Numerical matching of the sheath and presheath solutions for a spherical probe in radial-motion theory

Abstract

The theory of positive-ion collection by a probe immersed in a low-pressure plasma was reviewed and extended by Allen, Boyd, and Reynolds [Proc. Phys. Soc. 70, 297 (1957)]. For a given value of the ion current, the boundary values of the matched “nonneutral” or “sheath” solution Ṽnn(m)(r; rm) were obtained from the “quasineutral” or “presheath” solution Ṽqn(r) by choosing the small potential and electric-field values corresponding to some large “matching radius” rm. Here, a straightforward but efficient numerical method is presented for systematically determining an optimal value of the matching radius at which the presheath and sheath solutions are joined to yield the “matched” potential profile. Some suitable initial matching radius rm1 is chosen and the related potential and electric-field values of the quasineutral solution are calculated. Using these as boundary conditions, Poisson's equation is integrated to yield the matched nonneutral solution including the corresponding potential at the probe surface. This procedure is repeated for increasing values rm2, rm3,…. until the resulting potential at the probe surface becomes practically constant. The corresponding value of rm is taken as the “optimal” matching radius rmo at which the presheath and sheath solutions are ultimately joined to yield the “optimal” matched potential profile in the entire plasma-probe transition region. It is also shown that the Bohm criterion is inapplicable in the present problem.