Local analysis of extraordinary-mode stability properties for relativistic non-neutral electron flow in a planar diode

The extraordinary-mode eigenvalue equation is used to investigate the local stability properties of relativistic, non-neutral electron flow in a planar diode. The local stability analysis assumes gentle equilibrium gradients and short perturbation wavelengths. The lowest-order local dispersion relation is derived assuming that localized solutions for the eigenfunction exist, and stability properties are investigated numerically over a wide range of System parameters for perturbations with frequency small in comparison with the electron cyclotron frequency. It is found that the local dispersion relation supports three solutions in this frequency regime. One of the solutions corresponds to a stable diocotron mode driven by the local density gradient. The other two branches are found to exhibit instability over a wide range of electron density. These modes are electromagnetic in nature and require relativistic electron flow with velocity shear in order for instability to exist. Moreover, the growth rate of the unstable electromagnetic mode can be substantial (a few per cent of the electron cyclotron frequency).