A general theory of multipactor in orthogonal electric and magnetic fields is given. The model consists of two parallel plates of known secondary emission properties, across which a time varying voltage is applied, and between which a constant magnetic field is applied. Expressions are derived for the resonant phases at which the RF-driven cascades occur; these reduce to previously derived expressions in the limit of the vanishing magnetic field. In addition, this work obtains the conditions governing the stability of the motion about those phases, as well as a dynamic constraint from imposing the restriction that each impact on a plate is the first impact that is allowed by the equations of motion. Chaotic effects from the random ejection velocities of the secondaries are addressed for the first time. It is proven that the phase focusing effect from the radio frequency (RF) interaction will overcome the dispersive effect from the random emission, provided that the mean square emission velocity is sufficiently small; in that case stability of the multipactor is determined by the stability of an ‘‘ideal’’ multipactor with zero emission velocity. That reduces the multipactor parameter space to only two control parameters: the normalized RF electric and DC magnetic field strengths. The multipactor prone region is mapped over the entire parameter space by applying both dynamic and secondary yield criteria; previous studies have been limited to specific cases. Finally a simplified model is adopted to address collective effects and multipactor saturation from the space charge buildup. In addition to the mutual electron repulsion, we find that the previously neglected interaction of the electron sheath with the induced image charge on the plates is important to saturation.
An analytic theory is presented that yields the maximum transmittable current across an anodecathode gap that is embedded in an arbitrary transverse magnetic field (B). The limiting current is found to be relatively insensitive to B for all B < BH, where BH is the Hull cutoff magnetic field required for magnetic insulation. The classical Child-Langmuir solution is recovered in the limit B-+0.
It is shown that a small amount of dissipation, caused by current flow in a lossy external circuit, can produce a disruption of steady-state cycloidal electron flow in a crossed-field gap, leading to the establishment of a turbulent steady state that is close to, but not exactly, Brillouin flow. This disruption, which has nothing to do with a diocotron or cyclotron instability, is fundamentally caused by the failure of a subset of the emitted electrons to return to the cathode surface as a result of resistive dissipation. This mechanism was revealed in particle simulations, and was confirmed by an analytic theory. These near-Brillouin states differ in several interesting respects from classic Brillouin flow, the most important of which is the presence of a microsheath and a time-varying potential minimum very close to the cathode surface. They are essentially identical to that produced when ͑i͒ injected current exceeds a certain critical value ͓P. J. Christenson and Y. Y. Lau, Phys. Plasmas 1, 3725 ͑1994͔͒ or ͑ii͒ a small rf electric field is applied to the gap ͓P. J. Christenson and Y. Y. Lau, Phys. Rev. Lett. 76, 3324 ͑1996͔͒. It is speculated that such near-Brillouin states are generic in vacuum crossed-field devices, due to the ease with which the cycloidal equilibrium can be disrupted. Another novel aspect of this paper is the introduction of transformations by which the nonlinear, coupled partial differential equations in the Eulerian description ͑equation of motion, continuity equation, Poisson equation, and the circuit equation͒ are reduced to an equivalent system of very simple linear ordinary differential equations.
Analysis of temperature-limited flow, space-chargelimited flow, and the transition between them using a simple planar diode with a thermionic cathode, in which the cathode surface has spatially nonuniform emission properties, is presented. Our theoretical results, which are derived from a model based on solutions to the Vlasov and Poisson equations, compare well with the results of particle-in-cell simulations. We find that the location and the shape of the knee in the anode current versus temperature characteristic (Miram or "rollover" curve) are significantly affected by non-uniformities in the space-charge density in the A-K gap, but are relatively unaffected by the electron motion parallel to the electrode surfaces. In particular, emission from an actively emitting region is strongly affected by the forces (or lack thereof) exerted by the space-charge of the electrons emitted by their neighbors. Perhaps, most remarkably, we find that the limiting current reaching the anode is approximately given by the classical 1-D Child-Langmuir law, even if a significant fraction of the cathode surface is nonemitting.
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