Analytic formulas for current density profiles in small-orbit spiralling electron beams

Analytic expressions for the current density as a function of radius in small-orbit, spiralling electron beams, including guiding center spread, are derived using probability theory. The expressions are valid for beams with a small spread in Larmor radius compared to the guiding center spread and a space charge depression below 10%, which are typical for beams produced by modern magnetron injection guns. The current density profile is shown to exhibit a number of functional forms, depending on the relative size of the spread in guiding center radii compared to the average Larmor radius. Several applications for the new expressions are explored, including the calculation of the thermal loading experienced by beam-intercepting metal objects, as well as the computation of average electron density per unit volume (used in predicting the growth of electrostatic cyclotron instabilities). The analytic results are extended, in the vicinity of the beam edges, to include the effects of Larmor radius spread. The edge expressions are employed to predict the amount of current intercepted by a tunnel that closely surrounds the beam, which can be useful in selecting radial clearances.