Debye-sheath properties in the Tonks–Langmuir discharge with warm neutrals

Kos, L.; Tskhakaya, D. D.; Kühn, S.; Jelić, N.

doi:10.1017/s0022377813000949

Cited by 3 publications

(1 citation statement)

References 15 publications

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“…For a more rigorous description, one can account for the fact that ions are born with a finite kinetic energy described by a distribution similar to that of the neutral gas. Several models have been developed to extend the Tonks-Langmuir model to include sources with an energy distribution [14,15,16,17,22,34,35,36] Here, we consider the plasma-sheath model developed by Robertson [28] that models ion generation using a Maxwellian source. This model solves for the ion distribution function and electrostatic potential in a 1D domain assuming electrons are Boltzmann: n e = n o exp(−eφ/T e ).…”

Section: Role Of Ion Generation In and Near The Sheathmentioning

confidence: 99%

Extensions and applications of the Bohm criterion

Scheiner¹,

Yee²,

Hopkins³

et al. 2015

Plasma Phys. Control. Fusion

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Abstract.The generalized Bohm criterion is revisited in the context of incorporating kinetic effects of the electron and ion distribution functions into the theory. The underlying assumptions and results of two different approaches are compared: The conventional 'kinetic Bohm criterion' and a fluid-moment hierarchy approach. The former is based on the asymptotic limit of an infinitely thin sheath (λ D /l = 0), whereas the latter is based on a perturbative expansion of a sheath that is thin compared to the plasma (λ D /l 1). Here λ D is the Debye length, which characterizes the sheath length scale, and l is a measure of the plasma or presheath length scale. The consequences of these assumptions are discussed in terms of how they restrict the class of distribution functions to which the resulting criteria can be applied. Two examples are considered to provide concrete comparisons between the two approaches. The first is a TonksLangmuir model including a warm ion source [Robertson 2009 Phys. Plasmas 16 103503]. This highlights a substantial difference between the conventional kinetic theory, which predicts slow ions dominate at the sheath edge, and the fluid moment approach, which predicts slow ions have little influence. The second example considers planar electrostatic probes biased near the plasma potential using model equations and particle-in-cell simulations. This demonstrates a situation where electron kinetic effects alter the Bohm criterion, leading to a subsonic ion flow at the sheath edge.

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Section: Role Of Ion Generation In and Near The Sheathmentioning

confidence: 99%

Extensions and applications of the Bohm criterion

Scheiner¹,

Yee²,

Hopkins³

et al. 2015

Plasma Phys. Control. Fusion

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Introduction to the theory and application of a unified Bohm criterion for arbitrary-ion-temperature collision-free plasmas with finite Debye lengths

