# 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)

## 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 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.