Transition from statistical Coulomb interactions to averaged space–charge effect

The statistical Coulomb interaction effect and the averaged space–charge effect, which broaden electron-beam size, have been studied to clarify their applicable regions and their transition region as a function of beam current. The beam size of a single beam segment can be analytically estimated by two theories based on different approaches: the expression of the trajectory displacement effect based on stochastic electron–electron interactions, and the beam radius equation based on the Coulomb repulsion force by averaged space charge. Both theories were compared with Monte Carlo simulation results for a beam-current ranging from 0.1 nA to 10 mA. It was confirmed that the beam size at the end of the focusing segment could be predicted by the expression of the trajectory displacement effect in a low beam current and by the beam radius equation in a high beam current. In addition, the beam sizes predicted by these theories were found to result in almost the same for a medium beam current. The theories transited as the beam-current increased.