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^{1}, Kang T. Tsang

^{2}and Han S. Uhm

^{3}

### 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 ω^{2} _{ p b }(*x*)≪ω^{2} _{ c } and ‖ω−*k* *V* _{ y }(*x*)‖^{2}≪ω^{2} _{ c }. 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 *n* _{ b }(*x*)/ γ_{ b }(*x*) =*n*̂_{ b }/γ̂_{ b } =const within the electron layer (*x* ^{−} _{ b }<*x*<*x* ^{+} _{ b }). Here, *n* _{ b }(*x*) =*n*̂_{ b } cosh [κ̂(*x*−*x* ^{−} _{ b })]/ cosh[κ̂(*x* ^{+} _{ b }−*x* ^{−} _{ b })] is the equilibrium density profile, γ_{ b } (*x*)=cosh[κ̂(*x*−*x* ^{−} _{ b })] is the relativistic mass factor, and ω̂_{ D } =κ̂*c*=4π*n*̂_{ b } *e* *c*/*B* _{0} =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=ω̂_{ D }(*x* ^{+} _{ b }−*x* ^{−} _{ b })/*c*, and the geometric factors Δ_{ i }=*x* ^{−} _{ b }/*d*, Δ_{0}=(*d*−*x* ^{+} _{ b })/*d*, and Δ_{ b }=(*x* ^{+} _{ b }−*x* ^{−} _{ b })/*d*. It is found that the diocotron instability is completely stabilized by relativistic and electromagnetic effects whenever 2(Δ_{ i }Δ_{0})^{1} ^{/} ^{2}/Δ_{ b } <(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 Δ_{0} or Δ_{ i }).

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