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8.3 The potential ϕ ina planar sheath. Cold ionsare assumed to enter thesheath with a uniformvelocity u08.2 Sheaths271If u(x) is the ion velocity in the sheath, conservation of energy requires1212Mu2 ¼ Mu20 eϕðxÞð8:1Þ2eϕ 1=22u ¼ u0 Mð8:2ÞThe ion equation of continuity then gives the ion density ni in terms of the density n0in the main plasma:n0 u0 ¼ ni ð x Þ uð x Þð8:3Þ2eϕ 1=2ni ðxÞ ¼ n0 1 Mu20ð8:4ÞIn steady state, the electrons will follow the Boltzmann relation closely:ne ðxÞ ¼ n0 expðeϕ=KT e Þð8:5Þ" # d2 ϕeϕ2eϕ 1=2 1ε0 2 ¼ eðne ni Þ ¼ en0 expdxKT eMu20ð8:6ÞPoisson’s equation is thenThe structure of this equation can be seen more clearly if we simplify it with thefollowing changes in notation:eϕ,χ KT e1=2xn0 e 2ξ¼xλDε0 KT eMu0ðKT e =MÞ1=2ð8:7ÞThen Eq.
(8.6) becomes2χ 1=2χ ¼ 1þ 2 eχM00ð8:8Þwhere the prime denotes d/dξ. This is the nonlinear equation of a plane sheath, andit has an acceptable solution only if M is large enough. The reason for the symbol Mwill become apparent in the following section on shock waves.2728.2.38 Nonlinear EffectsThe Bohm Sheath CriterionEquation (8.8) can be integrated once by multiplying both sides by χ 0 :ðξ0ðξ ðξ2χ 1=2 001þ 2χ χ dξ1 ¼χ dξ1 eχ χ dξ1M00000ð8:9Þwhere ξ1 is a dummy variable. Since χ ¼ 0 at ξ ¼ 0, the integrations easily yield120202χ χ 0 ¼ M2"2χ1þ 2M1=2# 1 þ eχ 1:ð8:10Þ0If E ¼ 0 in the plasma, we must set χ 0 ¼ 0 at ξ ¼ 0.
A second integration to find χwould have to be done numerically; but whatever the answer is, the right-hand sideof Eq. (8.10) must be positive for all χ. In particular, for χ 1, we can expand theright-hand terms in Taylor series:2χ1 χ1M 1þ 2þ 1 þ 1 χ þ χ2 þ 1 > 02 M42M11 2χ 2þ1 >02M2M2 > 1orð8:11Þu0 > ðKT e =MÞ1=2This inequality is known as the Bohm sheath criterion. It says that ions must enterthe sheath region with a velocity greater than the acoustic velocity vs.
To give theions this directed velocity u0, there must be a finite electric field in the plasma. Ourassumption that χ 0 ¼ 0 at ξ ¼ 0 is therefore only an approximate one, made possibleby the fact that the scale of the sheath region is usually much smaller than the scaleof the main plasma region in which the ions are accelerated. The value of u0 issomewhat arbitrary, depending on where we choose to put the boundary x ¼ 0between the plasma and the sheath. Of course, the ion flux n0u0 is fixed by theion production rate, so if u0 is varied, the value of n0 at x ¼ 0 will vary inverselywith u0. If the ions have finite temperature, the critical drift velocity u0 will besomewhat lower.The physical reason for the Bohm criterion is easily seen from a plot of the ionand electron densities vs.
χ (Fig. 8.4). The electron density ne falls exponentiallywith χ according to the Boltzmann relation. The ion density also falls, since the ionsare accelerated by the sheath potential. If the ions start with a large energy, ni(χ)falls slowly, since the sheath field causes a relatively minor change in the ions’velocity. If the ions start with a small energy, ni(χ) falls fast, and can go below the necurve. In that case, ne ni is positive near χ ¼ 0; and Eq. (8.6) tells us that ϕ(x) mustcurve upward, in contradiction to the requirement that the sheath must repel8.2 Sheaths273Fig.
8.4 Variation of ionand electron density(logarithmic scale) withnormalized potential χ in asheath. The ion density isdrawn for two cases: u0greater than and u0 less thanthe critical velocityelectrons. In order for this not to happen, the slope of ni( χ) at χ ¼ 0 must besmaller (in absolute value) than that of ne( χ); this condition is identical with thecondition M 2 > 1.8.2.4The Child–Langmuir LawSince ne( χ) falls exponentially with χ, the electron density can be neglected in theregion of large χ next to the wall (or any negative electrode). Poisson’s equation isthen approximately00χ 2χ1þ 2M1=2Mð2χ Þ1=2ð8:12ÞMultiplying by χ 0 and integrating from ξ1 ¼ ξs to ξ1 ¼ ξ, we have12 pffiffiffi 0202χ χ s ¼ 2M χ 1=2 χ s1=2ð8:13Þwhere ξs is the place where we started neglecting ne.
