Accelerator Column Models for Low-Current Beams

This paper describes three analytic approaches used to model electrostatic accelerator columns in beam-transport codes for low-current beams and compares the results of each approach with the results obtained by numerically calculating the electric field based on charge distribution on equipotential surfaces. The three analytic approaches described are (1) a cubic energy-gain approximation, (2) a cubic longitudinal electric-field approximation, and (3) the aperture equation. The first two approaches calculate impulse approximations at the apertures, whereas the third is an integration of particle trajectories through the column field. The conditions under which the solutions tend to break down are discussed.