Upper bound of the beam energy broadening in acceleration region

Abstract

Using an expression of the local space-charge force derived in a previous paper, we analyze the broadening of axial energy distribution of beam particles in the acceleration region and calculate an upper bound of the energy broadening 〈‖ΔE‖〉UB. Assuming that the emitter is a sphere of radius R with potential V, we obtain the expression 〈‖ΔE‖〉UB ∼∝ m1/4 I1/2e V−1/4 T1/2eff α−1e R0, where m is the particle mass, Ie is the total emission current, V is the acceleration voltage, Teff is an effective temperature showing the kinetic energy of particles at the emitter surface, and αe is the beam semiangle at the emitter. Theory gives 〈‖ΔE‖〉UB ∼several eV for typical electron guns operating in space-charge unlimited conditions. The result is compared with experiments and a discussion is given of the problem we have to solve to calculate the magnitude of 〈‖ΔE‖〉.