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The plasmasheath boundary region

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2003 J. Phys. D: Appl. Phys. 36 R309

(http://iopscience.iop.org/0022-3727/36/22/R01)

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 36 (2003) R309R320 PII: S0022-3727(03)61082-X

TOPICAL REVIEW

The plasmasheath boundary regionR N Franklin

Oxford Research Unit, Open University, Boars Hill, Oxford OX1 5HR, UK

Received 19 March 2003Published 30 October 2003Online at stacks.iop.org/JPhysD/36/R309

AbstractIn this review an attempt is made to give a broad coverage of the problem ofjoining plasma and sheath over a wide range of physical conditions. We goback to the earliest works quoting them, where appropriate, to understandwhat those who introduced the various terms associated with the structure ofthe plasmasheath had in mind. We try to bring out the essence of theinsights that have been gained subsequently, by quoting from the literatureselectively, indicating how misunderstandings have arisen. In order to makeit accessible to the generality of those currently working in low temperatureplasmas we have sought to avoid mathematical complexity but retainphysical insight, quoting from published work where appropriate.Nevertheless, in clarifying my own ideas I have found it necessary to doadditional original work in order to give a consistent picture. In this way Ihave sought to bring together work in the late 1920s, the 1960s, and nowmindful of the commercial importance of plasma processing, work over thepast 15 years that adds to the general understanding.

1. Motivation

This review has been stimulated by controversy following thework of Godyak and Sternberg (1990a, b, 2002) and by arecent publication by Slemrod (2002) showing that the resultsobtained by Riemann (1997) in his treatment of a collisionalgenerationless plasmacollisionless sheath represent a solutionto that problem that in a mathematical sense has well-definedproperties. It follows by analogy that the solutions given bySu and Lam (1963), Lam (1965), and Franklin and Ockendon(1970) for similar related problems have the same properties.

Thus, it is opportune to survey the large body of work thatthere has been since Langmuir introduced the terms sheath(1923) and plasma (1928).

In what follows it is necessary to define some termsby generationless we mean that the creation rate of chargedparticles is negligibly small, by collisionless we mean thatthe motion of the ions is essentially inertial. These or theiropposites combined with plasma and sheath constitute theterritory we seek to describe.

For simplicity we will, in general, deal here with the one-dimensional situation, but we note that for a probe immersedin a plasma it is perfectly possible to envisage a situation thatis both collisionless and generationless.

2. Historical introduction

We begin by quoting relevant passages from Langmuir, givingthe original two definitions.

(1) Let us now assume that the plane electrode be charged toa negative potential of 100 volts. Electrons will thereforebe prevented from approaching close to the electrode,whereas positive ions will be drawn towards it. Therewill therefore be a layer of gas near the electrode wherethere are positive ions but no electrons, and in this regionthere will therefore be a positive ion space charge. Theouter edge of this sheath of ions will have a potential of 1and the positive ions pass through this outer edge with avelocity corresponding to 2 volts, Langmuir (1923).

We note that 100 V is large compared to kTe/e for atypical gas discharge, and the implication that the sheath isa region of effectively zero electron density, i.e. it containsonly positive ions.

(2) Except near the electrodes where there are sheathscontaining very few electrons, the ionized gas containsions and electrons in about equal numbers so that theresultant space charge is very small. We shall use thename plasma to describe this region containing balancedcharges of ions and electrons Langmuir (1928).

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Thus, there is an apparent significant difference inelectron density in plasma and sheath. However wherethey meet electron and ion densities must be comparable.The next significant development is usually described as

the criterion attributed to Bohm (1949), namely that the ionsleaving the plasma and entering the sheath are required tohave a speed v cs where c2s = kTe/M , Te is the electrontemperature and M the ion mass. But the basic concept isthere in the passage quoted above, and Langmuir (1932) waseven more explicit. The value of the wall potential used byBohm, no doubt influenced by earlier work, was also 100 V.

In their groundbreaking paper Tonks and Langmuir (1929)introduced two new concepts the sheath edge and the plasmabalance equation. The first represented the point whereplasma ended and sheath began, and they identified it witha point inside the singularity for the plasma solution where forthe full plasmasheath solution the difference in electron andion densities reached a certain fractional part. The secondrepresents the fact that in the steady state the volume generationrate of charged particles must equal the wall loss rate in theabsence of volume recombination. For their sheath edge it isclear from their figure 6 that they had in mind 1% as being acertain fractional part.

The plasma equations obtained by setting electron andion densities equal have infinite spatial derivatives in all thepotential, field and charged particle densities at a certain point,near where Tonks and Langmuir had put their sheath edge.Whereas the collisionless generationless sheath equations havethe property that at large distances from the wall, all physicalquantities have zero spatial derivatives when the criterion issatisfied.

The scale lengths of plasma and sheath are fundamentallydifferent, and this was made explicit by Caruso and Cavaliere(1962), but they did not take the further step of consideringhow to join plasma and sheath for finite plasma size L andfinite Debye length D.

The mathematical tool that allowed a suitable joining ofplasma and sheath, was the method of matched asymptoticexpansions, which had been developed primarily to deal withthe structure of boundary layers in fluid mechanics and inaerodynamics (Van Dyke 1964). The first application of itto plasma boundaries was by Su and Lam (1963), and Cohen(1963). This was followed rapidly by Lam (1965, 1967), andSu (1967)all concerned with electrostatic probes in plasmasof varying degrees of collisionality. Blank (1968) dealt withthe positive column in a collisional active plasma, and Franklinand Ockendon (1970) gave the corresponding treatment for acollisionless active plasma, in both the fluid approximation andthe free-fall approximation.

What emerged was that there was a fundamental differencein the structure depending on whether the spacecharge sheathwas collisionless or collisional. In the collision-dominatedcase, the plasma and collisional sheath joined smoothlywithout the need for a transition layer. But in all caseswhen the sheath was collisionless, there was the structure ofplasmatransition layercollisionless sheath, with the scale ofthe transition layer varying with the nature of the model usedto describe the plasma.

