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

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 < r 0 , region without reflected ions) and the “outward” one ( r ≥ r 0 , 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 r 0 ≤ 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 r 0 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)