Diocotron instability for intense relativistic non-neutral electron flow in planar diode geometry

Abstract

The extraordinary-mode eigenvalue equation is used to investigate detailed properties of the diocotron instability for sheared, relativistic electron flow in a planar diode. The theoretical model is based on the cold-fluid-Maxwell equations assuming low-frequency flute perturbations about a tenuous electron layer satisfying ω2pb(x)≪ω2c and ‖ω−kVy(x)‖2≪ω2c. The cathode is located at x=0; the anode is located at x=d; the outer boundary of the electron layer is located at x=x+b<d; and the inner boundary of the layer is located at x=x−b<x+b. The extraordinary-mode eigenvalue equation is solved exactly for the case where nb(x)/ γb(x) =n̂b/γ̂b =const within the electron layer (x−b<x<x+b). Here, nb(x) =n̂b cosh [κ̂(x−x−b)]/ cosh[κ̂(x+b−x−b)] is the equilibrium density profile, γb (x)=cosh[κ̂(x−x−b)] is the relativistic mass factor, and ω̂D =κ̂c=4πn̂bec/B0 =const is the diocotron frequency. The analysis leads to a transcendental dispersion relation for the complex eigenfrequency ω in terms of the wavenumber k (in the flow direction), the relativistic flow parameter θ=ω̂D(x+b−x−b)/c, and the geometric factors Δi=x−b/d, Δ0=(d−x+b)/d, and Δb=(x+b−x−b)/d. It is found that the diocotron instability is completely stabilized by relativistic and electromagnetic effects whenever 2(ΔiΔ0)1/2/Δb <(sinh θ)/θ, i.e., stabilization occurs whenever the flow is sufficiently relativistic (sufficiently large θ) and/or the vacuum regions are sufficiently narrow (sufficiently small Δ0 or Δi).