This has been confirmed by subsequent work summarizedby Riemann (1991), and more recently in Riemann (1997),

Slemrod and Sternberg (2001), Riemann (2000), Benilov andFranklin (2002), and Riemann (2003). There is now effectivelya full coverage of parameter space in terms of the ordering ofthe quantities, distance from the plasma centre to the wall, L,ion mean free path, i, and Debye length, D, using the method.Of course D L, for one to be able to talk of a plasma, buti ranges from the largest to the smallest length, and so interms of the method, which depends on expansions in termsof a small parameter, that varies from treatment to treatmentquoted above, as they range from i L to i D.

In recent years, considerable effort has gone intogeneralizing the Bohm criterion to take into account thevelocity distributions of both electrons and ions, the effect ofseveral species of both positive and negative particles, and theinclusion of ion temperatures. These we shall review below.

The increase in computing power has led to othertreatments solving the full plasmasheath equations again overa wide range of parameters (Franklin and Snell 2000, 2001).Also simulations have been carried out, though generally overa lesser range (Gozadinos et al 2001).

Thus, a comprehensive description of the structure of theplasmasheath can now be given, but it must be recognizedthat for radio-frequency (rf) generated plasmas a fully self-consistent treatment including generation is not yet available.However, we will attempt to relate the rf situation to the dc,indicating what is generally agreed, and what remains to bedone.

3. Plasmasactive and quiescent

Historically, the first plasmas studied were gas dischargeswhere the positive column provided examples of steadystate plasmas over a wide range of the similarity variablepressure plasma sizefrom 0.1 to 103 Pa m in a numberof electropositive gases. In general, ionization occurred byelectron impact in the volume, the volume generation beingbalanced by losses to the wall. Such plasmas have numberdensity distributions that are a function of position with acentral maximum and decreasing monotonically towards thewall. Such a plasma is described as active.

More recently, efforts have been made to create uniformplasmas by separating the generation region, and thusthe plasma being studied is generationless. Examples ofdevices producing such plasmas are Q-machines, thermallyproduced plasmas, and multipolar machines or bucketsources. Such plasmas are described as quiescent. Typically,these devices operate at 0.1 Pa m or less, and the presence ofmagnetic fields in some cases is over-riding in terms of thedescription of the plasma and its boundary regions.

4. Why is the Bohm criterion so universal?

To answer this question we will concentrate on establishingthe fact that in a simplistic fluid model the plasma equationsare singular where the criterion is satisfied, regardless of thecollision model for the ions and regardless of the generationmechanism for the charged particles.

We take the equations:

d(nivi)dx

= G(ne, x) (generation) (1)

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d(niv2i )dx

+ C(ni, vi, x) = nieEM

(ion momentum) (2)

ne = ne0 exp(

eV

kTe

)(Boltzmann relation) (3)

and using the plasma aproximation ne = ni = n, find that withE = dV/dx, and ne0 the electron density where V = 0,

dndx

= C + 2viGv2i (kTe/M)

anddvidx

= Cvi + G(v2i + kTe/M)

n(kTe/M v2i )Thus, both equations are singular, regardless of the functionalform of G and C the functions describing generation andcollisions, provided that the numerators are everywherepositive, and that is certainly so if both C and G 0, andone of them is >0. Equations of this form with C = 0 and forall geometries were given in Rosa and Allen (1970), and forgeneration other than electron impact in Johnson et al (1978).

It follows that there cannot be a collisionally modifiedBohm criterion as has been suggested (Godyak and Sternberg1990a, b).

In this simplified fluid model due to Woods (1965) wehave neglected the stress tensor.

We add here for completeness, an example of a situationwhere the Bohm criterion is not relevant. That is, in plasmadiodes where electrons and ions are generated in equal numbersat the hot boundaries, the gas being an alkali metal vapour(Fang et al 1969, Phelps and Allen 1976, Braithwaite and Allen1981).

We also note that in high current discharges it may benecessary to take into account the fact that the neutral densityis not constant but recycling of ions from the walls leads todensity depletion in the centre (Allen and Thonemann 1954,Allen and Coville 1961, Franklin 1963). The Bohm criterionstill applies as noted above even though G is a function of x.

5. Collisionless sheath boundary conditions

The usual conditions that are applied at large distances from thewall in a collisionless sheath are that the electric field and thenet space charge become vanishingly small. The potentialthus becomes constant at the plasma potential and the electronand ion densities become equal.

In some circumstances, it turns out to be more convenientto examine d/dV and treat it as the vanishingly small quantity.

Examples where this has proved particularly useful, relateto sheaths adjacent to longitudinally uniform plasmas. Thecase of a double sheath between two plasmas was treated byAndrews and Allen (1971). The structure of the double sheathassociated with a hot cathode was described by Prewett andAllen (1976), who obtained the expression for the equivalentof the Bohm criterion for this case. This agreed with that givenby Pak and Emeleus (1971) for ion waves in a beam-generatedplasma, in the limit 0.

The expansion of a plasma into a vacuum proved tobe another case where this boundary condition was relevant(Wickens and Allen 1979).

These examples show how in some situations it isboth convenient and insightful to use the potential as theindependent variable.

6. The structure of collisionless plasma andcollisionless sheath

In an earlier section we demonstrated how universal the Bohmcriterion is.

The essential problem in joining plasma and sheath undercollisionless conditions is how to pass beyond the singularityin the plasma solution where v2i = kTe/M . This is coupledwith the fact that the generationless, collisionless sheathsolution is such that all spatial derivatives go to zero atdistances that are many Debye lengths from the wall. Thetwo solutions are disjoint and it is necessary to introduceanother region to achieve smooth joining and it is this transitionlayer that has been the subject of Lam (1965), Franklin andOckendon (1970), Riemann (1977, 1991, 1997), and Slemrodand Sternberg (2001).

