Analytical expression for sheath edge around corner cathodes

Abstract

A simple analytical expression for the position of the sheath edge around a two-dimensional corner cathode with included angle θc has been discovered. This expression is valid for weakly collisional sheaths in the Child–Langmuir regime ϕc ≫ kTe/e, where −ϕc < 0 is the cathode bias and Te is the electron temperature. In polar coordinates (r, θ), the sheath edge is given by (r/s0)sin[πθ/(2π − θc)] = [π/(2π − θc)] where s0 is the planar sheath width far from the vertex of the corner. This result is verified by comparison with previous numerical solutions (Watterson P A 1989 J. Phys. D: Appl. Phys. 22 1300) for the knife edge (θc = 0) and convex square corner (θc = π/2). The observed agreement suggests that this expression gives the sheath edge for all corner angles, both concave and convex. The utility of this result is demonstrated by computing the full sheath solution for a knife-edge cathode with ϕc = 100kTe/e.