Synthesis of the Pierce gun

The design of a Pierce gun is uniquely defined by four parameters: beam voltage, current, waist radius, and cathode current density. The gun convergence angle, cathode spherical radius, anode-cathode spacing, and throw (distance to the waist) are calculated by an iterative procedure which typically converges in four cycles to 0.1°. The anode lens formula of Danielson, Rosenfeld, and Saloom is combined with inverted forms of the Langmuir-Blodgett solution for the spherical diode and the Universal Beam Spread equation for the tunnel region; the iterations match the trajectories at the anode plane. Correction for spherical aberration is made by the method of Frost and Purl as quantified by True. The inverted equations are simple enough to be solved in a few minutes with a programmable calculator. The experimental data of Frost and Purl is used to demonstrate the validity of the procedure for guns of medium perveance.