The various treatments have subtle differences dependingon the plasma model, but they do have one feature in common,namely that there is a transition layer and it has a scale lengthp

D0L1p where 1 > p > 0, with D0 the central Debye length,

and thus intermediate in size between plasma and sheath.For the particular case of the collisionless fluid ion model

we show the structure of the ion and electron spatial densitydistributions near the wall obtained by integrating the basicequations (1)(3) above with G = Zne, where Z is theionization rate, C = 0, and also taking into account Poissonsequation d2V/dx2 = e(ne ni)/0.

The central boundary conditions are ne = ne0, vi = 0,V = 0, E = 0, and these when fed into equations (1)(3) plusPoissons equation lead to the requirement in plane geometrythat ni0/ne0 = 1 + (ne0/ni0)2(D0Z/cs)2, while at the wall werequire that the ion directed flux and the electron random fluxshould be equal. A convenient parameter is = D0Z/cs andwe introduceX = xZ/cs, so that the eigenvalue of the problemis the value of X at the wall Xw LZ/cs.

The method of matched asymptotic approximations givesthis eigenvalue in plane geometry asXw = (/2 1)[1 + K1(/Xw)4/5 + K2(/Xw) + ] where the expansionparameter is /Xw D0/L. K1 is a universal constant4.4 . . ., and K2 depends on the ion mass because of the wallboundary condition. However, this method does not give asolution that is uniformly valid throughout all three regions,and that was recognized in Franklin and Ockendon (1970).

Nevertheless, we can use the results from it to obtainestimates of the electric field at different locations in the plasmasince the potential is normalized to kTe/e throughout and thusthe fields in the different regions scale to this quantity dividedby their extent. Furthermore, we can estimate the values ofthe field at the point where the plasma and transition layermerge by estimating the field in mid transition layer to bekTe/e4/5Ds L1/5 while that at the boundary between transitionlayer and sheath it is kTe/eDs. This points to the field atthe plasma edge being kTe/e3/5Ds L2/5 the value advancedby Riemann (1997).

Displayed in figure 1 are the computed solutions of thefull plasmasheath equations, without any approximation, for

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Figure 1. The ion and electron density distributions for a collisionless plasmacollisionless sheath for D0Z/cs = 0.002 showing(a) the plasma solution ( ), (b) the full computed solutions (), and (c) the sheath solutions (- - - -) obtained by integrating inwardsfrom the wall. The asymptotic behaviour is clearly seen and the problem of identifying a simple single plasmasheath boundary manifested.In order to make the structure visible we show only 7% of the total dimension, i.e. 0.93 < x/L < 1, but we give the full solutions for0 < x/L < 1 as an inset.

parameters = 0.002 andm/M = 1.36105 correspondingto argon. Also in the same figure is the plasma solutioncorresponding to = 0, and the sheath solutions obtainedby taking the wall values from the integration of the fullequations, and then, with appropriate rescaling, integrating thesheath equations inwards from the wall. This shows that thefull solution breaks away from the plasma solution near to,but shortly before, the singular point. That was the Tonksand Langmuir sheath edge and which, with present insights,is more appropriately named the plasma edge. After thetransition layer the full solution becomes asymptotic to thesheath solution and one might then talk of a sheath edgein this vicinity. This is where the sheath solution and thetransition region solution, echoing the words of Tonks andLangmuir, are such that the electron and ion densities of thetwo solutions are once again within a certain fractional part.It is important to note that it is within the transition layer thatthe Bohm criterion is satisfied, and that this point cannot belocated with precision. This fact is clearly demonstrated infigure 4.11 of Franklin (1976) where a relatively large value of (0.0144) was chosen to make it visible on a diagram.

Furthermore, for this floating wall situation, the electrondensity at the wall is 1.46% of its central value in hydrogen,0.23% in argon and 0.095% in mercury; i.e. the electron densitywhile small, is not vanishingly small.

For such a small value of as given above (0.002), theanalytical intermediate solution is effectively indistinguishablefrom the full computed solution, and this is borne out bythe relative values of the eigenvalue obtained analyticallyand computationally, viz 0.6030, cf 0.6033 for argon and = 0.002.

It has been suggested that neglect of volume ionizationgives rise to the problem of simply joining the plasma andsheath solution, usually described as patching, at some suitablyselected point. But in the intermediate region, while therelative magnitude of the terms concerned is reflected in therescaling in the analytic method, none is neglected, and it isfor this reason that the asymptotic behaviour shown in figure 1is as it is.

Figure 2. The plasma density N , the electron density Ne, and theion density Ni near the plasma edge for a collisionalplasmacollisional sheath for the constant ion mean free path case.The spatial coordinate Y (xp x)/4/5D0 L1/5. Thus, the plasmasolution is zero at x = xp N n/n0 (L/D0)4/5. The extent of thesheath is determined by the wall boundary conditions, scales as

4/5D0 L

1/5, and the sheath solutions join the plasma smoothly after a

few scale lengths.

We discuss in a later section the magnitude of the errorsintroduced when one introduces the expediency of simplypatching.

7. The structure of collisional plasmacollisionalsheath

In order to emphasize the difference between the collisionlesscase and the collision-dominated we show in figure 2 thestructure of the region near the wall for the collisional situationwith constant ion mean free path (Benilov and Franklin 2002).

It is clear that the plasma solution which variesparabolically near N = 0, and the sheath solution, joinsmoothly. N the normalized density here is (n/n0)(L/D)4/5.There is no need for an intermediate layer.

The scaling of the collisional sheath is similar to that ofthe collisionless transition layer in terms of the parameters

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involved, but that has to do with the fact that the density variesparabolically near the plasma edge in both cases, rather thanany deeper physical reason.

