Macroscopic extraordinary-mode stability properties of relativistic non-neutral electron flow in a planar diode with applied magnetic field

Abstract

Extraordinary-mode stability properties of relativistic non-neutral electron flow in a planar diode with applied magnetic field are investigated within the framework of the macroscopic cold-fluid-Maxwell equations. The eigenvalue equation is derived for flute perturbations (kz=0) about the general class of relativistic planar equilibria characterized by electron density profile n0b(x), sheared velocity profile V0y(x) =−cE0x(x)/B0z(x), and relativistic mass factor γ0b(x) =[1−E02x(x)/B02z(s)]−1/2. The full influence of equilibrium self-electric and self-magnetic fields is retained in the analysis, and the cathode is located at x=0 and the anode at x=d. The exact eigenvalue equation is simplified for low-frequency perturbations in the guiding-center limit of strongly magnetized electrons with m→0. In this regime, it is shown that (∂/∂x)[n0b(x)/γ0b(x)]≤0 over the interval 0≤x≤d is a sufficient condition for stability of the relativistic electron flow to extraordinary-mode perturbations. A specific example of stable oscillations [rectangular profile for n0b(x)/γ0b(x)] is analyzed in detail. Finally, the exact eigenvalue equation is solved numerically for a wide range of electron density corresponding to weak and strong instability driven by velocity shear with ∂V0y/∂x≠0.