A theory of magnetron injection guns

This paper is concerned with the theoretical properties of the axially symmetric electron beam arising from a space-charge limited cathode of conical shape, immersed in a uniform axial magnetic field. It is not the purpose of this paper to discuss the gun from an engineering point of view; hence, no advice is offered concerning the selection of design parameters for a particular application. The problem is treated nonrelativistically, and thermal velocities of emission are considered only in a very approximate, though fundamentally important, fashion. It is assumed that all trajectories are geometrically similar, so that one may obtain a given trajectory from any other by simple magnification. Because of this assumption, it follows that the flow is considered to be laminar, i.e., two trajectories never intersect. It is shown that a given gun design is completely specified when values have been assigned to three parameters, namely a) the half angle of the cathode θ0, b) a space-charge parameter , which involves the magnetic field strength, the absolute radius of a reference point on the cathode, and the emission density at that point, and c) an initial velocity parameter which, if properly chosen, will greatly simplify the problem of theoretically calculating correct electrode shapes. Following an introductory Section I, the space-charge flow equations are discussed in Section II; these equations are then specialized in Section III to the case of a conical cathode and a uniform axial magnetic field. Numerical solutions of these equations, for selected values of the parameters, are presented in Section IV, properties of the beam remote from the cathode are discussed in Section V, and the problem of electrode design is treated in Section VI. In Section VII, perturbation equations are derived, useful for predicting the effect of varying either the magnetic field strength or the gun anode potential separately. Finally, Section VIII is concerned with the properties of a thin hollow beam generated by a magnetron injection gun, but far from the gun, i.e., in a region where the potential may be very different from the beam potential in the gun proper.