Relaxation of a collisionless ion matrix sheath

An analytical fluid model describing the time evolution of a collisionless ion sheath in front of a plane absorbing wall biased to pulsed high negative voltage is presented. The model utilizes the Lagrangian formulation of hydrodynamics and allows one to extend and generalize previous results for the matrix relaxation process obtained from the usual Euler formulation of hydrodynamics. Investigating the sheath evolution, the different phases of the matrix extraction and sheath expansion are characterized and treated separately. Considering the matrix extraction phase, a description is developed that, in principle, is suitable to account for arbitrary inhomogeneous initial conditions. Explicit results are given for two initial conditions of special interest. The end of the matrix extraction phase manifests itself in a distinct 'kink' in the numerical solutions. By an investigation of the transition between the matrix extraction and sheath expansion phases, it is shown that this kink is due to a changing type of the ion orbits striking the wall. In order to describe the late quasistatic sheath expansion phase, the sheath boundary evolution is approximated and corresponding solutions are presented. The analytical results are compared with numerical solutions of the fluid equations as well as of a kinetic particle-in-cell/Monte-Carlo simulation and show convincing agreement