Theoretical and experimental studies of kinetic equilibrium and stability of the virtual cathode in an electron injected inertial electrostatic confinement device

This paper explores the electron-electron two-stream stability limit of a virtual cathode in spherical geometry. Previous work using a constant density slab model [R. A. Nebel and J. M. Finn, Phys. Plasmas 8, 1505 (2001)] suggested that the electron-electron two-stream would become unstable when the well depth of the virtual cathode was 14% of the applied voltage. However, experimental tests on INS-e have achieved virtual cathode fractional well depths ∼ 60 % with no sign of instability. Here, studies with a spherical gridless particle code indicate that fractional well depths greater than 90% can be achieved without two-stream instabilities. Two factors have a major impact on the plasma stability: whether the particles are reflected and the presence of angular momentum. If the particles are reflected then they are guaranteed to be in resonance with the electron plasma frequency at some radius. This can lead to the two stream instabilities if the angular momentum is small. If the angular momentum is large enough it stabilizes the instability much the same way as finite temperature stabilizes the two-stream instability in a slab.