Kinetic stability theorem for relativistic non-neutral electron flow in a planar diode with applied magnetic field

Abstract

A kinetic stability theorem is developed for relativistic non-neutral electron flow in a planar high-voltage diode with applied magnetic field. The effects of strong inhomogeneities and intense self-electric and self-magnetic fields are retained in the analysis in a fully self-consistent manner. Use is made of global (spatially averaged) conservation constraints satisfied by the fully nonlinear Vlasov–Maxwell equations, assuming electromagnetic perturbations with extraordinary-mode polarization, and space-charge-limited flow with E0x(x=0)=0 at the cathode. It is also assumed that the y-averaged, x-directed net flux of particles, y momentum, and energy, vanish identically at the cathode (x=0) and at the anode (x=d). It is shown that the class of self-consistent Vlasov equilibria f0b(H,Py) is stable for small-amplitude perturbations, provided f0b is a monotonic decreasing function of H−VbPy, i.e., provided ∂f0b/∂(H−VbPy)≤0. Here, H is the energy and Py is the canonical y momentum. The generality of this sufficient condition for stability should be emphasized. First, the derivation of the stability theorem has not been restricted to a specific choice of f0b(H−VbPy). Moreover, the fully non-neutral electron equilibria are generally characterized by strong spatial inhomogeneities and intense self-electric and self-magnetic fields. For the class of equilibria with ∂f0b/∂(H−VbPy)≤0, it is also shown that the density profile n0b(x)=∫ d3p f0b and x–x pressure profile P0b(x) =∫ d3p vxpxf0b decrease monotonically from the cathode (x=0) to the anode (x=d) provided the applied magnetic field at the anode (Ba) is sufficiently strong that (Vb/c)Ba ≥4πe ∫d0 dx′ n0b(x′).