title: Beyond the Child-Langmuir limit.
abstract: 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).
authors.affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095, USA.
authors.forename: R E
authors.initials: RE
authors.lastname: Caflisch
authors.forename: M S
authors.initials: MS
authors.lastname: Rosin