Beyond the Child-Langmuir Limit

TL;DR

This paper introduces an exact analytic solution formulation for nonlinear, unsteady electron beam flow in diodes, revealing new solutions beyond the classical Child-Langmuir limit, including virtual cathodes and periodic fluxes.

Contribution

It provides a novel implicit solution approach for nonlinear diode flow, clarifies the origin of the Child-Langmuir limit, and discovers new unsteady and periodic solutions exceeding this limit.

Findings

Derived an exact implicit solution for nonlinear diode flow.

Identified unsteady solutions with virtual cathodes for flux above the Child-Langmuir limit.

Discovered time-periodic solutions with average flux exceeding the classical maximum.

Abstract

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 $J_max$.