Boundary conditions for the Child-Langmuir sheath model

A collision-free space-charge sheath formed by cold ions at a negative surface is considered. The method of matched asymptotic expansions is applied, small parameter being the ratio of the electron temperature to the sheath voltage. Two expansions are considered, one describing bulk of the sheath where the electron density is exponentially small as compared to the ion density, and another describing the outer section of the sheath in which the densities are comparable. Boundary conditions for equations describing the bulk are found such that the model have exponential accuracy. A physical meaning of these conditions is that the ions are accelerated in the outer section from the Bohm velocity to twice the Bohm velocity and that the voltage drop, in the outer section equals (3/2)(kT_e/e). The model predicts the electric field and ion velocity at the surface and the thickness of the ion layer to the accuracy of several percent for sheath voltages exceeding 3(kT_e/e)