Self-consistent Vlasov equilibria for intense hollow relativistic electron beams

This paper discusses the procedure for constructing intense hollow relativistic electron beam equilibria within the framework of the steady-state (∂/∂t = 0) Vlasov–Maxwell equations. It is assumed that the electron beam propagates parallel to a uniform axial guide field B ext 0 = B 0 ê z inside a grounded cylindrical conductor, and that the positive ions provide a partially neutralizing background with density n 0 i (r) = fn 0 e (r), where f = const. = fractional neutralization. The equilibrium properties are calculated for the specific choice of electron distribution function , where H, P θ , and P z are the energy, canonical angular momentum, and axialcanonical momentum, respectively, for an electron moving in the equilibrium fields, and , and P 0 are constants. For this choice of distribution function, the mean axial velocity of the electron beam is equal to , and the beam density profile is hollow with inner and outer radii, R 0 and R 1 , determined self-consistently from nonlinear boundary conditions that involve the equilibrium parameters P o , γ b , γ 0 , etc., and the equilibrium self fields. Closed expressions for Ro and B1 are obtained for the case where the collisionless skin depth c/ω pe is large in comparison with the characteristic radius of the electron beam, and the beam density is sufficiently low that . The two cases, P o >0 and P o <0, are considered.