# Boundary-condition refinement of the Child-Langmuir law for collisionless dc plasma sheaths

## Abstract

An exact solution to the problem of collisionless, space-charge-limited flow of cold ions across a one-dimensional (planar) dc plasma sheath of negligible electron density is derived for general values of the presheath ion velocity v0 and electric field E0. For a given ion current density J and sheath thickness d, the exact solution reduces to the classical Child-Langmuir model in the case that v0=0 and E0=0. When either v0 or E0 is sufficiently large, however, the exact solution may differ appreciably from the Child-Langmuir law. The existence of a closed-form expression for the spatial variation of the sheath potential is shown to be contingent upon the satisfaction of a simple inequality relating v0 and E0 to J. When v0 obeys the Bohm criterion and the magnitude of E0 suggests that the Bohm energy is acquired over a distance not less than one Debye length, this inequality is indeed satisfied.