Virtual Cathode Behavior in Crossed-Field Gaps at the Limiting Current

Crossed-field devices, in which the magnetic field B is perpendicular to the electric field, are often operated near the limiting current density, J_c¹. A virtual cathode (VC) corresponding to the potential minimum, arises for nonzero injection velocity, u₀. While VCs have been examined without magnetic fields, they are less studied for crossed-field gaps with thermionic emitters. We examine VC formation for nonzero u₀ for current density J near J_c for B below the Hull cutoff for magnetic insulation. VC formation for u₀ < 2/π depends on B. Each u₀ has an associated cutoff magnetic field B_cutoff such that a stable VC forms for J ≤ J_c for B < B_cutoff but vanishes for B > B_cutoff. Increasing B for B < B_cutoff with a fixed u₀ causes stable VC formation at lower J/J_c. Increasing u₀ for a fixed B < B_cutoff reduces J/J_c further. Particle-in-cell simulations support the theoretical predictions. For B < B_cutoff, a stable VC forms at the predicted J. For B>B_cutoff, stable VC formation does not occur. For both cases, oscillating VCs form at J = J_c.

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