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

Ralf Peter Brinkmann

J. Phys. D: Appl. Phys. 44, 042002 (2011)


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 is small compared with the ion gradient length , a first order differential equation is established for the ion density as a function of the transformed spatial coordinate . 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.


Tags: Bohm criterion, collisional sheath, plasma sheath