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).

Relativistic stabilisation of the diocotron instability in a pulsar “cylindrical” electrosphere

“…Indeed, when the plasma approaches the light cylinder, the guiding centre motion becomes relativistic and magnetic perturbations become significant. The relativistic aspect of the diocotron instability has already been investigated in the planar diode geometry by Davidson et al (1987Davidson et al ( , 1988. They clearly demonstrate the stabilisation due to electromagnetic effects.…”

“…For completeness, we recall the main results. For a detailed discussion, see Davidson et al (1987Davidson et al ( , 1988. The plasma is drifting in the y-direction at a speed V y and is located between x = X 1 and x = X 2 .…”

Stability of Brillouin flow in planar, conventional, and inverted magnetrons

“…When the system is brought to the planar limit, the negative mass instability disappears and only the diocotron instability remains. Interestingly, finite thickness of the electron beam, specifically an increase in the velocity gradient across the beam, leads to decreased perturbation growth and even stabilization of the negative mass instability [40], [41], and of the diocotron instability [42], [43].…”

“…It should be stressed that the Brillouin flow, because of its strong velocity shear, is itself subjected to a diocotron-like instability, which is the same instability mechanism as the diocotron instability applied to a thick beam instead of a thin beam. The stability of Brillouin flow in the planar magnetron and the conventional magnetron configuration has been studied extensively by Buneman [34], Swegle [64], Antonsen [35], Davidson [26], [43], [65], [66], and Tsang [67]. There are even experiments testing microwave production of smooth-bore magnetrons [68].…”

Exact solution of the dispersion equation for electromagnetic waves in sheared relativistic electron beams

“…This in turn, generates the so-called slipping or diacotron instability, the energy source of which is the relative sheared motion between two adjacent concentric layers of beam particles (see e.g. Harrison 1963;Levy 1965;Zhelyazkov 1970;Davidson, Tsang & Uhms 1988).…”