THEORY OF THE SHEATH EDGE IN A WEAKLY IONIZED COLLISION DOMINATED PLASMA

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THEORY OF THE SHEATH EDGE IN A WEAKLY IONIZED COLLISION DOMINATED PLASMA

K. Riemann

To cite this version:

K. Riemann. THEORY OF THE SHEATH EDGE IN A WEAKLY IONIZED COLLISION DOMINATED PLASMA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-831-C7-832.

�10.1051/jphyscol:19797401�. �jpa-00219401�

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JOURNAL DE _PHYSIQUE CoZZoque C7, suppldment au n07, Tome 40, J u i l l e t 1979, page C7- 831

W O R Y OF THE YlEATH EDGE IN A WEAKLY IONIZED COLLISION DOMINATED PLASMA

K.U. Riernann.

I n s t i t u t fiir fieoretische Physik, Ruhr-Universit2-t Bochwn, 0-4630 Bochwn, Fed. Rep. Germany.

I. Introduction

Whereas the wall layer of a collisionless plasma was treated long time ago [I], there is little in- formation on the transition from a collision dominated plasma to the collision free space charge sheath. In a former paper [ 23 we have given the analytic presheath and sheath solutions for a simple plasma model. These solutions refer to two distinct scales : The sheath with a typical dimen- sion of a Debye length AD and the presheath on the scale of the mean free path A . Due to the singularity of the problem, however, they cannot simply be matched to a uniformly valid potential variation.

11. Presheath and sheath

We consider a weakly ionized plasma in contact with an absorbing negative wall. The electrons are in equilibrium with the selfconsistent potential distribution. The ion kinetics is dominated by charge exchange collisions (A = const) with cold (To<< T-) neutrals. We refer to 21 and use dimen- sionless quantities

(Z = space coordinate, plasma: z<0, wall: z=0;

Mv 2 /2=ion energy ; U=potential). The normalized particle densities are given by

j+ represents the ion current density and k(x)=x is the inverse function of the potential variation X(x)

.

From Poisson1 s eq.

we expect that for AD/A +O the potential is de- termined by the quasineutrality condition.

This is true on the scale x=0(1) of the presheath. From [ 23 we take the solution

singularity kf(0)=O at x = 0 indicates the

X

necessity of a second scale . -

- 1 / 2 A D

5 = X/E with E = j+ ^-A

-2 -1 X 0

Fig. 1

representing a thin layer (sheath) where space charges are essential. The sheath solution [ 2

J

is shown in fig. 2. On this scale the qua- sineutral region is infinitely remote and the potential tends to X=O+for 5 -t -rn. On the other hand, the sheath is "compressed"

into a vertical line (corresponding to the field singularity) on the scale of the

which is shown in fig. 1 . The field

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797401

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111. Transition at the sheath edge

From the potential variation near the sheath edge

x

= 0 it is clear, that the "outer" (presheath) and

"inner" (sheath) solutions cannot be matched to a sensible uniformly valid solution for small but fi- nite values of E. We have therefore to reinvesti- gate the problem on an intermediate scale for

x

^-t0 accounting for collisions

and

for space charge.

From eqs. (4) and (6) it follows

on the outer and

Fig. 3 on the inner scale. Comparing the leading terms we figs. (4a,b) on the scale of the presheath (x) and find the appropriate scale transformation of the sheath (5). The curves demonstrate the

for the intermediate problem. Physically the transition region is dominated by ions emerging from charge exchange collisions: The high density contribution of these ions favours the formation of a positive space charge. On the other hand the loss of fast ions due to collisions can be neglected, to lowest order, on this scale. Consequently, collisions have nearly the same effect as an ionization process. This is the reason, why eq.(9) agrees with the corresponding scaling [3]for the collisionless Tonks-Langmuir model. The systematic development of eqs.(2) and (3) leads to the nonlinear integro- differential equation ( K < ~ J =

<>

for the transition region. Using a simple ansatz to evaluate the integral this equation can, to a good approximation, be replaced by

w(5) = wo (5) for 5 < TA

The influence of the arbitrary limit CA is only weak; GA x

-

1.6 gives the best agreement with eq.

(10). w(5) is plotted in fig. 3. We see that it can easily be matched with the outer (wo) and inner

(wi) solution to construct the wanted continous potential variation. The result is shown in

approach to the asymptotic case on both scales.

I

-

Figs. (4a,b) References

[I]

S.A. Self, Phys. Fluids - 6, 1762 (1962)

[z]

K.-U. Riemann, J. Nucl. Mat. 76 & 77, 579(197 [3] R.N. Franklin and F.R. Ockendon, J.Plasma

Phys.

4

, 371 (1970)