Ion sheet beams with parabolic density profiles

A model for a warm nonrelativistic ion sheet beam described by a reduced Vlasov–Maxwell approximation is extended to include parabolic beam densities. A single‐humped ion distribution function is found analytically for a stationary beam, whose parabolic density at the beam edge is zero. For a special model of a rippled sheet beam with a nonlinear focusing force, the ion distribution function, the condition for a matched beam, and the emittance for the beam are determined.