If the ion motion model is of constant collision frequencythen the density varies linearly and the sheath dimension is oforder 2/3D0 L1/3 (Blank 1968).

8. When is a sheath thick?

In the situation of the positive column at low pressure witha floating wall the structure of the plasmasheath is in thenotation used by Lam (1967)plasma, transition layer, thinelectron sheath.

If the wall is an electrode whose potential is determinedby other external variablesthis could equally well be anegatively biased probethen the structure has additionallywhat Lam called a thick ion sheath. For such a situation, theChildLangmuir description is a good approximation of thatregion, and this yields an expression for the extent of the thickion sheath25/4/3(eVs/kTe)3/4Ds or 25Ds for Vs the biasvoltage 100kTe/e. For such a situation, the thick ion sheathmay well be more extensive than the transition layer.

This specific case is the subject of the asymptotictreatment of Benilov (2000) which contains additional insights,effectively identifying the scale of the thin electron sheath.

When the sheath is thick the ions strike the wall withenergies such that secondary electron emission may occur andthis adds a further degree of complication. Such a situationwas analysed by Prewett and Allen (1976), though the specificcase they covered was an electron emitting cathode adjacent toa plasma. This has been followed by further work by Benilovand Coulombe (2001) in relation to high pressure lamps.

9. Patching plasma and sheath

There have been many attempts over the years to modify theplasma and sheath solutions to enable them to be simply joined.Basically what is envisaged is that the plasma solution is takento be valid up to some specific point and the generationlesssheath solution is joined to it and then integrated to thewall. An interesting example is to be found in Allis andBuchsbaum (1967)historically they were post-Bohm, butwithout explicit recognition of the transition region, they wereunable to achieve their object of a smooth joining of plasmaand sheath.

It appeared that the problem had been satisfactorilyresolved with the work of Franklin and Ockendon (1970),and Riemann (1977, 1991). However, Godyak and Sternberg(1990a, b) intervened asserting that the transition was notnecessary, and that the mathematical ideas behind it introducedunphysical complexity.

This has caused a considerable degree of confusionespecially since that paper also advanced the concept of acollisionally modified Bohm criterion.

As we have indicated above the plasma solution issingular where the Bohm criterion is satisfied regardless ofthe collisionality.

What happens as the plasma becomes collisional is thatthe density at the plasma edge falls, the corresponding Debyelength increases, and at some point ions undergo at least

Table 1. Normalized wall values of the electron density, ion density,ion flux, ion speed, potential, field, and physical size, obtained bycomputing the full plasmasheath equations and by patching. Thegas is hydrogen and the parameter = ZD0/cs = 0.002. Thedifference in percentage terms is indicated as the error. The effect ofignoring the transition layer is evident. The wall values for thepatched case correspond to those given in figure 3. w is theeigenvalue ZLw/cs.

New Niw Fluxw Uw w Ew wComputed 0.0298 0.1989 0.5102 2.566 3.513 411.5 0.5999Patched 0.0292 0.1935 0.5000 2.584 3.536 534.5 0.5776Error % 2.1 2.8 2.0 0.7 0.6 29.9 3.9

one collision in traversing the transition region and theyconsequently do not pass through the Bohm speed. Thenthe structure of plasmatransition layercollisionless sheathgives way to that of collisional plasmacollisional sheath asdescribed for the probe situation by Su (1967) and Blank (1968)for the active positive column.

This picture has been confirmed by recent extensivecomputations carried out by Franklin and Snell (2000, 2001).

In order to be specific we give in table 1 the error in usingpatching, for the case of a floating wall and = 0.002, the gaschosen was hydrogen, and the patch point at the point whereE = kTe/eDs. It is seen that the error is tolerable in somequantities, but not in others. Again, it must be emphasized thatthese are wall values.

Since the wall potential is determined by flux equality, itshows the least error, the ion speed at the wall Uw is likewiselittle in error. However, the fact that the transition layer hasbeen excised affects the densities and the ion flux becauseof ionization occurring in the transition layer and the totaldimension is significantly in error. But the field at the wall isthe quantity that is most seriously in error because the potentialdifference has been forced to occur over a significantly shorterdistance.

We give as figures 3(a)(c) the spatial variations inthe volume of the charged particle densities, ion speed andpotential, and finally, the field. The gradient of the field, ornet space charge, as seen in figure 3(c), is zero on the sheathside of the patch point but kTe/e2D0 on the plasma side. Itis far from continuous, and so the idea that patching can givesmooth solutions is not well-founded.

10. When is the ion motion inertial and collisional?

The simplified ion fluid equation of motion that we have beenusing is

vi dvidx

+

(Zne

ni

)vi + ivi = eE

M(4)

and the first term on the left-hand side represents inertia andthe second and third friction, the fluid speed vi is continouslyrenormalized to ensure that momentum is conserved and theions can be described by a single speed. We can thereforetalk in terms of inertial motion or collisional, depending onwhich terms dominate. This section and the subsequent onewill concentrate on what can be learned from this ion fluidequation of motion.

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(a)

(b)

(c)

Figure 3. Patched solutions showing (a) charged particle densities(b) ion speed and field, and (c) the field gradient or space charge.While (a) and (b) are continuous, (c) is discontinuous at the patchpoint.

In the plasma ne = ni, and thus we are comparingU dU/dX with (1 + i)U , where U = vi/cs, i = i/Z andX = xZ/cs.

From the plasma solution, i.e. (1)(3) with ne = ni wecan derive expressions for both of these quantities and identifya value of U = U where they are equal. This effectivelymarks the transition from collisional motion to inertial, andwe follow the ion motion thereafter asking whether, within theplasma, once the ions move out of collisional equilibrium withthe field, they ever return. U can readily be shown to be givenby [i/2(1 + i)]1/2, and the singular point occurs at U = 1.

