The exact theoretical expressions involved in the formation of sheath in front of an electron emitting electrode immersed in a low-density plasma have been derived. The potential profile in the sheath region has been calculated for subcritical, critical, and supercritical emissions. The potential profiles of critical and supercritical emissions reveals that we must take into account a small, instead of zero, electric field at the sheath edge to satisfy the boundary conditions used to integrate the Poisson's equation. The I-V curves for critical emission shows that only high values of plasma-electron to emitted-electron temperature ratio can meet the floating potential of the emissive electrode. A one-dimensional fluid like model is assumed for ions, while the electron species are treated as kinetic. The distribution of emitted-electron from the electrode is assumed to be half Maxwellian. The plasma-electron enters the sheath region at sheath edge with half Maxwellian velocity distribution, while the reflected ones have cut-off velocity distribution due to the absorption of super thermal electrons by the electrode. The effect of varying emitted-electron current on the sheath structure has been studied with the help of a parameter G (the ratio of emitted-electron to plasma-electron densities).

The kinetic theory of an electron emitting electrode immersed in a low-density isotropic plasma is developed for the first time to include the theory of formation of a virtual cathode in this scenario. In addition to virtual cathode solution for supercritical emission, the potential profile solution for subcritical and critical emission is also included. The plasma-electron and emitted-electron are assumed to have half Maxwellian velocity distributions at the sheath entrance and electrode surface, respectively, while the plasma ions are assumed to be cold. Poisson's equation is then solved numerically for charge densities in the assumed negative sheath structure. The resulting potential profiles in the sheath for the floating and current carrying electrode/wall are calculated numerically. These potential profiles show a smooth transition from subcritical to critical and to supercritical emissions with increasing emitted-electron temperature Te,em (decreasing parameter α = Te,pl/Te,em, plasma-electron to emitted-electron temperature ratio). The numerical solution of potential profiles for supercritical emission confirms the formation of a virtual cathode. The structure of the virtual cathode is dependent on the chosen boundary values. These results also show that the virtual cathode potential profile structure exists around α < 5 to α = 1.5 but the solution at α = 1 does not exist in this scenario. It indicates that the present model is applicable only to the situation where the sheath potential is negative relative to plasma potential.

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.

The theory of positive-ion collection by a probe immersed in a low-pressure plasma was reviewed and extended by Allen et al. [Proc. Phys. Soc. 70, 297 (1957)]. The numerical computations for cylindrical and spherical probes in a sheath region were presented by F. F. Chen [J. Nucl. Energy C 7, 41 (1965)]. Here, in this paper, the sheath and presheath solutions for a cylindrical probe are matched through a numerical matching procedure to yield “matched” potential profile or “M solution.” The solution based on the Bohm criterion approach “B solution” is discussed for this particular problem. The comparison of cylindrical probe characteristics obtained from the correct potential profile (M solution) and the approximated Bohm-criterion approach are different. This raises questions about the correctness of cylindrical probe theories relying only on the Bohm-criterion approach. Also the comparison between theoretical and experimental ion current characteristics shows that in an argon plasma the ions motion towards the probe is almost radial.