Kinetic description of betatron oscillation instability for nonrelativistic nonneutral electron flow

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The linearized Vlasov–Poisson equations are used to investigate the electrostatic stability properties of nonrelativistic nonneutral electron flow in a planar diode with cathode located at x=0 and anode at x=d. The electron layer is immersed in a uniform applied magnetic field B0êz, and the equilibrium flow velocity V0yb(x) is in the y direction. Stability properties are calculated for perturbations about the choice of self-consistent Vlasov equilibrium f0b (H,Py) =(n̂b/2πm) δ(H) δ(Py), which gives an equilibrium with uniform electron density (n̂b =const) extending from the cathode (x=0) to the outer edge of the electron layer (x=xb). Assuming flute perturbations, the eigenvalue equation is simplified and solved analytically for long-wavelength, moderately high-frequency perturbations satisfying k2x2b ≪1 and (ω−kVd)2 ≊ω2v =ω2c −ω̂2pb. The present analysis is restricted to electron densities below the Brillouin flow (ω̂2pb <ω2c) and the nonzero electric field at the cathode. The eigenvalue equation leads to a fourth-order algebraic dispersion relation for the complex eigenfrequency. The dispersion relation is solved numerically, and detailed stability properties are investigated as a function of system parameters for both the upshifted branch (ω−kVd ≊+ωv) and the downshifted branch (ω−kVd≊−ωv). For a sufficiently thin electron layer, it is found that only the upshifted branch exhibits instability. Typically, instability exists for a range of ω̂2pb/ω2c.