Plasma-sheath transition in the kinetic Tonks-Langmuir model

The famous Tonks-Langmuir model of a plane symmetric discharge is supplemented by a kinetic theory of the plasma-sheath transition on the “intermediate scale.” The mathematical structure of the transition is identical to that found for the collision dominated charge exchange model [K.-U. Riemann, J. Phys. D: Appl. Phys. 36, 2811 (2003)]. It reveals the significance of the asymptotic sheath edge and clarifies the role of the kinetic Bohm criterion. From the matching conditions an approximation for the plasma eigenvalue problem is obtained that is correct up to terms of the order $λD∕L$⁠. Matched asymptotic potential curves are constructed and compared with numerical solutions. They show an excellent agreement, exceeding by far the expected range of validity in $λD∕L$⁠.