Steady-state isothermal bounded plasma with neutral dynamics

The Tonks-Langmuir, Godyak, and Schottky steady-state isothermal models are extended when the dynamics of the neutral gas is taken into account. Exact analytic quadratures for the electron temperature, densities, and potential profiles are obtained for these three models within the plasma approximation. It is shown that, contrary to the uniform neutral pressure case, the particle and the power balance are both necessary to determine the electron temperature when neutral dynamics is included. When neutral dynamics is governed by collisions with ions, the neutral density that results from ionization is predicted to have a minimum at the center of the discharge, as indeed is observed in experiment. It is found that even a small amount of ionization can result in a rather strong neutral depletion. However, when the drag on neutrals due to collisions with ions is negligible, the predicted profile of the neutral density that results from intense ionization is reversed and exhibits an unexpected maximum at the center of the discharge.