Paraxial Solutions for Accelerated and Decelerated Axially Symmetric Space-Charge Flow

Abstract

Particle trajectories in the presence of space charge are calculated for prescribed axial potential variations, including linear, parabolic, and sinusoidal variations, using the paraxial‐ray equation including space charge for axial symmetry. Solutions are also presented for the potential as a function of distance corresponding to prescribed hyperbolic and parabolic trajectories. The off‐axis equipotentials are calculated for some representative cases. These two‐dimensional results are interpreted in the light of one‐dimensional space‐arge theory in which exact solutions can be obtained for the potential as a function of distance. Some of the limitations of the present theory in which either the potential or the trajectory is prescribed are pointed out.