On the effects of thermal velocities in two-dimensional and axially symmetric beams

The effects of thermal velocity in two-dimensional and axially symmetric beams are evaluated, assuming a Maxwellian distribution of velocities at the cathode. The linear treatment is similar to that of Cutler and Hines, but the approach is somewhat different. Analytic expressions and curves are given for the current density in terms of the error and modified Bessel functions; the arguments of these functions are the solutions of the usual paraxial equations, neglecting thermal effects. Our predictions agree with those of Cutler and Hines for the case they considered. An approximation is given for the nonlinear effects of the electro-magnetic fields; these nonlinear effects may be due to aberrations in the system, or to changes in the space-charge fields due to the linear effects of thermal velocities. As an example, the problem of a beam with a finite transverse temperature in a field-free drift space is considered. Analytic expressions are given for the current density in such a beam.