In this comprehensive review article, the regions between quasi-neutral plasma and wall are investigated by defining sheath and presheath. Bohm criterion, which is the required condition to form an electrostatic sheath vanishing in the plasma or the condition for an electrostatic potential to satisfy the boundary condition at the wall, is examined with a cold ion fluid model to illustrate the traits in between the plasma-sheath transition. To match the solutions, including singularity, on the both sides of the sheath boundary, a transition layer is inserted. The sheath thickness is defined with Debye length for the thin sheath approximation. The region in which ions are accelerated specifies presheath. The Bohm velocity states the acoustic velocity and ions should be accelerated to supersonic velocities to satisfy the Bohm criterion. This is a similar approach as we see in fluid dynamics in the concept of breaking the sound barrier. The passing from subsonic to supersonic velocities implies there should be strong space charge formation.
To avoid singularity in the solution, singular perturbation theory is applied with two-scale approximation. For this method, the sheath is planar and collisionless. According to the potential variations, presheath is assumed quasi-neutral but to satisfy this condition, the only way is to demonstrate that the ion current density increases approaching the wall and ions experience friction in the presheath region. There are some conditions that can create the presheath with pre-described features: geometric presheath in spherical probe, collisional presheath with ion friction, ionizing presheath and the magnetic presheath. If ions are considered as warm rather than cold, the pressure contribution should be included into the momentum equation. Then, the Boltzmann equation is solved with collision, convection and source terms. After solving Poisson’s equation, the collision-free Boltzmann relation is obtained for planar geometry. There are some density variations due to slow ions in presheath region. Those are taken into account with the appropriate boundary conditions. The wall is assumed absorbing at the beginning but then ion emission and specular reflection are added to the calculation. Ions are treated with half Maxwellian distribution. The plasma and sheath solutions are overlapped with a rearranged scaling.
To sum up, the Bohm criterion is satisfied at the sheath edge with the asymptotic limit (Debye length > L, characteristic length). The limit is valid for collisionless sheath but not for collisionless plasma. It is seen that the Bohm criterion determines the wall sheath formation in any structure.
Reference: Riemann, K-U. “The Bohm criterion and sheath formation.” Journal of Physics D: Applied Physics24, no. 4 (1991): 493.
Course: AME60637