Upper bound of the beam energy broadening in acceleration region

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 /sub UB/. Assuming that the emitter is a sphere of radius R with potential V, we obtain the expression /sub UB/ approx./sub //sup proportional/ m1/sup //4 I/sup 1/2//sub e/ V-1/sup //4 T/sup 1/2//sub eff/ α-1/sub e/ R0, where m is the particle mass, I/sub e/ is the total emission current, V is the acceleration voltage, T/sub eff/ is an effective temperature showing the kinetic energy of particles at the emitter surface, and α/sub e/ is the beam semiangle at the emitter. Theory gives /sub UB/ approx.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