J. Plasma Fusion Res. SERIES, Yol.4 (200t) 13-22

Sheaths, Double Layers and Dust Levitation
ALLEN, John Edward
Department of Engineering Science, university of Oxford, U.K.
(Received: 5 December 2000 / Accepted: 3 September 2001)

Abstract
This paper contains an overview of certain sheath phenomena, namely the criterion for sheathformation, some examples of double layers and a description of some recent experiments on thesuspension ofdust particles in a sheath. A detailed review is not attempted and results obtained at Oxford
are used to illustrate the last two topics.

Keywords:
sheaths, Bohm criterion, negative ions, double layers, dusty plasmas

Introduction
The first topic to be discussed is that of the spacecharge sheath at the boundary of a low pressure plasma.

Relatively high electric fields exist in such a sheath, but
the positive ions, on entering the sheath, have alreadytravelled through weak but long-range electric fields inthe plasma. The latter is quasi-neutral, the small
imbalance between the electron and ion densities isimportant, being associated with a potential differenceof the order of the electron temperature in electron volts.
Early work on the subject was carried out by Tonks andLangmuir; in 1949, however, Bohm published a simple
criterion for sheath formation. This stated that thevelocity of the ions on entry to the sheath is given by v
> (kT"l@rt2, the theory being relevant to the case where
the Debye distance is small compared with the plasma
dimension. It can be demonstrated that an equality sign,rather than an inequality sign, is usually valid. The
Bohm velocity is the same as that of an ion acoustic
wave in a plasma, this fact has led to an analogy
between the plasma boundary and the Mach surface of
fluid mechanics. A generalized Bohm criterion that
included the distribution of ion energies has been shown
to be consistent with this concept. The Bohm criterion
can be readily extended to several other cases including
a plasma containing a mixture of positive ions, and a
Corresponding author's e-mail: john.allen@ eng.ox.ac.uk

plasma containing negative ions.

The second topic to be discussed is that of double
layers. Space charge regions are to be found not only
adjacent to electrodes or discharge chamber walls, butalso between two different plasmas. An example isgiven of a double layer, formerly termed a double
sheath, at a constriction in a discharge tube. That isfollowed by a detailed description of experiments

carried out to study a "free double layer", i.e. oneformed between two plasmas in a tube without
constrictions. It was found that the double layer was infact a triple layer of charge, but the term double layer isstill usually employed. The experiments showed that thelayer was formed when the electron drift velocity
approached the electron thermal velocity, a result inagreement with simulations found in the literature.

The third topic discussed here is that of dust
levitation in a sheath. This subject is of much current
interest following reports in 1994 that "plasma crystals",
constituted of dust particles, had been produced in thelaboratory. In these experiments, however, the dust
particles are pulled out from the plasma by their weightand reside in the sheath beneath. The strong electricfield in the sheath enables the weight of the particle tobe balanced by an electric force. The dust particles@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchAllen J.E. Sheaths, Double Layers and Dust Levitation

behave rather like miniature Langmuir probes and attaina floating potential, and a certain charge, such that thereach it. The geometry considered is plane and the wall

electron and ion currents become equal in magnitude.
One method of determining the charge acquired is toobserve vertical oscillations of the particle about its
equilibrium position. A description is given here of anexperiment on the damped oscillations of particles
introduced into the plasma. Dust charges of the order of
10,000 electrons have been determined in this way, forparticle radii of the order of microns. A knowledge ofthe potential variation in the sheath is required in orderto determine the charge. A number of sheath models

back into the plasma. Bohm made three assumptions:firstly, that ionization in the sheath is negligible'
secondly that the electric field at the plasma boundary isvery small and thirdly that the energy distribution of theis so negative that most of the electrons are reflected

positive ions may be neglected (i.e. mono-energetic ionsare considered). In addition it is assumed that theelectron distribution is Maxwellian so that their density
can be described by the Boltzmann relation.

have been examined and it was found that the potential
profile is almost parabolic over much of the sheath in all
the cases examined. This is of interest and considerable

convenience in analysing the experimental data.

