Kinetic equilibrium properties of relativistic non-neutral electron flow in a cylindrical diode with applied magnetic field.

Abstract

The equilibrium properties of a relativistic non-neutral electron layer confined in a magnetically insulated cylindrical diode are investigated within the framework of the steady-state (\ensuremath{\partial}/\ensuremath{\partial}t=0) Vlasov-Maxwell equations. The analysis is carried out for an infinitely long cylindrical electron layer with axis of symmetry parallel to an applied magnetic field ${B}_{0}$e\ensuremath{\rightarrow}${^}_{z}$, which provides radial confinement of the electrons. The theoretical analysis is specialized to the class of self-consistent Vlasov equilibria ${f}_{b}^{0}$(x\ensuremath{\rightarrow},p\ensuremath{\rightarrow}) in which all electrons have the same canonical angular momentum (${P}_{\ensuremath{\theta}}$=${P}_{0}$=const) and the same energy (H=${\mathrm{mc}}^{2}$), i.e., ${f}_{b}^{0}$=(n${^}_{b}$${R}_{c}$/2\ensuremath{\pi}m)\ensuremath{\delta}(H-${\mathrm{mc}}^{2}$ )\ensuremath{\delta}(${P}_{\ensuremath{\theta}}$-${P}_{0}$). One of the most important features of the analysis is that the closed analytic expressions for the self-consistent electrostatic potential ${\ensuremath{\varphi}}_{0}$(r) and the \ensuremath{\theta} component of vector potential ${A}_{0}$(r) are obtained. Moreover, all essential equilibrium quantities, such as electron density profile ${n}_{b}^{0}$(r), total magnetic field ${B}_{0z}$(r), perpendicular temperature profile ${T}_{\ensuremath{\perp}b}^{0}$(r), etc., can be calculated self-consistently from these potentials. As a special case, the equilibrium properties of a planar diode are investigated in the limit of large aspect ratio, further simplifying the functional form of the electrostatic and vector potentials. Detailed equilibrium properties are investigated numerically for a cylindrical diode over a broad range of system parameters, including diode voltage ${V}_{0}$, cathode electric field, electron density n${^}_{b}$ at the cathode, diode polarity, and applied magnetic field ${B}_{0}$.