Boundaryhyphen;condition refinement of the Childndash;Langmuir law for collisionless dc plasma sheaths

摘要

An exact solution to the problem of collisionless, spacehyphen;chargehyphen;limited flow of cold ions across a onehyphen;dimensional (planar) dc plasma sheath of negligible electron density is derived for general values of the presheath ion velocityv0and electric fieldE0. For a given ion current densityJand sheath thicknessd, the exact solution reduces to the classical Childndash;Langmuir model in the case thatv0=0 andE0=0. When eitherv0orE0is sufficiently large, however, the exact solution may differ appreciably from the Childndash;Langmuir law. The existence of a closedhyphen;form expression for the spatial variation of the sheath potential is shown to be contingent upon the satisfaction of a simple inequality relatingv0andE0toJ. Whenv0obeys the Bohm criterion and the magnitude ofE0suggests that the Bohm energy is acquired over a distance not less than one Debye length, this inequality is indeed satisfied.