Equilibrium solutions for partially immersed relativistic electron beams

The equations relating the beam radius and axial electron velocity to easily measurable external beam parameters are developed for solid, relativistic beams. The beam parameters are tunnel voltage, beam current, axial magnetic field, cathode magnetic field, tunnel radius, and cathode radius. The equations are sufficiently complex to warrant the use of a digital computer if many cases are to be evaluated. Error indicators are formulated to prevent use of the relations beyond their range of validity. The equilibrium solutions for relativistic electron beams from unshielded cathodes show that the required magnetic focusing field is lower than that computed from nonrelativistic formulas. The angular magnetic field produced by the beam itself aids in focusing the electrons. The potential depression due to space charge is analyzed. Correction curves are given which allow the use of nonrelativistic equations in predicting equilibrium behavior. The maximum possible microperveance of relativistic beams is shown to be lower than the classical value of 25.4.