Singular particle distributions associated with a steady state, asymmetric virtual cathode in the collision-less sheath facing a Langmuir plasma probe

We present singular weak solutions of the steady state, two species Vlasov-Poisson equations governing a surface bound plasma in which a minimum of the finite amplitude electric potential (i.e. a virtual cathode) appears in steady state conditions and exhibits strong spatial asymmetry in the sheath facing a Langmuir probe. The energy distribution of the finite mass, mobile ions is shown to be log singular at the position of the potential minimum. We prove that the electron energy distributions on opposite sides of this minimum are not equal. This leads to a jump discontinuity of the electron distribution across its separatrix. A simple relation exists between the difference of these two electron distributions and that of the ions. The distributions of both species meet smooth boundary conditions at one plasma end; they are given in terms of elementary functions and allow a simple evaluation of the probe's current-voltage response curve.