Pinching limits and scaling laws for high current diodes

A maximum current density achievable in a diode has been determined from a previously obtained steady state relativistic constant of the motion for the beam fluid~ This current density limit has been used together with the purely radial and purely axial flow limits of the fluid equations to establish a minimum pinch radius for the beam which can be expressed in terms of macroscopic diode parameters. This expression has been combined with a Child-Langmuir emission law to obtain a necessary condition for optimal pinching. In this paper we list and discuss several electron beam machines, both present and proposed, in the light of this pinching condition.