1. Sheaths
1.1 The Work of Langmuir
The first topic to be discussed is that of the spacecharge sheath that forms at the boundary of a low-pressure plasma. Such a sheath is to be foundsurrounding a probe immersed in a plasma; otherFig. 1 Diagram of the path of a charged particle passing

situations are those of sheaths forming at the wall of thedischarge vessel or adjacent to an electrode. Langmuir,
the pioneer of this subject, first assumed that the sheathadjoined a perfectly neutral plasma. Figure 1 shows adiagram from one ofLangmuir's early papers [1]. tn thisdiagram a charged particle moves in a straight line until
it enters the space charge region. Thereafter it isthrough a sheath surrounding a cylinder or
sphere [1].

8 : !->
q'8so?

deflected by the electric field; the particle illustrated isdeflected but not captured by the probe. Other particles
would hit the probe and contribute to the probe current.

0.2 0.4 06 0.8\

0.2

Later it was realized that this picture was simply
not self consistent. The plasma is a quasi-neutal rcgion
that sustains an electric field due to the small but finiteT

0.4/ cm. - ->limbalance between the positive ion and electron
densities. Langmuir never corrected his probe theory butlater Tonks and Langmuir [2] derived a theory of thei06

low-pressure positive column. In this case the spacecharge sheath forms at the tube wall and a radial electricfield is set up in the quasi-neutral plasma. The potential
distribution in the plasma is shown in Figure 2, together
with the early probe measurements of Killian [3].

0.8

1.0

\
s

\

\
\x

J-

\
\

t.0

1.2 The Bohm Griterion for Sheath Formation
Bohm gave a very simple derivation of a sheatht.2
Fig. 2 Radial potential distribution for a low-pressure arc

criterion, which I shall cite here. The case considered byin a discharge tube. Experimental points arelor a

Bohm [4] was that of a negatively charged wall (or= 3.1 cm, I" = 38,800 K. mercury vapour pressure= 0.2 x 10-3 mm Hg. ion free path = 31 cm [3].

probe) which acts as a sink for the charged particles thatt4Allen J.E. Sheaths, Double Layers and Dust Levitation

The result found by Bohm is the following,v2Q=+VQ.V(VO.Vd)'
2a2

eVo2L P7.
where eVe is the ion energy

This equation is mathematically equivalent to thefundamental equation for compressible flow with abeing the sound speed (Bohm speed). Thus we canadopt results from the theory of fluid mechanics andapply them to the problem of the plasma boundary. In
particular we can utilize the concept of the Machsurface. In a region of supersonic flow the Mach surface
is such that the fluid velocity component normal to thesurface is equal to the sound speed c. Following this lineof reasoning Stangeby and Allen [7] identified theplasma sheath boundary as a closed Mach surface. An- "n,., to the sheath.

It is seen that the positive ions must have an energy ofI kT. or more before a space charge sheath can form.
This result is known as the Bohm criterion for sheathformation. This kind of result was first found in 1929 byTonks and Langmuir, but not in such a concise form.
Indeed Langmuir stated in 1923 [5] that the potential at
the sheath edge is - I volt if the electrons have anenergy of 1 volt!
The above criterion contains an inequality sign but
it can be shown, by performing a plasma calculation
rather than a sheath calculation, that an equality sign isvalid in many cases.

application of this idea was to the flow of plasmaaround a Langmuir probe; some of the plasma ions areabsorbed by the probe, after travelling through the

1.3 The Work of Caruso and Cavaliere

By using the technique of the stretching ofvariables Caruso and Cavaliere [6] extended the theoryof Tonks and Langmuir (for the plane case) byconsidering both the plasma region and the sheathregion of space. The theory applies when the ratio of the
Debye distance to the size of the plasma is vanishingly
small. We are dealing here with a mathematical model,
but one that is very useful and sufficiently accurate formany purposes.
Figures 3 and 4 show the potential distribution inplasmatheand in the sheath, respectively. In the firstcase we can interpret the result as showing that theelectric field at the plasma boundary is large comparedFig.3 The plasma potential distribution for theLangmuir model [6].

with the average field in the plasma. The second
diagram, figure 4, illustrates that the electric field at theplasma boundary is small compared with the averagelro-0.854 -----------

field in the sheath. In these calculations, as in those ofTonks and Langmuir, the ions have a distribution of

10

15

----

energies corresponding to their different points of origin.

In the plane case it turns out that the potential at theplasma boundary, -0.854 kT"/e, is independent of theionization process, e.g. single stage or multi-stage
ionization.