But, in general, X = U/(1 + i) + (2 + i)/(1 + i)3/2 tan1 U(1+i)1/2 so that we can readily determine the numberof collisions made between disequilibrium setting in and theplasma edge being reached. It is given by i(Xw X), andwe show this function in figure 4. Clearly if the ions moveout of equilibrium within the plasma, they continue in thatmode since the number of collisions is always less than one,regardless of the collisionality.

Figure 4. The number of collisions after disequilibrium sets in, andbefore the plasma edge over the full range of collisionality for theplasma solution. Note that it is always less than one.

However, this raises another pertinent question as towhether the sheath can be sufficiently long and collisionalthat the ions once again return to collisional equilibrium withthe field in a thick ion sheath. It is well-known from sheathsolutions, be they collisionless or collisional (see, e.g. Cobine(1941)) that the ion speed increases monotonically as does theelectric field and the potential. And it is perfectly possible forions to be travelling in collisional equilibrium with the fieldin the sheath and their speed exceed the ion sound speed forthe plasma from which they derived, though their motion atthis stage is not linked in any way to the ion sound speed.Experimental data on ion drift shows this to be true for Teof the order of a few electronvolts (McDaniel 1964, Brown1966).

In order to gain further enlightenment, we solved the fullplasmasheath equations for situations where the sheath lengthis comparable to the plasma dimension, plotting the ratio of theinertial term to the electric field, i.e. the ratio of the first termin (4) to the right-hand side, for a wide range of values ofi. What is immediately obvious from figure 5 is that thereis a significant break point in the solutions near the plasmasheath boundary, and we will examine that further later. Butit is also true that for values of i 100, while there is someinfluence on the ion motion near the boundary, the ions do notmove significantly out of equilibrium as the electron densitygoes to zero, and soon return to that condition. However,the rapid change in field in this region means that there isan influence on the ion motion that is still discernible at quitehigh collisionalities.

We then sought to characterize the point of maximumdisequilibrium by determining the electric field at that point.The results are shown in figure 6 where it is seen that,over a wide range of collisionality, the field is given byeED0/kTe 1. This is reminiscent of the value proposedby Godyak and Sternberg (1990a, b) for patching of plasmaand sheath. But it is important to note that here we have thecentral plasma Debye length, whereas the Debye length at theplasma edge Ds tends to infinity as D01/4i as i becomeslarge. Thus, we are not simply recovering by another routeeEDs/kTe = 1.

However, on the basis of this evidence it is possible toconclude that in situations where there is a thick ion sheath,it may be possible to obtain solutions of sufficient accuracyfor some purposes by patching plasma and sheath. This

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Figure 5. Values of 1 R where R is the ratio of the inertial term in the positive ion fluid momentum equation to the electric field forcollisionality i/Z = 1, 10, 30, 100, 1000, and 10 000. R Mv(dv/dx)/eE. In the first three cases, the Bohm criterion is satisfied and theions remain in disequilibrium even though the thick ion sheath is comparable in size with the plasma. In the second three, inertia neverdominates and for 10 000 the sheath edge is scarcely detectable, i.e. the ions are everywhere in collisional equilibrium. = 0.002.

Figure 6. The value of the quantity eED0/kTe at the point ofmaximum disequilibrium over a wide range of collisionality1 < i/Z < 1000 showing that this quantity varies by less than afactor of 2 over the range.

clearly requires the thick ion sheath to be much larger thanthe combined extent of the transition layer and the thinelectron sheath when the plasma is collisionless, and verymany Debye lengths long when collisional. Furthermore, itdoes not negate the existence of the detailed structure foundat low collisionality (Franklin and Ockendon 1970, Riemann1991, 1997, Slemrod and Sternberg 2001, Kaganovich 2002a).And the changes in the physical quantities in these regionsshould be properly taken into account, before patching.

Thus, unless one recognizes that the plasma edge and thesheath edge are at different spatial positions albeit only a fewDebye lengths apart, there will be problems arising from thedifferent conceptions of what constitutes plasma and whatsheath. It is for this reason that the recognition of a transitionlayer as necessary to achieve a smooth variation of all physicalquantities is so important, and why we have pointed up theconceptual differences between quotations (1) and (2).

11. Determining the return to collisional equilibrium

In this section we examine in greater detail the situation inthe sheath when it is so extensive that the ions do return tocollisional equilibrium.

This involves solving the ion equation of motion in thesituation where there is no generation because the electrondensity has gone to zero and thus the relevant equations arevi dvi/dx + ivi = eE/M , or vi dvi/dx + v2i /i = eE/M , anddE/dx = nie/0, nivi = constant = nsvs. Not surprisinglyit turns out to be easier, and mathematically more elegant, totreat this problem in time in the constant collision frequencycase.

We give that case first, substituting d/dt for vi d/dx.The combined differential equations reduce to the relativelysimple form

vi + ivi =

nisvise2

0M= 2pisvis (5)

where denotes differentiation with respect to time. Thiscan readily be integrated to givevi = vis + 2pisvist/i +(vis

2pis/

2i c2s /D0i) [eit 1] and integrating this again

with respect to time allows one to relate vi and E to distancein the thick sheath. Clearly, the motion settles down toequilibrium with a time constant 1/i, and does so from aboveif vis > c2s i/D02pis or vis > 2Dsi/D0, and otherwise frombelow. Indeed the long time behaviour is vi 2pisvist/i asexpected from (5) when vi vii. Since then the inertial termis no longer of significance in determining the ion motion.

The constant ion mean free path case has correspondinglyvi dvi/dx+v2i /i = eE/M , nivi = nisvis and dE/dx = nie/0.This gives vi + 2vivi/i = 2pisvis with a corresponding larget solution vi = visi/(i + 2vist) + pis(visti)1/2 i/4t + for t i/vis.

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The constant collision frequency model allows a smoothtransition as it increases to the case considered by Schottky(1924) with constant ion mobility. However, the constantion mean free path is more accurate when charge exchangecollisions are the dominant collisions that the ions undergo.

