Bridging Child–Langmuir and Warren: exact and approximate solutions for the unipolar sheath of intermediate pressure

Abstract

The stationary unipolar ion sheath is investigated in the regime of intermediate gas pressure. A mathematical model is consulted which describes the motion of a single species of positive ions under the influence of the self-generated electric field and accounts for ion inertia and collisional ion-neutral friction assuming a constant mean free path. In the collisionless limit, the model reduces to the power law of Child and Langmuir (1911 Phys. Rev. 32 492; 1913 Phys. Rev. II 2 450); in the collisional case, to that of Warren (1955 Phys. Rev. 98 1658). The general case, however, resists a power law ansatz, and an exact analytical solution is currently not known. This study will close the gap by constructing such a solution in terms of Airy functions. An approximation based on the quadratic order harmonic mean of the limiting velocities and an earlier potential interpolation by Riemann et al (1992 J. Phys. D: Appl. Phys. 25 620) are critically assessed.