side of the fusion pellet to its core. One one hand, a relativistic electron beam passing through a plasma is a highly unstable system. On the other hand, the pellet core is denser than its side by four orders of magnitude so that the beam makes its way through a important density gradient. We here investigate the effect of this gradient on the instabilities. It is found that they should develop so early that gradient effects are negligible in the linear phase.

We will consider relativistic electron beam interacting with plasma and study the electromagnetic instabilities obtained for arbitrarily oriented wave vectors ranging from two-stream to filamentation instabilities. For these unstable modes, we will study every temperature effects, namely beam and plasma normal, and parallel temperatures. Temperatures are supposed to be non-relativistic and modeled through water bag distributions. It is found that only normal beam temperature and parallel plasma temperature have a significative influence over the growth rates for wave vector making an angle with the beam larger than a critical angle θc which is determined exactly. The largest growth rate being reached for a wave vector making an angle with the beam smaller than θc, it is not damped by any kind of temperatures. We finally explore collisions effects and show they can reduce the largest growth rate.

We investigate intermediate unstable modes between two stream and filamentation instabilities. We detail the problem of the angle between the wave vector and its electric field and use an electromagnetic formalism allowing for any value for this angle. We display analytical results for 3 different models: cold beam-cold plasma, cold beam-hot plasma and cold relativistic beam-hot plasma. We demonstrate that plasma temperature prompts a critical angle for which waves are unstable at any k and show that for a relativistic beam, the most unstable waves are obtained for wave vectors which are neither normal nor perpendicular to the beam.