Of course there are physical limitations to thesedescriptions but they are not significant until the ion energiesreach 1020 keV when the effective ion collision cross-sectiondecreases with energy, and thus there is the possibility of ionrunaway (Massey and Burhop 1952).

12. The stress equation for collisionless plasma andfor collisionless sheath

If one takes the Poissons equation for a collisionless,generationless sheath and integrates it, one derives a stressequation that describes the physical situation.

Thus, we begin with d2V/dx2 = e(ne ni)/0 andthen substituting ne = ns exp(eV/kTe) and nivi = nsvs,with v2s = kTe/M and v2i = v2s eV/M , obtain anequation which reads2nskTe = nskTe exp(eV/kTe)+nskTe(12 eV/Mv2s )1/2 0/2(dV/dx)2.

This is the stress balance equation for a collisionless sheath(Allen 1974) and holds throughout the sheath from the plasmato the wall.

However, it does not give credence, or validity, to thesuggestion that the point where 0(dV/dx)2 = nskTe has thesignificance suggested by Godyak and Sternberg. The threeterms on the right-hand side are such that the first goes rapidlyto zero, while the others grow at a rate that is limited by the left-hand side. Thus, on this basis, the so-called Godyak point isnot significant. Throughout the sheath once the electron termhas gone to zero, 0(dV/dx)2 and nskTe may be of similarmagnitude, but where they are equal is of no significance.

Equally, one can compare the magnitudes of the termsnekTe, niMv

2i , and 0E2/2 in the plasma solution. The ratio

of the first two is simply U 2 and that of the first and last canbe shown to be (2/2Ne)(U/(1 U 2))2. Thus, the electricstress within the plasma is of significance only within a Deybelength or two of the boundary, i.e. where we have placed theplasma edge.

13. Criteria for the development of a sheath

Here we give comments stimulated by the recent work ofValentini (2000) who sought to characterize space chargeseparation and how and why it occurred.

His standpoint then was to ask the question Whatconditions must be fulfilled for the electron density to fall morerapidly than the ion density while still essentially within theplasma?.

In order to answer the question he posed, it is necessaryonce again to separate the equations involved into a hierarchy.Thus, we maintain an electron distribution given by ne =ne0 exp(eV/kTe), but describe the ions by their equations ofmotion and of generation, viz vi dvi/dx + vii + viZne/ni =eE/M , d(nivi)/dx = neZ, and then ask under what conditionsis |dne/nedx| > |dni/nidx|, or rather, |dne/nedx| >|dni/nidx|. This, when we take into account the fact that vi

is monotonically increasing, i.e. dvi/dx > 0 gives the twoconditions

eE

M

(i +

Zne

ni

)> vi >

eE(1 v2i /c2s )M(i + 2Zne/ni)

We conclude that a sheath always develops if there arecollisions, and also if Mv2i > kTe at some point. This is, initself, a significant insight. It demonstrates that there are tworegimes namely the collision-dominated, and the collisionless.

The latter is characterized by the Bohm criterion, but theformer is one where the Bohm criterion has no significance.

14. Generalizations of the Bohm criterion

The descriptions given above have all been limited by theirassumption that the ions behave as a simplified fluid,i.e. have a common speed, and the electrons obey aBoltzmann relation. These restrictions have been relaxed byHarrison and Thompson (1959) who showed that for an iondistribution function fi(vi) the corresponding expression was

0 fi(vi)/Mv2i dvi

0 fi(vi)/kTe dvi.

The further generalization to allow the electrons to havea non-Maxwellian distribution was given by Riemann (1995),and can be written as

0 fi(vi)/Mv2i dvi

0 fi(vi) dvi+

(1/mve)fe(ve)/ve dve

fe(ve) dve= 0

These expressions place requirements on the behaviour of bothfi and fe near vi = 0 and ve = 0.

Other generalizations can be found in situations wherethere are more than one species of charged particle. Atlow pressures when one has two negatively charged speciesobeying Boltzmann distributions with different temperaturesTe and Tn, Braithwaite and Allen (1988), considering theplane sheath equations for an electronegative plasma, andalso the probe-sheath equations, showed that the analogousexpression was

Mv2i =kTeTn(ne + nn)

neTn + nnTe(6)

Equally, they showed that for a bi-Maxwellian electrondistribution the expression was (6), with Th replacing Te and Tcreplacing Tn, because if the negative particles obey their ownBoltzmann relation they are independent of mass. Franklinand Snell (1992), considering the active plasma equations,showed that they were singular where (6) is satisfied. Theseexpressions can be rewritten in terms of density ratios andtemperature ratios, and the density ratios in particular, maywell be local values rather than quantities that are constantthroughout the plasma. Further subtleties of such plasmas willbe considered in a later section.

The case of two positive ion species was considered byTokar (1994), but had been examined earlier (Cooke 1981).Designating the species 1 and 2, the sheath equations allowone to obtain (n1 + n2)/kTe = n1/M1v21 + n2/M2v22 . Thesame expression can be shown to correspond to a singularityin the plasma equations (Franklin 2000), but again the densitiesare local quantities and so the position where this compositeBohm criterion is satisfied will not in general coincide with

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either species satisfying its own Bohm speed. That occursfor collisionless active plasmas generated by electron impact(Franklin 2000), but not for collisional (Franklin 2001).Further expressions for multi-species cases have been givenby Valentini (1988) and Benilov (1996).

Other workers have examined the ion distribution nearthe wall at a level of detail where it is possible todistinguish those ions that have made a collision in thesheath region and those that have fallen through the fullplasmawall potential. In this connection we mention theexperimental and theoretical work of Riemann et al (1992),the analytical/computational kinetic treatments of Vasenkovand Shigal (2002a, b) and of Riemann (2003).

