Energy lost by a relativistic electron beam due to two-stream instability

The two-stream instability between a monoenergetic relativistic electron beam of energy γ0 mc2 and density nb and an unmagnetized, homogeneous plasma of density ne is investigated. For weak interactions, S=β20γ0(nb/2ne)1/3<0.45, the two-stream spectrum is predominantly one-dimensional, and the fractional beam energy loss is ΔE=S (1+S)−5/2. However, for strong interactions, S≳0.45, efficient coupling occurs to waves with perpendicular wavenumbers 0≳k⊥< (nb/ne)1/3 ωe/c. Independent of S ΔE≃0.18 for strong interactions. The analysis is supported by one- and two-dimensional computer simulations. In addition, it is observed that nonlinear wave particle interactions lead to rapid heating of the plasma in two-dimensional space. Symmetric bulk heating and high energy tail formation of plasma electrons is observed perpendicular to the beam direction. Plasma heating parallel to the beam is asymmetric (v≳0), since the large amplitude waves are generated as a result of beam trapping in the forward direction.