1.4 The Plasma Boundary as a Mach Surface
If one considers cold ions, a quasi-neutral plasma
can be formed in a converging system in the absence of
local ionizing processes. This differs from the case inwhich ions are produced locally [2,6]. Assuming that theelectrons have a Maxwellian distribution, and that theelectric field is conservative. one can readilv obtain thefollowing equation

Fig. 4 The sheath potential distribution [6].

t5

201

Allen J.E. Sheaths, Double Layers and Dust Levitation

sheath, and some of them are deflected but not captured

andby the probe.

)z'w'

1.5 The Generalized Sheath Criterion
One can generalize the sheath criterion to allow fora distribution ofion energies. The first attempt at such aderivation was made by Boyd [8] and a later study was
made by Harrison and Thompson [9]. The result is thefollowingI u(,

2\t2 '\-'=lbut these latter results are not general. The first criterion
above appears to be less useful than that for a single ion
species. When any particular plasma model is developed
the criterion is found to hold at the plasma boundary,"*"1=but one cannot use it in a completely independent sheathmodel. One must cany out a plasma calculation first, inorder to determine the ion densities and ion velocities

kr

appearing on the LHS of the equation.

where we have assumed the equality sign since, as
briefly discussed above, this can be justified byapproaching the plasma-sheath boundary from theplasma side. It is clear that the criterion is not one
simply concerned with energy; it is concerned insteadwith the ion velocity distribution function. The

1.7 The Bohm Criterion in the Presence ofNegative lons
Many plasmas used in plasma processing containelectronegative gases; for this reason the subject ofplasmas that have a mixture of positive and negativeions is of considerable interest. Since the plasma isquasi-neutral there is a coupling between the different
ion species; this results in a changed Bohm criterion.
Early work on the subject was carried out by Boyd andThompson [3], but they did not discuss the multiple
solutions that are found in the mathematical model.
Braithwaite and Allen considered the problem, being
unaware of the earlier work, and obtained the results

distribution function is not specified by the sheath
criterion, but it must be of a form that satisfies thecriterion.

By considering the propagation of longitudinal
waves in a plasma Allen [10] was able to show that theabove criterion corresponded to a quasi-neutral wave
with a zero phase velocity in the laboratory frame ofreference, i.e. an ion acoustic wave is unable to travelback into the plasma. Thus once again we arrive at theconcept of sonic flow at the plasma boundary. It was

shown in Figure 5 t141. The resulting ion energy
required for sheath formation is decreased in theverified that the criterion held for all of the different
cases studied by Tonks and Langmuir [2), and for the
spherical probe theory of Bohm et al [1 1].

1.6 Extension of the Bohm Criterion to a
Mixture of Different lons.
The extension of the Bohm criterion to a mixture ofdifferent ions is straightforward and was derived by
Cooke in a D. Phil. thesis [2]. A convenient method to

obtain the result is to consider once again the
propagation of longitudinal waves (of long wavelength)

in a plasma, and simply allow for different ion
populations. In such waves the variation of the total ion
space charge follows the variations in electron density.

thereby maintaining quasi-neutrality in the plasma. The
result is that, at the sheath edge,

n,Zi

nrZ)

-

I

Fig. 5 Plot of the normalized sheath edge potential

n.vl M, n.vl M, kT.

against the ratio of the negative-ion density toelectron density in the bulk plasma for threewhere the ion and electron densities are the local ones.

In some cases

values of gamma, the ratio of electron

temperature to negative-ion temperature. The
arrowed dotted line indicates the necessary jumpin the solution [14].

lu,rl=l z, kr.
22t6

Allen J.E. Sheaths, Double Layers and Dust Levitation

presence of negative ions. When the population of thenegative dip reflecting some of the electrons comingfrom the LHS. Figure 1l is an example of a numericallysimulated profile based on this picture [20].
Figure 12 is a schematic diagram showing theposition of an ion energy analyser placed in the wall of
the discharge tube. Measurements showed that ion
energies of tens of volts were acquired on the anode side

latter is very high the temperature of the negative ionsreplaces that of the electrons in the Bohm criterion. In
general the required ion energy is between the electron
temperature and the negative ion temperature, measured

in eV. When the ratio of the electron temperature to thenegative ion temperature exceeds a certain value
multiple solutions are found. Braithwaite and Allen
proposed that, in this case, the lower value should be
employed, as shown in Figure 5. The subject is one offurther investigation at the present time.

diretioE of the el$trcn flow +1.8 The Bohm Griterion in the Presence ofRadio Frequency Fields
It has been shown that the criterion remains valid in

the case of capacitively coupled RF plasmas, thelongitudinal RF fields being largely screened out of theplasma tl5,l6l. The criterion is a very general one, forlow-pressure plasmas, the only exception known to thepresent writer being that of the thermally produced

Fig. 6 Double sheath at a constriction in a gas-discharge
tube [18].

