The Estimation of Thermal Velocity Effects in Thick Steady Beams†

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Abstract
Previous analyses for estimating thermal velocity effects in paraxial, axially symmetric, beams are extended to cover thick beams. By using a perturbation of the zero-temperature behaviour, a 6 × 6 matrix is set up relating the exit parameters of the beam with the entrance ones. By suitable assumptions at the cathode, e.g. assuming a half-Maxwellian current distribution, expressions are derived for the current density emerging from the system. It is shown that in the limiting case of a paraxial, solid beam, the expressions simplify to ones developed earlier.