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.