The investigation of the high-frequency interaction of a relativistic electron beam and a plasma is extended to include arbitrary variation of the plasma density. Analysing the coupled linearized Vlasov—Maxwell equations by means of a low-temperature expansion of the orbit integrals, a general equation for the electric field accurate to first order in the plasma temperature, beam temperature, and betatron frequency is obtained. This equation is applied to the investigations of transverse and longitudinal modes. A new transverse mode with |ω − kV0| ∼ ωβ is found to be collisionally unstable. The electrostatic instability is found to be slowed down by both low plasma temperature and low beam temperature, but the betatron oscillations increase the growth rate. A new longitidinal mode with |ω − kV0| ∼ ωβ, is found to be unstable for nonzero beam temperatures. The lowest order correction to the electrostatic growth rate due to a small plasma nonuniformity is obtained. The sign of this correction is found to depend critically on the shape of the inhomogeneity.
1D3V PIC simulation of propagation of relativistic electron beam in an inhomogeneous plasma
