Sheath formation in low-pressure discharges, the Bohm criterion and the consequences of collisions

The space charge density in low-pressure discharges results from the generation of charged particles, the momentum transfer from these particles to the neutral gas and the electric field. A simplified model is used to treat this process analytically and numerically across the whole plasma. The effect of the electric field alone can cause the formation of the space charge sheath if the ion drift velocity υ_i to the wall exceeds the modified Bohm velocity υ_C = υ_B × (n_i/n_e)^1/2, where υ_B is the Bohm velocity and n_i and n_e are the number densities of the ions and the electrons, respectively. However, a domain with υ_i ⩾ υ_C can occur only if the effect of collisions is weak. This domain is very narrow and does not come up to the wall. Limits of the electric field strength determining the sheath formation are given. It is shown that the electric field strength cannot be set equal to zero at υ_i = υ_B or υ_C under collisional conditions. The sheath extends from the region near the wall towards the centre and a result of that is to lower υ_i with respect to υ_B as the collisionality rises. These results are used to take into consideration various sheath criteria. The Bohm criterion takes into account the effect of the electric field only and reveals a well-defined sheath edge at υ_i = υ_B. This criterion remains a useful approximation of the sheath edge in almost collisionless plasmas as well. Under collisional conditions the definition of the sheath edge becomes more difficult and a little arbitrary. This paper takes into account new sheath criteria modified for the case of finite collisionality. The divergence between the densities of the ions and the electrons, the gradients of the space charge density and of the generalized Bohm speed υ_C are studied as functions of υ_i or the distance from the wall. These criteria are compared with the collisionally modified Bohm criteria proposed by Godyak (1982 Phys. Lett. A 89 80), Valentini (1996 Phys. Plasmas 3 1459), Chen (1998 Phys. Plasmas 5 804) and Brinkmann (2011 J. Phys. D: Appl. Phys. 44 042002). (paper)