Effects of Nonlinear Space Charge and Magnetic Forces on Electron Beams Focused by a Solenoid Lens

Combined effects of both nonlinear space charge and magnetic forces are examined theoretically and experimentally for 5 keV, 80 - 200 mA electron beams focused by a short solenoid lens. A radial trajectory equation is derived which describes the path of electrons in the field of a short solenoid, including nonlinear magnetic forces up to third order and nonlinear space-charge forces. A computer code is described which calculates electron trajectories by solving this nonlinear equation. Trajectories launched far from the axis are strongly focused and cross the axis. Trajectories launched nearer to the axis are more weakly focused and are repelled by a strong space-charge force and do not cross the axis. As a result, a discontinuity develops in the particle distribution. This discontinuity is best seen in a phase-space representation where it appears as a "tearing" of the electron distribution. The beam breaks up into two components, the inner core and an outer "halo" that behave in distinctively different ways. Radial current density profiles are computed which show an initially uniform beam becomes markedly nonuniform near the waist. A test stand designed to experimentally see these effects is described. Photographs of beam images on a phosphor screen are presented to give a qualitative experimental picture of the beam profiles. More quantitative measures made with a Faraday cup are also provided. Both forms of measurements show the same basic features as the computed profiles, and a direct comparison reveals good agreement between experiment and theory. Applications to periodic focusing of electron beams in a long solenoid channel are briefly discussed.