Analytic-Numerical Matching of the Sheath and Plasma Solutions for a Spherical Probe in a Low-Density Plasma

摘要

Finding the optimum matching between the numerically realizable part of the (space-charge dominated) sheath solution (i.e., potential distribution) and the (quasineutral) presheath plasma solution is quite a challenging problem in general. Here, an analytic-numerical matching procedure is proposed for the sheath-plasma transition related to a spherical probe in a low-density plasma. First, a fairly general spherical-probe scenario based on trajectory integration of the Vlasov equation is formulated and specialized to the particular situation considered in [I. B. Bernstein and I. N. Rabinowitz, Physics of Fluids 2, 112 (1959)] (B&R), in which the incident ions are monoenergetic and isotropic. Then, this newly developed formalism is used for finding the potential profile in the entire “plasma-probe transition (PPT)” region. The complete “sheath” solution, which by definition satisfies Poisson's equation, consists of the “inward” sheath solution (r r₀, region without reflected ions) and the “outward” one (r ≥ r₀, region with reflected ions), but only the inward sheath solution can be realized numerically. The outward sheath solution, on the other hand, is approximated for r₀ ≤ r ≤ r_mtch (where r_mtch is the “matching” radius) by the (second-order) “expanded” sheath solution, and for r r_mtch by the “plasma” solution, which by definition satisfies the quasineutrality conditon. The “optimum” values of r_mtch and r₀ are simultaneously determined by requiring that at r = r_mtch both the values and the first derivatives of the (second-order) expanded sheath and plasma solutions are equal, respectively. While the inward sheath solution was also given by B&R, the expanded outward sheath and plasma solutions, the quasineutral solution and the related matching procedure represent genuinely new results (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)