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 E⁰_x(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 f⁰_b(H,P_y) is stable for small‐amplitude perturbations, provided f⁰_b is a monotonic decreasing function of H−V_bP_y, i.e., provided ∂f⁰_b/∂(H−V_bP_y)≤0. Here, H is the energy and P_y 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 f⁰_b(H−V_bP_y). 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 ∂f⁰_b/∂(H−V_bP_y)≤0, it is also shown that the density profile n⁰_b(x)=∫ d³p f⁰_b and x–x pressure profile P⁰_b(x) =∫ d³p v_xp_xf⁰_b decrease monotonically from the cathode (x=0) to the anode (x=d) provided the applied magnetic field at the anode (B_a) is sufficiently strong that (V_b/c)B_a ≥4πe ∫^d₀ dx’ n⁰_b(x’).