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 (∂/∂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→${^}_z$, which provides radial confinement of the electrons. The theoretical analysis is specialized to the class of self-consistent Vlasov equilibria $f_b^0$(x→,p→) in which all electrons have the same canonical angular momentum ($P_{\theta }$=$P_0$=const) and the same energy (H=${\mathrm{mc}}^2$), i.e., $f_b^0$=(n${^}_b$$R_c$/2πm)δ(H-${\mathrm{mc}}^2$ )δ($P_{\theta }$-$P_0$). One of the most important features of the analysis is that the closed analytic expressions for the self-consistent electrostatic potential ${\phi }_0$(r) and the θ 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 $T_{\perp b}^0$(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 $B_0$.

• Received 22 October 1984

DOI:https://doi.org/10.1103/PhysRevA.31.2556