et al. 2018

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At present, identifying and characterizing the common plasma–sheath edge (PSE) in the conventional fluid approach leads to intrinsic oversimplifications, while the kinetic one results in unusable over-generalizations. In addition, none of these approaches can be justified in realistic plasmas, i.e., those which are characterized by non-negligible Debye lengths and a well-defined non-negligible ion temperature. In an attempt to resolve this problem, we propose a new formulation of the Bohm criterion [D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields (McGraw-Hill, New York, 1949)], which is here expressed in terms of fluid, kinetic, and electrostatic-pressure contributions. This “unified” Bohm criterion consists of a set of two equations for calculating the ion directional energy (i.e., the mean directional velocity) and the plasma potential at the common PSE, and is valid for arbitrary ion-to-electron temperature ratios. It turns out to be exact at any point of the quasi-neutral plasma provided that the ion differential polytropic coefficient function (DPCF) of Kuhn et al. [Phys. Plasmas 13, 013503 (2006)] is employed, with the advantage that the DPCF is an easily measurable fluid quantity. Moreover, our unified Bohm criterion holds in plasmas with finite Debye lengths, for which the famous kinetic criterion formulated by Harrison and Thompson [Proc. Phys. Soc. 74, 145 (1959)] fails. Unlike the kinetic criterion in the case of negligible Debye length, the kinetic contribution to the unified Bohm criterion, arising due to the presence of negative and zero velocities in the ion velocity distribution function, can be calculated separately from the fluid term. This kinetic contribution disappears identically at the PSE, yielding strict equality of the ion directional velocity there and the ion sound speed, provided that the latter is formulated in terms of the present definition of DPCFs. The numerical values of these velocities are found for the Tonks–Langmuir collision-free, plane-parallel discharge model [Phys. Rev. 34, 876 (1929)], however, with the ion-source temperature extended here from the original (zero) value to arbitrary high ones. In addition, it turns out, that the charge-density derivative (in the potential “space”) with respect to the potential exhibits two characteristic points, i.e., potentials, namely the points of inflection and maximum of that derivative (in the potential space), which stay “fixed” at their respective potentials independent of the Debye length until it is kept fairly small. Plasma quasi-neutrality appears well satisfied up to the first characteristic point/potential, so we identify that one as the plasma edge (PE). Adopting the convention that the sheath is a region characterized by considerable electrostatic pressure (energy density), we identify the second characteristic point/potential as the sheath edge (SE). Between these points, the charge density increases from zero to a finite value. Thus, the interval between the PE and SE, with the “fixed” width (in the potential “space”) of about one third of the electron temperature, will be named the plasma–sheath transition (PST). Outside the PST, the electrostatic-pressure term and its derivatives turn out to be nearly identical with each other, independent of the particular values of the ion temperature and Debye length. In contrast, an increase in Debye lengths from zero to finite values causes the location of the sonic point/potential (laying inside the PST) to shift from the PE (for vanishing Debye length) towards the SE, while at the same time, the absolute value of the corresponding ion-sound velocity slightly decreases. These shifts turn out to be manageable with employing the mathematical concept of the plasma-to-sheath transition (different from, but related to our natural PST concept), resulting in approximate, but sufficiently reliable semi-analytic expressions, which are functions of the ion temperature and Debye length.

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Modeling and simulations of plasma and sheath edges in warm-ion collision-free discharges

et al. 2018

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It has been shown recently by Kos et al. [Phys. Plasmas 25, 043509 (2018)] that the common plasma-sheath boundary is characterized by three well defined characteristic points, namely the plasma edge (PE), the sheath edge (SE) and the sonic point. Moreover, it has been shown that the sheath profiles, when properly normalized at the SE, as well as the potential drop in the plasma–sheath transition region (PST), (region between between PE and SE) in collision-free (CF) discharges are rather independent of discharge parameters, such as the plasma source profile, ion temperature and plasma density, providing that the sheath thickness is kept well bellow the plasma length. While these findings were obtained by theoretical means under idealized discharge conditions, the question arises whether and to which extent they are relevant under more complex physical scenarios. As a first step toward answering this question the CF discharge with warm ions is examined in this work via kinetic simulation method in which some of the model assumptions, such as independence of time and the Boltzmann distribution of electrons can hardly be ensured. Special attention is payed to effects of ion creation inside the sheath. It is found that only with considerably increased sheath thickness the sonic point always shifts from SE towards the wall. Whether the absolute value of ion directional velocity at the sonic point will increase or decrease depends on the ion temperature and the source strength inside the sheath. In addition preliminary comparison of results obtained under CF assumption with the representative ones obtained with strongly enhanced Coulomb collisions (CC), indicate the relevancy of hypothesis that the VDF of B&J can be considered as a universal one in future reliable kinetic modeling and solving the plasma boundary and sheath problem in both collisional and collision-free plasmas.

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Debye-sheath properties in the Tonks–Langmuir discharge with warm neutrals

Cited by 3 publications

References 15 publications

Extensions and applications of the Bohm criterion

Extensions and applications of the Bohm criterion

Introduction to the theory and application of a unified Bohm criterion for arbitrary-ion-temperature collision-free plasmas with finite Debye lengths

Modeling and simulations of plasma and sheath edges in warm-ion collision-free discharges

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