Beyond the Child-Langmuir Limit

This paper describes a new solution formulation for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables - i.e., variables that follow particle paths - the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming velocity $v_0$, electric field $E_0$ and current $J_0$. For steady solutions, this approach clarifies the origin of the maximal current $J_max$, derived by Child and Langmuir for $v_0=0$ and by Jaffe for $v_0>0$. The implicit formulation is used to find (1) unsteady solutions having constant incoming flux $J_0>J_max$, which leads formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of $J_max$.