Stabilization of diocotron instability by relativistic and electromagnetic effects for intense nonneutral electron flow

Abstract

The extraordinary-mode eigenvalue equation is used to investigate the diocotron instability for sheared, relativistic electron flow in a planar diode. The cold-fluid model assumes low-frequency flute perturbations about a tenuous electron layer satisfying ω 2 pb ( X) « ω 2 c and |ω− kV y ( x)| 2«ω 2 c. The locations of the cathode, anode, outer layer boundary, and inner layer boundary are at x=0, x= d, x= x + b < d, and x= x - b < x + b , respectively. The eigenvalue equation is solved analytically for the case where n b(x) γ b(x) = n ̂ b gamma; ̂ b = const within the electron layer. This leads to a transcendental equation for the complex eigenfrequency ω in terms of the wavenumber k, flow parameter θ= ω ̂ D (x + b − x − b) c , and geometric factors Δ i= x − b d , Δ 0 = (d−x + b) d , and Δ b = (x + b − x − b) d . Here, ω D = 4π n ̂ bec B 0 = const is the diocotron frequency. The diocotron instability is completely stabilized by relativistic and electromagnetic effects whenever sinh (θ) θ > 2(Δ iΔ 0) 1 2 Δ b .