Caesium plasma [17].

2. Double Layers
Space charge layers form not only at solid surfaces,
but can form between one plasma and another. The term

double sheaths is to be found in early papers on thesubject, but the terminology changed and the term
double layers took its place. One example of a double
layer separating two plasmas is shown in Figure 6; inthis case the double layer effectively increases thecollecting area for electrons, directing them towards theconstriction in the discharge tube. Calculations ofpotential and space charge within such a double layer
are shown in Figure 7 [8]. Once again the ions enter
the space charge region with a certain energy that could
be described by a modified Bohm condition.

A discharge tube used in Oxford to study double

Fig. 7 Spatial variation of normalized charge density andnormalized potential, for a certain set oflayers is shown in Figure 8. In this case the vapour

parameters [18].

pressure was carefully controlled and sharp constrictionsin the tube were avoided. The double layer was thenobserved in the middle of the discharge tube, separating

two different plasma regions. Probe measurements
showed that in fact a triple layer was formed, sometypical results are given in Figure 9 t191. The term
double layer has been retained, in common usage, toinclude multiple layers of space charge, a potential
difference existing across the assembly of layers.

Fig.8 Discharge tube: 1, Hg pool electrode; 2, port todiffusion pump; 3, cooling jacket; 4, ionization

A diagram illustrating the motion of ions andelectrons in such a triple layer is shown in Figure 10. It(pressure) gauge; auxiliary electrode; 6, Langmuirprobe. The overall length is S m [19].

is seen that five groups of particles are involved, the17Allen J.E. Sheaths, Double Layers and Dust Levitationof the layer (the double layer conveniently drifted past
the analyser in these experiments). The energies here
were associated with the velocity component normal to
the tube wall. I shall not attempt to explain the results in

Potentiqt (V)!

the present paper. There is no mystery about the source=/of the energy, however, it derives from that gained by

!/

,l

,l

the electrons that cross the layer from the cathode side
to the anode side.
Figure 13 shows the ratio of electron drift velocity

to thermal velocity as the pressure is lowered (moving

v{

,lu
,I

\

-8 E RE

Distqnce/
Fig. 11 Numerically simulated potential profile [20].

OoobtrFig. 9 The axial distribution of floating potential

Di*tnrs tuh

loyrr

throughout the triple layer at different distancesfrom the wall: (a) 0 mm (at the wall); (b) 2 mm; (c)10 mm; (d) 13.5 mm (centre); pressure 1-2x 1O'a
torr. The triple layer slowly drifted past the probe

Fig. 12 Schematic diagram showing the ion energy
analyser. The double layer slowly drifts from left

to right [19].

11e1.

let

fr o-lfcrro

fFig. 13 Variation of the ratio between the electron drift
velocity and the thermal velocity as the pressure

Fig. 10 Schematic plot showing five groups of particles.
Subscripts refer to electron (e) or ion {i), free (f)or trapped (t), in the first (1) or second branch (2)is reduced by lowering the temperature of thecooling jackets. The discharge current is held at
5.6 A [19].

t201.

18

Allen J.E. Sheaths, Double Layers and Dust Levitation

Fig. 14 lon and electron densities at the front of a

Fig. 15 Experimental set-up

plasma expanding into a vacuum [22].

to study damped

oscillations of dust particles in a sheath [25].

from right to left) at a fixed current. The ..double layer"
appears when this ratio approaches unity; this result is in

J\r^agreement with simulations performed by De Groot et al

Dandls: lJ2p
PF$: ll3l P.

5-Y:.t*t''

fl#wm^^"-*

I2rl.
Another example of a double layer in space, notadjacent to a solid surface, is shown in Figure 14. This
diagram illustrates the electron and ion densities at theboundary of a plasma expanding into a vacuum [22];quasi-neutrality is again approached behind the "front".

Dcd Rdis a,l5 p6

g8

DtdRdl'r LtrtF

P!6'c 13.31 P.

Pr6tW 5,57 P!
a

The field existing in the double layer accelerates a small
number of ions to high energies.

