Kinetic extraordinary-mode eigenvalue equation for relativistic nonneutral electron flow in planar geometry

Abstract

Use is made of the Vlasov–Maxwell equations to derive an eigenvalue equation describing the extraordinary–mode stability properties of relativistic, non-neutral electron flow in high-voltage diodes. The analysis is based on well-established theoretical techniques developed in basic studies of the kinetic equilibrium and stability properties of nonneutral plasmas characterized by intense self fields. The formal eigenvalue equation is derived for extraordinary-mode flute perturbations in a planar diode. As a specific example, perturbations are considered about the choice of self-consistent Vlasov equilibrium , where . is the electron density at the cathode (x = 0), H is the energy, and Py is the canonical momentum in the Y-direction (the direction of the equilibrium electron flow). As a limiting case, the planar eigenvalue equation is further simplified for low-frequency long-wavelength perturbations with |ω − kvd, ≪ ωυ where and and ⋯c = eB0/mc, and B0ệz is the applied magnetic field in the vacuum region xb < x ≤ d. Here, the outer edge of the electron layer is located at x = xb; ω is complex oscillation frequency; k is the wavenumber in the y-direction; ωυ is the characteristic betatron frequency for oscillations in the x′-orbit about the equilibrium value x′ = x0 = xb/2; and Vd is the average electron flow velocity in the y-direction at x = x0. In simplifying the orbit integrals, a model is adopted in which the eigenfunction approximated by , where x′(t′) is the x′-orbit in the equilibrium field configuration. A detailed analysis of the resulting eigenvalue equation for , derived for low-frequency long-wavelength perturbations, is the subject of a companion paper.