Macroscopic guiding center stability theorem for nonrelativistic non-neutral electron flow in a cylindrical diode with applied magnetic field

Abstract

A sufficient condition is derived for electrostatic stability of nonrelativistic non-neutral electron flow in a cylindrical diode with applied magnetic field B0êz. The analysis is based on a cold-fluid guiding-center model that treats the electrons as a massless, strongly magnetized fluid with ω2pb(r)/ ω2c =4πn0b (r)mc2/ B20 ≪1, and flow velocity Vb (x, t)=−c∇φ(x, t)×êz/B0. Making use of global conservation constraints satisfied by the fluid-Poisson equations, it is shown that ∂n0b(r)/∂r ≤0 over the interval a≤r≤b is a sufficient condition for stability to small-amplitude electrostatic perturbations. Here n0b(r) is the electron density profile, and the cathode is located at r=a and the anode at r=b. Space-charge-limited flow with E0r (r=a)=0 is assumed. The analysis illustrates the major generality and flexibility of using global conservation constraints to determine a sufficient condition for stability. Nowhere is it necessary to make direct use of a detailed normal-mode analysis or eigenvalue equation.