Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature
Kos, Leon, 1966- ; Tskhakaya, D. D. ; Jelić, Nikola
A plasma-sheath transition analysis requires a reliable mathematical expression for the plasma potential profile ?(x) near the sheath edge xs in the limit ? ? ?D/l = 0 (where ?D is the Debye length ...and l is a proper characteristic length of the discharge). Such expressions have been explicitly calculated for the fluid model and the singular (cold ion source) kinetic model, where exact analytic solutions for plasma equation (? = 0) are known, but not for the regular (warm ion source) kinetic model, where no analytic solution of the plasma equation has ever been obtained. For the latter case, Riemann [J. Phys. D: Appl. Phys. 24, 493 (1991)] only predicted ageneral formula assuming relatively high ion-source temperatures, i.e., much higher than the plasma-sheath potential drop. Riemanns formula, however, according to him, never was confirmed in explicit solutions of particular models (e.g., that of Bissell and Johnson [Phys. Fluids 30, 779 (1987)] and Scheuer and Emmert [Phys. Fluids 31, 3645 (1988)]) since the accuracy of the classical solutions is not sufficient to analyze the sheath vicinity [Riemann, in Proceedings of the 62nd Annual Gaseous Electronic Conference, APSMeeting Abstracts, Vol. 54 (APS, 2009)]. Therefore, for many years, there has been a need for explicit calculation that might confirm the Riemanns general formula regarding the potential profile at the sheath edge in the cases of regular very warm ion sources. Fortunately, now we are able to achieve a very high accuracy of results [see, e.g., Kos et al., Phys. Plasmas 16, 093503 (2009)]. We perform this task by using both the analytic and the numerical method with explicit Maxwellian and water-bag ion source velocity distributions. We find the potential profile near the plasma-sheath edge in the whole range of ion source temperatures of general interest to plasma physics, from zero to practical infinity. While within limits of very low and relatively high ion source temperatures, the potential is proportional to the space coordinate powered by rational numbers ? = 1/2 and ? = 2/3, with medium ion source temperatures. We found ? between these values being a non-rational number strongly dependent on the ion source temperature. The range of the non-rational power-law turns out to be a very narrow one, at the expense of the extension of ? = 2/3 region towards unexpedicly low ion source temperature.
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