Space charge compensation of thermal beams

Summary

The space charge compensation of round electron as well as ion beams is treated, using the equivalence of the KV envelope equation to the paraxial-ray equation. The compensation of cold beams is simulated first, solving the radial Poisson equation for a beam with uniform charge distribution and a compensating particle distribution according to Boltzmann's law. For maximum compensation a simple relation is obtained between the temperature of the compensating particles and the central degree of compensation. In contrast to simple expectations, the compensating particles concentrate closer to the beam for a higher degree of compensation. The focusing of thermal beams then is treated by extending the Pierce-Walker theory by an emittance term, finding the self-consistent distribution functions for thermal beams. The compensation of these beams gives similar results to those found for cold beams.