Kinetic stability properties of nonrelativistic non-neutral electron flow in a planar diode with applied magnetic field.

Abstract

The linearized Vlasov-Poisson equations are used to investigate the electrostatic stability properties of nonrelativistic non-neutral 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 ${B}_{0}$e${^}_{z}$, and the equilibrium flow velocity ${V}_{\mathrm{yb}}^{0}$(x) is in the y direction. Stability properties are calculated for perturbations about the choice of self-consistent Vlasov equilibrium ${f}_{b}^{0}$(H,${P}_{y}$)=(n${^}_{b}$/2\ensuremath{\pi}m)\ensuremath{\delta} (H)\ensuremath{\delta}(${P}_{y}$), which gives an equilibrium with uniform electron density (n${^}_{b}$=const) extending from the cathode (x=0) to the outer edge of the electron layer (x=${x}_{b}$). Assuming flute perturbations (\ensuremath{\partial}/\ensuremath{\partial}z=0) of the form \ensuremath{\delta}\ensuremath{\varphi}(x,y,t)=\ensuremath{\delta}\ensuremath{\varphi}${^}_{k}$(x)exp(iky -i\ensuremath{\omega}t), the eigenvalue equation for \ensuremath{\delta}\ensuremath{\varphi}${^}_{k}$(x) is simplified and solved analytically for long-wavelength, low-frequency perturbations satisfying ${\mathrm{kx}}_{b}$\ensuremath{\ll}1 and \ensuremath{\Vert}\ensuremath{\omega}-${\mathrm{kV}}_{d}$${\ensuremath{\Vert}}^{2}$\ensuremath{\ll}${\ensuremath{\omega}}_{v}^{2}$\ensuremath{\equiv}${\ensuremath{\omega}}_{c}^{2}$ -\ensuremath{\omega}${^}_{\mathrm{pb}}^{2}$. This gives a quadratic dispersion relation for the complex oscillation frequency \ensuremath{\omega}. Defining \ensuremath{\mu}=\ensuremath{\omega}${^}_{\mathrm{pb}}^{2}$/${\ensuremath{\omega}}_{v}^{2}$ and g=d/(d-${x}_{b}$), it is shown that the necessary and sufficient condition for instability (Im\ensuremath{\omega}>0) is given by (1+\ensuremath{\mu}+g)(\ensuremath{\mu}+g)>2(1+\ensuremath{\mu})(1+\ensuremath{\mu}${/4\phantom{\rule{0ex}{0ex}})}^{2}$. It is found that the maximum growth rate in the unstable region can be substantial. For example, for d=2${x}_{b}$ and g=2, the maximum growth rate is (Im\ensuremath{\omega}${)}_{\mathrm{max}}$\ensuremath{\simeq}0.25(${\mathrm{kx}}_{b}$)${\ensuremath{\omega}}_{c}$, which occurs for \ensuremath{\mu}\ensuremath{\simeq}2.3.