R4

lliiftnnnne.* flnrr.*\iv-'11u

3. Dust Levitation*Hf;li.'*E

Another field of research in which the sheath is ofgreat importance is that ofdust levitation. The subject offlfinnn^-:-"dusty plasmas" has attracted a great deal of interest in
the last few years; one reason for the surge of interest
being the discovery that "plasma crystals", formed ofcharged dust particles, could readily be formed inlaboratory experiments t23,241. In many experiments,

o.o

o-2

0.4

Du tediw: 6,85 1r

M

13.33h\A^^^0..

o* (Hn*)

o,

0.4

06

08

Fig. 16 Dust oscillation data and harmonic theory; thefull lines are theoretical damped harmonic

however, dust particles of a few microns in diameter are

curves. Each column of plots is for the same

pulled out of the plasma by their weight, andpressure. while each row of plots is for the same

particle. Typical position errors are of the orderof 10 microns, not visible on this scale [25].

subsequently reside in the space charge sheath situatedbelow the plasma. In the sheath the strong electric field
produces a force acting on the dust particle that canbalance its weight. Needless to say, many of theparticles of different sizes. Much of the trajectory can betheoretical papers on dusty plasmas consider a neutral

fitted to the curves describing simple damped harmonic
oscillations, as shown in the diagram. This means thatplasma and not a space charge sheath!

Figure 15 illustrates an experimental set-up that has
been used to determine the charge residing on a dust

the electric potential in the sheath must be almost
parabolic over much of the sheath. The total potential
including the gravitational potential is illustrated in
Figure 17. The curve is, of course, a parabola, theminimum being where the equilibrium position of theparticle is situated. A number of different sheath models

particle [25]. Dust particles are dropped into an RF
plasma and then oscillate around their equilibrium
positions in the sheath. The particles are illuminated by
a HeNe laser and their trajectories are recorded using a
CCD camera. Some results are shown in Fieure 16 forhave been examined, that shown in Figure 18 being onlyt9Allen J.E. Sheaths, Double Layers and Dust Levitation

q

-.
ti
0.4
.o
H
cl

c)
()

arl

0.2

0.0

0.0

as a runction or,^:::'::;"
particle",radius. The data are for melamine

0.2 0.4 0.6 0.8 1.0

Fig. 1e Varues

Height above Electode (arbihary units)

formaldehyde particles suspended in an argon
plasma-sheath [25].

Fig. 17 Schematic diagram showing the parabolic total

potential resulting from the addition of a linear
gravitational potential and a parabolic electricalpotential. The location of the sheath edge isshown by the vertical full line and theone example 126l.lt is most remarkable that, over muchof the sheath, a parabolic fit is an excellent

equilibrium position of the dust is indicated by
the vertical dotted line. In this model the electric

approximation. In the example shown the fit is to within
l7o.We must not deduce that the ion density is uniform,
perhaps we can introduce the idea of quasi-unifurmity ofpotential (dot-dash curve) is constant to the rightofthe sheath edge [25].

charge density as an analogy of quasi-neutrality in theplasma.

Figure 19 shows some results obtained foresbemelamine formaldehyde particles in an argon plasma125).lt is seen that the charge on a dust particle is of theorder of 10,000 electronic charges, for radii of the orderof microns, and that it is a non-linear function of theparticte radius. The latter fact illustrates that the floating
potential is greater for the larger particles, assuming thatthe effective capacitance of the dust particle is that of asphere in a vacuum. A comparison of the experimental
results with existing theories is shown in Figure 20, thecurves showing the ratio of the experimental value tod@ot

the theoretical value. The theories employed here are theABR (radial motion) theory [27) and the OML (orbital

aJ

E
e

Elc
.AH

oo

i>

24681012

motion limited) theory [28]. The agreement is betterwith the ABR theory. It should be remembered,

Height above Electrode (mm)however, that these theories were developed for probes
in a neutral plasma; what is now required is a viable
theory of charging in the sheath. The problem was first

Fig. 18 Diagram showing the close fit between aparabola and a model of the sheath [25]. The
horizontal (dot-dash) line shows the potential at
the plasma boundary. allowing for a finite field atconsidered by Nitter [29].

the boundary. The lower curve shows theAn alternative method of determining the charge
residing on a dust particle is that of simply using thedifference between the model and the fitted
parabola, i.e. less than 1%.
20

A-llen J.E. Sheaths, Double Layers and Dust Levitation

balance between the electric and gravitational forces[26]. The data used are the dust mass and the observed
suspension height; the same model of the sheath is thenemployed to determine the electric field at the positionof the dust. Some results are shown in Figure 21,
together with those obtained using the previous method.
It is seen that there is good agreement with the damped
oscillation method of determining the charge. It is[]

27,449 (1924).