We can redefine the zero of χ0so that χ s ¼ 0 at ξ ¼ ξs. We shall also neglect χ s , since the slope of the potentialcurve can be expected to be much steeper in the ne ¼ 0 region than in the finite-neregion. Then Eq. (8.13) becomes02χ ¼ 23=2 Mχ 1=20χ ¼ 23=4 M 1=2 χ 1=4ð8:14Þordχ=χ 1=4 ¼ 23=4 M 1=2 dξ:ð8:15Þ2748 Nonlinear EffectsIntegrating from ξ ¼ ξs to ξ ¼ ξs + d/λD ¼ ξwall, we have4 3=4χ ¼ 23=4 M 1=2 d=λD3 wð8:16Þpffiffiffi 3=24 2 χw 2λM¼9 d2 Dð8:17ÞorChanging back to the variables u0 and ϕ, and noting that the ion current into the wallis J ¼ en0u0, we then find 4 2e 1=2 ε0 jϕw j3=2J¼9 Md2ð8:18ÞThis is just the well-known Child–Langmuir law of space-charge-limited current ina plane diode.The potential variation in a plasma–wall system can be divided into three parts.Nearest the wall is an electron-free region whose thickness d is given by Eq.
(8.18).Here J is determined by the ion production rate, and ϕw is determined by the equalityof electron and ion fluxes. Next comes a region in which ne is appreciable; as shownin Sect. 1.4, this region has the scale of the Debye length. Finally, there is a regionwith much larger scale length, the “presheath,” in which the ions are accelerated tothe required velocity u0 by a potential drop |ϕ| ½KTe/e. Depending on the experiment, the scale of the presheath may be set by the plasma radius, the collision meanfree path, or the ionization mechanism.
The potential distribution, of course, variessmoothly; the division into three regions is made only for convenience and is madepossible by the disparity in scale lengths. In the early days of gas discharges, sheathscould be observed as dark layers where no electrons were present to excite atoms toemission. Subsequently, the potential variation has been measured by the electrostatic deflection of a thin electron beam shot parallel to a wall.Recent theories of discharges in finite cylinders show that the sheaths on theendplates play an essential rôle in moving plasma created at the radial edge towardsthe axis, so that the final density profile is peaked at the center. It has also beenshown that the density profile in such a plasma, including the presheath, tends to fallinto a universal shape independent of pressure and discharge diameter.8.2.5Electrostatic ProbesThe sheath criterion, Eq.
(8.11), can be used to estimate the flux of ions to anegatively biased probe in a plasma. If the probe has a surface area A, and if theions entering the sheath have a drift velocity u0 (KTe/M )1/2, then the ion currentcollected is8.2 Sheaths275I ¼ ns eAðKT e =MÞ1=2ð8:19ÞThe electron current can be neglected if the probe is sufficiently negative (severaltimes KTe) relative to the plasma to repel all but the tail of the Maxwellian electrondistribution. The density ns is the plasma density at the edge of the sheath.
Let usdefine the sheath edge to be the place where u0 is exactly (KTe/M )1/2. To accelerateions to this velocity requires a presheath potential |ϕ| ½KTe/e, so that the sheathedge has a potential12ϕs ’ KT e =eð8:20Þrelative to the body of the plasma. If the electrons are Maxwellian, thisdetermines ns:ns ¼ n0 eeϕs =kT e ¼ n0 e1=2 ¼ 0:61n0ð8:21ÞFor our purposes it is accurate enough to replace 0.61 with a round number like 1/2;thus, the “saturation ion current” to a negative probe is approximately12I B ¼ n0 eAðKT e =MÞ1=2ð8:22ÞIB, sometimes called the “Bohm current,” gives the plasma density easily, once thetemperature is known.If the Debye length λD, and hence the sheath thickness, is very small compared tothe probe dimensions, the area of the sheath edge is effectively the same as the areaA of the probe surface, regardless of its shape.
At low densities, however, λD canbecome large, so that some ions entering the sheath can orbit the probe and missit. Calculations of orbits for various probe shapes were first made by I. Langmuirand L. Tonks—hence the name “Langmuir probe” ascribed to this method ofmeasurement. Though tedious, these calculations can give accurate determinationsof plasma density because an arbitrary definition of sheath edge does not have to bemade. By varying the probe voltage, the electron distribution is sampled, and thecurrent–voltage curve of a Langmuir probe can also yield the electron temperature,if the electrons are Maxwellian, or their velocity distribution if they are not. TheLangmuir probe is the first plasma diagnostic and is still the simplest and the mostlocalized measurement device.
Material electrodes can be inserted only inlow-density, cool plasmas. Nonetheless, these include most non-fusion laboratoryplasmas, and an extensive literature exists on probe theory. The problem with usinga large probe, to which Eq. (8.22) applies, is that it collects from an ill-definedsheath edge that surrounds it, and it may also disturb the plasma. Thin cylindricalprobes with radii λD are more commonly used.2768 Nonlinear EffectsProblems8.1 A probe whose collecting surface is a square tantalum foil 2 2 mm in area isfound to give a saturation ion current of 100 μA in a singly ionized argonplasma (atomic weight ¼ 40). If KTe ¼ 2 eV, what is the approximate plasmadensity? (Hint: Both sides of the probe collect ions!)8.2 A solar satellite consisting of 10 km2 of photovoltaic panels is placed insynchronous orbit around the earth.
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