Oblique electromagnetic instabilities for an ultra relativis - tic 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 which value is given analytically. The growth rate of modes propagating in this direction decreases like k −1/3. The interaction of a relativistic electron beam with a plasma is a subject of relevance from many fields of physics ranging from inertial confinement fusion [1] to some astrophysical scenarios [2–4]. The linear analysis of the interaction reveals an highly unstable situation which has been investigated for a long time. Modes propagating along the beam are unstable within a certain range of wave-vector and form the electrostatic two-stream instability. The so-called filamentation instability is found for modes propagating transversely to the beam and finally, it can be proved that some modes propagating at arbitrary angle to the beam are also unstable [5]. As far as the growth rate is concerned, we see that it is eventually a function of the parallel and perpendicular wave vector components. As long as the beam is not relativistic, the largest growth rate are found for wave vectors parallel to the beam which means that the two-stream instability dominates the linear evolution of the system in this regime [6]. The situation evolves when considering a relativistic electron beam. Because relativistic electrons are harder to move in the direction of their motion than in the transverse direction, the two-stream growth rate is much more reduced than the growth rate of the modes propagating transversely, or even obliquely. If we denote γ b the beam relativistic factor, the maximum two-stream growth rate is scaled like γ −1 b , the filamentation growth rate like γ −1/2 b whereas the growth rate of the most unstable oblique wave vector is scaled like γ −1/3 b [5]. This shows that oblique instabilities should dominate all the more than the beam is relativistic. The ultra relativistic limit is relevant for astrophysical settings such as high energy cosmic rays or gamma ray bursts production scenarios, for which some authors consider relativistic factors up to 10 2 [2] and even 10 7 [3]. These unstable oblique modes were first investigated in …