We conclude this section by noting that some workers havelinked the Bohm criterion with limits in the propagation of ionwaves. The simplest observation is that the long wavelengthdispersion is /k = vD cs in a uniform plasma and thusthe zero frequency, i.e. dc limit coincides with vD = cs. Moredetailed consideration has been given by Stangeby and Allen(1970) and Allen (1976), where the concepts of a Mach surfaceand sonic flow are introduced and discussed. For rf generatedplasmas, Riemann (1992) and Allen and Skorik (1993) haveconsidered the propagation of ion waves and their limits.

15. Can the Bohm criterion be satisfied at more thanone point in a plasma?

The fact that this is so has been known for some time, andcould have been deduced in connection with electronegativeplasmas by Boyd and Thompson (1959), but it was not madeexplicit until Braithwaite and Allen (1988). They pointedout that for certain values of density ratios and temperatureratios there was a choice to be made because the criterionin terms of local variables was apparently satisfied for threedifferent sets of parameters. But the significance of the ratioshad already been appreciated in Wickens and Allen (1979) fortwo electron temperature plasmas, Teh and Tec where the ratioof Teh/Tec = 5 +

24 10 being a critical value appeared for

the first time.The same ambiguity was noted for an active electronega-

tive plasma in Franklin and Snell (1992). Sato and Miyawaki(1992) treated the situation by introducing a double layerbetween two different plasmas, but their treatment thoughimaginative was not fully self-consistent. And it was thework of Kono (1999) on probes in electronegative plasmasthat really brought to attention the fact that at low pressures,where Boltzmann relations apply for all negative species, therecan exist separate plasma regions each terminating in its ownBohm criterion. The colder (heavier) negative species is themore confined. In the region between the two different cri-teria being satisfied, there exists a space charge region whichhas been called a quasi-plasma. It is essentially a hot neg-ative species-positive ion plasma, but because particles enterit under conditions determined by the inner plasma, it is notin equilibrium, and this gives rise to spatial oscillations in thevariables concerned. The wavelength of the oscillations is ofthe order of the local Debye length, but their magnitude andextent depends on the model employed. For instance, the free-fall (TonksLangmuir) model imposes monotonicity of poten-tial and then they are almost tantamount to a double layer. This,

as noted above, was anticipated in the model described by Satoand Miyawaki (1992). A collisionless fluid model gives themfree rein, while it is known that collisions, diffusion and iontrapping can all contribute to damping the oscillations (Kono(2003) and references therein).

A treatment of this problem giving the insights thatmatched asymptotic approximations can give, albeit for aparticular model of ion motion, was carried out by Benilovand Franklin (1999).

Simulations have revealed the same phenomenon, butto date there have been no experiments under the necessaryconditions of low collisionality, significant temperature ratio,and a significant proportion of the colder (heavier) negativespecies to test theory.

16. Non-local plasmas

One of the most significant developments in the understandingof gas discharges in the late twentieth century has been that oftreating the positive column as a situation where all physicalquantities, not only the charged particle densities, vary withinthe spatial dimension of the plasma.

Much of this development was due to Tsendin (1974)who concentrated on the electron energy distribution and itsvariation with position. The revision of basic ideas in thisconnection has been considerably assisted by Ingold (1997)and his attention to detail.

But since our concern here is with the boundary regionswe concentrate on recent results revealing the nature of theelectron energy distributions near the wall. In particular werefer to the work of Uhrlandt (2002) which shows that, evenwith this refinement, the electron energy distribution near thewall is closely Maxwellian.

Thus, it seems probable that an equivalent Bohm criterioncould if necessary, be called into play. However, methods ofsolution for non-local treatments do not require the criterionto be invoked in order to solve the problem since Poissonsequation is retained throughout.

But they do indicate why in so many real plasmas,experimentalists relying on the visual aspect, may have judgedthe sheath to be more extensive than a simple generationlesscollisionless sheath is.

So one question for the future is When is the Bohmcriterion a useful concept?

17. Rf plasmasheaths

The situation in the positive column plasmasheath with theelectric field that determines the basic plasma parameters beingorthogonal to the radius makes it have a degree of simplicitywhich does not necessarily apply when the plasma is generatedby rf fields. The case where the plasma is generated betweenparallel plate electrodes is usually described as capacitivelycoupled, and that we will treat briefly.

The fact that the driving field is alternating introduces anew element in that there are time variations, which have tobe taken into account. These include the driving frequency, ,the electron plasma frequency, pe, the ion plasma frequency,pi, the ion collision frequency, i, and the ion sheath transittime i. Now i Ds/cs 1/pi, which reduces the number

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of parameters. In the most usual situation pe > > pithough the inequalities may not be strong, but this enables oneto construct a model which separates out the motion of thecharged particles so that the ions are described by their timeaveraged behaviour while the electrons respond to the drivingfield. This has the effect that the sheath expands and contractswith the driving field. Furthermore, the rf fields supply theenergy that causes ionization and thus one has to consider themechanisms by which the electrons gain their energy.

One such mechanism that is operative at higher pressuresis Ohmic heating deriving from the electrons colliding with thebackground gas, but at lower pressures the sheath dynamicsof the electrons is an important heating mechanism, usuallyreferred to as stochastic heating. Early work (Lieberman1988) suggested that this heating might be analogous to onemechanism believed to be of significance in the accelerationof cosmic rays, however the hard wall model (Godyak 1976)on which this was based has recently undergone significantrevision (Gozadinos et al 2001a, b), although the need hadbeen recognized for some time (Skorik et al 1992).

The problem of describing how the electric field penetratesthe plasma has not yet been solved in a totally self-consistentmanner. Early work was done by Riemann (1992) andAllen and Skorik (1993) and progress is currently being made(Kaganovich 2002b, Slemrod 2003).

It has been established for some time that the Bohmcriterion is still relevant at the plasma edge, even though itis time-varying in position (Riemann 1995).

