Beyond the Child-Langmuir limit

This article presents a new formulation of the solution 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 to formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of J(max).