Synthesis of a hollow-beam pierce gun

An explicit procedure is developed for the design of a Pierce-type electron gun to deliver a hollow beam of specified voltage, current, inside and outside radii at the exit plane, and cathode current density. It is shown that the solution involves Euclid´s Golden Ratio in an unexpected but valuable simplification. A comparison is made between a design obtained by this method and an analysis of that design using a well-established ray-tracing program. Agreement is satisfactory for engineering purposes.