In the fluid description of the classical Tonks-Langmuir (TL) model [Tonks and Langmuir, Phys Rev., 34: 876, 1929], arises the “closure problem.” The continuity and the momentum equations need, in addition, a closure equation. Usually, this closure equation assumes zero ion pressure (pi = 0), or a constant ion polytropic coefficient (γi). These simplified assumptions are likely to produce incorrect results, because (i) the ions have a non-zero temperature, even if the ion source is assumed to be cold, and (ii) the polytropic coefficient is, in fact, a function of space (or potential), and is far from constant (Kuhn et al., AIP Conference Proceedings, 1306: 216, 2010). Here, the concept of polytropic coefficient function (PCF) is applied to a TL-type discharge with a Kappa-distributed ion source. The ion density and temperature are calculated numerically, for different values of the spectral index κ. The polytropic coefficient function is then calculated using the relation γ = 1 + (n/T) (dT/dn). Striking deviation from results of the classical case are observed. It is shown here that, for Kappa-distributed ion sources, the ion PCF is not a constant, but is a function of the potential. Moreover, the sheath-edge potential differs significantly from the classical case. It is concluded that in order to close the set of fluid equations in an appropriate manner, better approximations to the PCF are needed.
Polytropic coefficient function Kappa distribution TL discharge
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The authors thank Dr. D. Tskhakaya for his valuable suggestions.
This study is supported and provided by the Higher Education Commission (HEC), Pakistan, under project no. 7632/Federal/NRPU/R&D/HEC/2017 and by the National Magnetic Confinement Fusion Program under Grant no. 2013GB104004 and Fundamental Research Fund for Chinese Central Universities. It is also supported and provided by HEC, Pakistan, under project no. 7659/Balochistan/NRPU/R&D/HEC/2017.
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