The plasma–sheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion

Abstract

The plasma–sheath transition in stationary low temperature plasmas is investigated for arbitrary levels of collisionality. The model under study contains the equations of continuity and motion for a single ion species, Boltzmann's equilibrium for the electrons and Poisson's equation for the field. Assuming that the electron Debye length λD is small compared with the ion gradient length l = ni/(∂ni/∂x), a first order differential equation is established for the ion density ni as a function of the transformed spatial coordinate q = ∫ni dx. A characteristic feature of this novel sheath equation is an internal singularity of the saddle point type which separates the depletion-field dominated sheath part of the solution from the ambipolar diffusion-controlled plasma. The properties of this singularity allow us to define, in a nonarbitrary way, a collisionally modified Bohm criterion which recovers Bohm's original expression in the collisionless limit but also remains meaningful when collisions are included.A comparison is made with the collisionally modified Bohm criteria proposed by Godyak (1982 Phys. Lett. A 89 80), Valentini (1996 Phys. Plasmas 3 1459) and Chen (1997 Phys. Plasmas 5 804) as well as with the approaches of Riemann (1991 J. Phys. D: Appl. Phys. 24 493) and Franklin (2003 J. Phys. D: Appl. Phys. 36 2821), who argued that the definition of a collisionally defined Bohm criterion is not possible.