Kinetic description of betatron oscillation instability for nonrelativistic nonneutral electron flow
Abstract
The linearized VlasovPoisson 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 B_{0}ê_{z}, and the equilibrium flow velocity V^{0}_{yb}(x) is in the y direction. Stability properties are calculated for perturbations about the choice of selfconsistent Vlasov equilibrium f^{0}_{b} (H,P_{y}) =(n̂_{b}/2πm) δ(H) δ(P_{y}), which gives an equilibrium with uniform electron density (n̂_{b} =const) extending from the cathode (x=0) to the outer edge of the electron layer (x=x_{b}). Assuming flute perturbations, the eigenvalue equation is simplified and solved analytically for longwavelength, moderately highfrequency perturbations satisfying k^{2}x^{2}_{b} ≪1 and (ωkV_{d})^{2} ≊ω^{2}_{v} =ω^{2}_{c} ω̂^{2}_{pb}. The present analysis is restricted to electron densities below the Brillouin flow (ω̂^{2}_{pb} <ω^{2}_{c}) and the nonzero electric field at the cathode. The eigenvalue equation leads to a fourthorder 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 (ωkV_{d} ≊+ω_{v}) and the downshifted branch (ωkV_{d}≊ω_{v}). For a sufficiently thin electron layer, it is found that only the upshifted branch exhibits instability. Typically, instability exists for a range of ω̂^{2}_{pb}/ω^{2}_{c}.
 Publication:

Physics of Fluids
 Pub Date:
 July 1986
 DOI:
 10.1063/1.865564
 Bibcode:
 1986PhFl...29.2273D
 Keywords:

 Electron Mobility;
 Magnetohydrodynamic Stability;
 Plasma Diodes;
 Plasma Dynamics;
 Plasma Oscillations;
 Vlasov Equations;
 Eigenvalues;
 Electrostatics;
 Planar Structures;
 Poisson Equation;
 Nuclear and HighEnergy Physics