A further factor which arises is that the sheath acts asa rectifier and thus there can be at one or both electrodesdepending on the driving field arrangements, a significant dcbias voltage and the sheath is many Debye lengths long, thoughthe electrons sweep in and out of this sheath, so that it is not asimple replication of the dc sheath.

This, in turn, leads to ions striking the electrodes withenergies sufficient to give rise to secondary emission ofelectrons, indeed one mode of such discharges known as thegamma mode (Raizer 1997) relies on secondary emission togenerate the plasma.

The principal parameter describing the oscillating sheath,in what has come to be called the moving step model becausethe electron density is taken to go to zero on a scale shorterthan others under consideration, has been given a differentsymbol by different workers, but once one recognizes thatthey are related, then it is possible to understand and comparethe underlying physics. The quantity H used by Lieberman(1988), used by Godyak and Sternberg (1990a, b) and usedby Benilov (2003) are simply related by2H 2 = = 4 and = (J0/Ji)(pi/) that is the ratio of the oscillating rf currentto the dc ion current through the sheath times the ratio of theion plasma frequency to the driving frequency, and so it has arelatively simple physical interpretation. This quantity, whichis proportional to the ratio of the sheath oscillation amplitude tothe Debye length, is usually greater than one, and the treatmentof Benilov is notable because it solves the problem to higherorder using the method of matched approximations and thus isable to bridge 1. A different approach is that of Skorikreported in Allen (1995).

Slemrod (2003) takes the problem one stage furtherand seeks to give a description of the whole plasmasheath.

He recovers results that link directly to the fluid model for thedc case as given in Franklin and Ockendon (1970), but it has tobe recognized that the generation rate is treated as a separatelydetermined constant and independent of position.

Kaganovich (2002b) has sought to give a description ofthe penetration of the rf field into the plasma, but introducesan arbitrary parameter being the ratio of the ion density in thesheath to that in the bulk plasma.

Thus, the treatments available to date are not fully self-consistent and it is for this reason that much use has beenmade of simulations in this area, however we will not attemptto review them here, basically because of the present heavycomputational cost in carrying out simulations over a sufficientrange of parameters to give sufficient physical insight to linkthem back into more simple treatments.

18. Experimental work

It is intrinsically difficult to make accurate measurements of therelevant quantities in the plasmasheath region. Introducingany material physical probe causes a disturbance on the scaleof interest and thus it is necessary to try to devise means thatare not so intrusive.

The first technique that we are aware of was the use ofelectron beams to measure the electric field, and this was usedin relation to the hot cathodeplasma sheath by Crawford andCannara (1965), and Goldan (1970) in a more conventionalplasmasheath situation. Goldan obtained good agreementwith the full TonksLangmuir (1929) description coveringthe region from plasma to within 2.2 mm = 4.4D of thewall. In the hot cathode case, theory followed experiment,but good agreement was found by Prewett and Allen (1976),who developed the relevant theory.

The technique of laser-induced fluorescence (LIF) allowsthe possibility of measuring ion velocity distributions in theplasmasheath boundary region and from the ion mean speedto deduce the ion density. This has been applied using argonions by Goeckner et al (1992) and Severn et al (2003). Theselatter have made comparison with the theoretical descriptionof Riemann (1997) for the case of a single ion species. Andfor two positive ion species, comparison can be made with itsextension (Franklin 2003).

It should be noted that good agreement with Riemann(1997), which described a quiescent plasmasheath fora single ion species, had been reported earlier using acombination of more conventional probe techniques by Oksuzand Hershkowitz (2002).

A recent diagnostic of the plasmasheath region hasarisen in connection with dusty plasmas, or more correctly, insituations where there is dust in plasmas, when the motion ofdust particles can be used to effectively diagnose the potentialdistribution, we mention in this connection the work reported inTomme et al (2000). This brings into perspective the relativelysmall differences that exist so far as the potential variation withdistance is concerned, and it is approximately parabolic in themany treatments we have described.

The LIF technique has been used with effect in rfdischarges to give both space and time resolution by Czarnetzkiet al (1999) and a more general discussion of its use in plasmadiagnostics is given in Freegarde and Hancock (1997).

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19. Conclusions

The structure of the boundary region between plasma andsheath is more complicated than Langmuir envisaged whenhe introduced the two terms. Interestingly plasma becamea universal term, whereas sheath got translated into otherlanguages.

The structure varies with pressure in a significant way,particularly according as the Debye length associated with theedge of the plasma Ds is greater than or less than the ion meanfree path i. There are also other subtleties associated with thenature of the plasma itself.

The Bohm criterion has proved useful in understandingstructure, and provides a check for more complicatedtreatments than the monoenergetic ion model that he used,as it often applies in some limit. Though, in spatial terms,precisely where it is met, when D and i are finite, is noteasy to locate. However, the fact that most practical plasmasare such that D/L is a small number, and that in the limit ofD/L 0 it survives as a singularity in the plasma equationfor so many different models of plasmas, has given it anenduring significance, particularly once its generalized formsare appreciated.

Practising plasma physicists need to ask themselves thequestion How much of the detail of the structure is needed inmy particular application?.

This paper is intended to assist in answering that question,and to indicate where to look for help.

Acknowledgments

I would like to thank those with whom I have interacted over theyears in developing the understanding of the plasma boundaryregions. Some of them are contributors to this volume, somehave predeceased me, and some as research students weredemanding at the time.

I have tried to make the description as comprehensive andup-to-date as possible, and am grateful for early sight of someof the work referred to.

I acknowledge several fruitful discussions with ProfessorJ E Allen.

Note added in proof A recent paper by Sternberg andGodyak (2003) recovers all of the results given for planegeometry and the fluid model for a collisionless plasma byFranklin and Ockendon (1970) and therefore represents aconsiderable retraction on their part.

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