Approximate analytical solutions for continuously focused beams and single-species plasmas in thermal equilibrium

Edward A. Startsev, Steven Lund

Abstract: Simple, two-dimensional Vlasov–Poisson descriptions of thermal equilibrium have been applied to both an unbunched ion beam propagating in a continuous linear focusing channel and an unneutralized, single-species plasma confined in a Penning–Malmberg trap geometry. In scaled variables, these two thermal equilibrium systems result in an identical nonlinear equation that must be solved to describe the radial density and/or potential of the equilibrium. Numerous publications have been based on numerical solutions of the highly nonlinear equilibrium equation. Following a brief review of the thermal equilibrium model, we construct approximate, closed-form analytical solutions to the equilibrium equation that are accurate over a wide range of system parameters.