Synthesis of a hollow-beam gun based on the Pierce design

This paper presents an explicit sequence of equations for the design of a hollow-beam Pierce gun, given the required beam voltage and current, the outer and inner beam radii at the exit plane (beam waist), and the cathode current density at which the gun is to operate. Discussion of the derivation of the first equation shows that the normal at the inner edge of the cathode ring, extended to meet the axis, is cut by the anode sphere in Euclid´s Golden Ratio; from this it follows that the inner diameter of the cathode is equal to 1.618 times the inner diameter of the exit beam, regardless of the other parameters. A comparison is given between a specific gun design based on this method and an analysis of that design using the large ray-tracing program of Dr. Richard True.