The solution of the plasma equation in plane parallel geometry with a Maxwellian source

Bissell, Robert; Johnson, P. C.

doi:10.1063/1.866328

Cited by 120 publications

(86 citation statements)

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“…This particular comparison of exact potential profiles with finite e's was obtained here for the Maxwellian distributed ion source with temperature T n =T e ¼ 10 via a highly reliable numerical computational method developed by Kos et al 6,20 for dealing with the complete T&L collisionless discharge model as well as with its special case of quasineutral plasma known as Bissell and Johnson (B&J) model. 2,3 This method has proved as superior to others known by us in obtaining results with an extremely high resolution and at the same time being applicable to a wide (practically unlimited) range of ion source temperatures.…”

Section: Introductionmentioning

confidence: 89%

“…Achieving this goal requires an exhaustive analysis consisting of many consecutive steps, e.g., rescaling the potential profile appropriately in the so-called "intermediate scale," in whose a) representation the potential profiles are nearly independent on discharge parameter e. Riemann has recently 4 summarized the "state of the art" of plasma-sheath analysis showing that there are three models that should be distinguished, namely (1) fluid model, (2) kinetic model with ions generated with zero initial velocity (so-called "cold" or "singular" ion source), and (3) kinetic model with ions generated with a finite initial velocity (the so-called "warm" or "regular" ion source). While proper analysis was performed within fluid approximation and the kinetic model with "cold" ion sources, Riemann states that "the structure of the plasma-sheath transition for models with hot ion source was never analyzed!"…”

Section: Introductionmentioning

confidence: 99%

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Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

2011

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A plasma-sheath transition analysis requires a reliable mathematical expression for the plasma potential profile UðxÞ near the sheath edge x s in the limit e k D =' ¼ 0 (where k D is the Debye length and ' 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 (e ¼ 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 a general formula assuming relatively high ion-source temperatures, i.e., much higher than the plasma-sheath potential drop. Riemann's 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 ]. Therefore, for many years, there has been a need for explicit calculation that might confirm the Riemann's 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 a ¼ 1=2 and a ¼ 2=3, with medium ion source temperatures. We found a 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 a ¼ 2=3 region towards unexpectedly low ion source temperatures.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

2011

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“…The complete plasma-sheath equation was solved numerically. Solving the plasma equation for a Maxwellian particle source S i ( , ) ξ η ∼ exp( ) − η , Bissel et al [5,6] showed that the final result is essentially sensitive to the choice of the source function.…”

Section: Introductionmentioning

confidence: 99%

Boundary function method to find the asymptotic solution of the plasma—sheath equation

Filippychev¹

2006

Comput Math Model

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We consider the asymptotic solution of the Tonks-Langmuir integro-different equation with an Emmert kernel, which describes the behavior of the potential both inside the main plasma volume and in a thin boundary layer. Equations of this type are singularly perturbed due to the small coefficient at the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the regular expansion series and in the boundary function expansion. The equation for the first coefficient of the regular series has only a trivial solution. Second-order differential equations are obtained for the first two boundary functions. The equation for the first boundary function is solved numerically on a discrete grid with locally uniform spacing. An approximate analytical expression for the first boundary function is obtained from the linearized equation. This solution adequately describes the behavior of the potential on small distances only.

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“…The inodilisd inotlcl of TJissd and J o l i i i~o~i [7] 1i;rs it "aornial" sourcc. A surprising contradiction between kinetic arid hydrodynamic analysis occurs in lhe case of supersonic ion flow.…”

Section: I1 Bohm Criterion and Sheath Edge Field Singularitymentioning

confidence: 99%

Plasma—sheath transition and Bohm criterion

Riemann

1992

Contrib. Plasma Phys.

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In typical boundary layer problems a quasineutral plasma is shielded from a negative wall by a "thiu" positive space charge rcgion ("sheath") with a characteristic extension of several electron Debye lengths AD. In the usual case XD < L -where L is the smallesl competing characteristic length relevant lor the boundary layer -the formation of a positive space charge is impeded by ion wall losses, and a stationary sheath can exist only if the h h i n critctiori [ I , 21 is fulfilled. In its simplest form it requires that the ions enter the sheath region with a high directed velocity v, 2 c, = (kTe/mi)"* exceeding the the ion sound velocity c, (I; = Bolzmann's constant, T, = electron temperature, mi = ion mass). Consequently, the ions must be preaccelerated by a non-shielded residual field in the quasineutral "presheath" region with an extension L > AD. Apart from ion inertia, the presheath is tloniiriatetl by (at least) one or Lhe following processes:

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The solution of the plasma equation in plane parallel geometry with a Maxwellian source

Cited by 120 publications

References 9 publications

Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

Boundary function method to find the asymptotic solution of the plasma—sheath equation

Plasma—sheath transition and Bohm criterion

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