Thermal equilibrium properties of an intense relativistic electron beam

The thermal equilibrium properties of an intense relativistic electron beam with distribution function f⁰_b=Z⁻¹_bexp[−(H−β_bcP_z−ω_bP_ϑ) /T] are investigated. This choice of f⁰_b allows for a mean azimuthal rotation of the beam electrons (when ω_b≠0), and corresponds to an important generalization of the distribution function first analyzed by Bennett. Beam equilibrium properties, including axial velocity profile V⁰_zb(r), azimuthal velocity profile V⁰_ϑb(r), beam temperature profile T⁰_b(r), beam density profile n⁰_b(r), and equilibrium self‐field profiles, are calculated for a broad range of system parameters. For appropriate choice of beam rotation velocity ω_b, it is found that radially confined equilibrium solutions [with n⁰_b(r→∞) =0] exist even in the absence of a partially neutralizing ion background that weakens the repulsive space‐charge force. The necessary and sufficient conditions for radially confined equilibria are ω⁻_b<ω_b<ω⁺_b for 0⩽ (2ω̂²_pb /ω²_cb) (1−f−β²_b) ⩽1, and 0<ω_b<ω_cb for (2ω̂²_pb/ω²_cb) (1−f−β²_b) <0. Here, ω_cb=eB₀/γ_b mc is the relativistic cyclotron frequency, ω_pb= (4πn̂_be²/γ_bm)^1/2 is the on‐axis (r=0) plasma frequency, f=n⁰_i(r)/n⁰_b(r) = const is the fractional charge neutralization, β_bc= (1−1/γ²_b)^1/2c is the mean axial velocity of the beam, and ω^±_b= (ω_cb/2) {1±[1−(2ω̂²_pb/ω²_cb) (1−f−β²_b)] ^1/2} are the allowed equilibrium rotation frequencies in the limit of a cold electron beam(T→0) with uniform density n̂_b.