Plasma-sheath transition in the magnetized plasma-wall problem for collisionless ions

A model for the collisionless plasma-wall problem under the action of an applied magnetic field is developed. The behavior of its solution is examined and found to be qualitatively consistent with experiment. The plasma and the sheath are then modeled separately to obtain the position of the quasi-neutral plasma boundary and the position of the edge of the electron-free sheath. It is shown that the plasma boundary can be specified as the point where the component of the ion velocity normal to the wall reaches the ion sound speed (Bohm criterion), and the sheath edge is specified as the point corresponding to Godyak´s condition for the electric field. Studying the behavior near the plasma boundary and the sheath edge, the plasma solution and the solution of the space charge region are patched together to approximate the solution of the plasma-wall problem.