12) L. Tonks and I. Langmuir, Phys. Rev. 34, 876(1929).

t3l T.J. Killian, Phys. Rev. 35, 1238 (1930).

[4] D. Bohm, The Characteristics of Electrical
Discharges in Magnetic Fields, ed A. Guthrie andperhaps surprising that this simple method ofdetermining the charge on a dust particle, in a space

R.K. Wakerling, Ch. 3, (New York: McGraw-Hill,
1949).

[5] I. Langmuir, Gen. Elecr. Rev. 26, j31 (1923).
t6l A. Caruso and A. Cavaliere, Nuovo Cimento 26,

charge sheath, does not appear to have been used before.

J--r

References
I. Langmuir and H. Mott-Smith, Gen. Elect. Rev.

r38e (1962).
t7l P.C. Stangeby and J.E. Allen, J. Phys. A: Gen.
Phys. 3, 304 (1970).
l8l R.L.F. Boyd, Proc. R. Soc. A 201,329 (1950).
t9l E.R. Harrison and W.B. Thompson, Proc. phys.
Soc.74, 145 (1959).
J.E.
Allen, J. Phys. D: Appl. Phys. 9,2331, (1976).
[10][11] D. Bohm, E.H.S. Burhop and H.S.W. Massey, The,,--,--,-4-'f
f

to'l

9

'7 n'zCharacteristics of Electrical Discharges in
Magnetic Fields, ed A. Guthrie and R.K.
Wakerling, Ch. 2, (New York: McGraw-Hill,.E tor'5e01949).

t12l M.J. Cooke, D. Phil. thesis. Universitv of Oxford(re80).
[13] R.L.F. Boyd and J.B. Thompson, Proc. R. Soc. A
2s2, r02 (19s9).
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[15] K.U. Riemann, Phys. Fluids B 4,2693 (1992).
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243 /1993)'

lor

01234561
DNt Radiu (micrcG)

Fig. 20 Comparison of the measured dust charge withthat obtained from the ABR and OML (plasma)theories. The plots show the ratio of theexperimental value to the theoretical value [25].

r P- 1333 h--*- Eqtrrdot(t).-... Rcf.pU

b, P.6.5?Pr

Egdlod(l)-*--o-. Rcf.lltl

g

g
s
6
E

o2lfDrsrRrrtirl1dcnr)'46
Fig. 21 Values of the dust charge determined by two different methods. The points labelled [31] refer to the method of

damped ocillations, whereas those labelled (1)were obtained from the simple force balance between thegravitational and electric forces [26].

21Allen J.E. Sheaths, Double Layers and Dust Levitation[23] J. Chu and L.I., Phys. Rev. Lett.72,4009 (1994)'
t24lH. Thomas et al.,Phys. Rev. Lett. 73,652 (1994).

t17l A.D.R. Phelps and J.E. Allen, Proc. R. Soc. A 34E,

22r ft976.
tl8l J.G. Andrews and J.E. Allen, Proc. R. Soc. A 320,t25l E.B. Tomme, B.M. Annaratone and J.E. Allen'
Plasma Sources Sci. Technol. 9, 87 (2000).
l26jE.B. Tomme et al., Phys. Rev. Lett. 85' 25 18

4s9 (r97r).

t19l H.S. Maciel and J.E. Allen, J. Plasma Phys. 42,321(2000).

(1e8e).

t27lI.E. Allen, R.L.F. Boyd and P. Reynolds, Proc.
Phys. Soc. 8 70, 297 (1957).

t20l H.S. Maciel and J.E. Allen, Proc. Second
Symposium on Plasma Double Layers and Related

t28l J.E. Allen, Physica Scripta 45,497 (1992).
l29lT. Nitter, Plasma Sources Sci. Technol. 5, 93

Topics, 21 8 (Innsbruck, 1984).
[21] De Groot et al., Phys. Rev. Lett. 38, 1283 (1977).
l22lJ.E. Crow, P.L. Auer and J.E. Allen, J. Plasma(l996).

Phys. 14, 65 (1975).

22