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5.2 Illustration of thedefinition of cross sectioncollision with a heavy atom, the electron can lose twice its initial momentum, sinceits velocity reverses sign after the collision. The probability of momentum loss canbe expressed in terms of the equivalent cross section σ that the atoms would have ifthey were perfect absorbers of momentum.In Fig.
5.2, electrons are incident upon a slab of area A and thickness dxcontaining nn neutral atoms per m3. The atoms are imagined to be opaque spheresof cross-sectional area σ; that is, every time an electroncomes within the area blocked by the atom, the electron loses all of its momentum. The number of atoms in the slab isnn A dxThe fraction of the slab blocked by atoms isnn Aσ dx=A ¼ nn σ dx5.1 Diffusion and Mobility in Weakly Ionized Gases147If a flux Γ of electrons is incident on the slab, the flux emerging on the other side is0Γ ¼ Γð1 nn σ dxÞThus the change of Γ with distance isdΓ=dx ¼ nn σΓorΓ ¼ Γ0 enn σx Γ0 ex=λmð5:1ÞIn a distance λm, the flux would be decreased to 1/e of its initial value. The quantityλm is the mean free path for collisions:λm ¼ 1=nn σð5:2ÞAfter traveling a distance λm, a particle will have had a good probability of making acollision.
The mean time between collisions, for particles of velocity v, is given byτ ¼ λm =vand the mean frequency of collisions isτ1 ¼ v=λm ¼ nn σvð5:3ÞIf we now average over particles of all velocities v in a Maxwellian distribution, wehave what is generally called the collision frequency ν:ν ¼ nn σv5.1.2ð5:4ÞDiffusion ParametersThe fluid equation of motion including collisions is, for any species,dv∂vþ ðv ∇Þv ¼ en E ∇ p mnνvm n ¼ mndt∂tð5:5Þwhere again the indicates the sign of the charge. The averaging process used tocompute ν is such as to make Eq.
(5.5) correct; we need not be concerned with thedetails of this computation. The quantity ν must, however, be assumed to be aconstant in order for Eq. (5.5) to be useful. We shall consider a steady state in which∂v/∂t ¼ 0. If v is sufficiently small (or ν sufficiently large), a fluid element will notmove into regions of different E and ∇p in a collision time, and the convective1485 Diffusion and Resistivityderivative dv/dt will also vanish. Setting the left-hand side of Eq. (5.5) to zero, wehave, for an isothermal plasma,1ðen E KT∇nÞmnνeKT ∇n¼Emνmv nv¼ð5:6ÞThe coefficients above are called the mobility and the diffusion coefficient:μ jqj=mνD KT=mνMobilityð5:7ÞDiffusion coefficientð5:8ÞThese will be different for each species. Note that D is measured in m2/s.
Thetransport coefficients μ and D are connected by the Einstein relation:μ ¼ jqjD=KTð5:9ÞWith the help of these definitions, the flux Γj of the jth species can be writtenΓ j ¼ nv j ¼ μ j n E D j ∇nð5:10ÞFick’s law of diffusion is a special case of this, occurring when either E ¼ 0 or theparticles are uncharged, so that μ ¼ 0:Γ ¼ D∇nFick, s lawð5:11ÞThis equation merely expresses the fact that diffusion is a random-walk process, inwhich a net flux from dense regions to less dense regions occurs simply becausemore particles start in the dense region.
This flux is obviously proportional to thegradient of the density. In plasmas, Fick’s law is not necessarily obeyed. Because ofthe possibility of organized motions (plasma waves), a plasma may spread out in amanner which is not truly random.5.25.2.1Decay of a Plasma by DiffusionAmbipolar DiffusionWe now consider how a plasma created in a container decays by diffusion to thewalls. Once ions and electrons reach the wall, they recombine there. The densitynear the wall, therefore, is essentially zero. The fluid equations of motion and5.2 Decay of a Plasma by Diffusion149continuity govern the plasma behavior; but if the decay is slow, we need onlykeep the time derivative in the continuity equation.
The time derivative in theequation of motion, Eq. (5.5), will be negligible if the collision frequency ν is large.We thus have∂nþ ∇ Γj ¼ 0∂tð5:12Þwith Γj given by Eq. (5.10). It is clear that if Γi and Γe were not equal, a seriouscharge imbalance would soon arise. If the plasma is much larger than a Debyelength, it must be quasineutral; and one would expect that the rates of diffusion ofions and electrons would somehow adjust themselves so that the two species leaveat the same rate.
How this happens is easy to see. The electrons, being lighter, havehigher thermal velocities and tend to leave the plasma first. A positive charge is leftbehind, and an electric field is set up of such a polarity as to retard the loss ofelectrons and accelerate the loss of ions. The required E field is found by settingΓi ¼ Γe ¼ Γ. From Eq. (5.10), we can writeΓ ¼ μi n E Di ∇n ¼ μe n E De ∇nE¼Di De ∇nμi þ μe nð5:13Þð5:14ÞThe common flux Γ is then given byΓ ¼ μi¼Di De∇n Di ∇nμi þ μeμi Di μi De μi Di μe Di∇nμi þ μe¼ð5:15Þμi De þ μe Di∇nμi þ μeThis is Fick’s law with a new diffusion coefficientDa μi De þ μe Diμi þ μeð5:16Þcalled the ambipolar diffusion coefficient. If this is constant, Eq.
(5.12) becomessimply∂n=∂t ¼ Da ∇ 2 nð5:17ÞThe magnitude of Da can be estimated if we take μe μi. That this is true can beseen from Eq. (5.7). Since ν is proportional to the thermal velocity, which is1505 Diffusion and Resistivityproportional to m1/2, μ is proportional to m1/2. Equations (5.16) and (5.9)then giveDa Di þμiTeDe ¼ Di þ DiμeTið5:18ÞFor Te ¼ Ti, we haveDa 2Dið5:19ÞThe effect of the ambipolar electric field is to enhance the diffusion of ions by afactor of two, and the diffusion rate of the two species together is primarilycontrolled by the slower species.5.2.2Diffusion in a SlabThe diffusion equation (5.17) can easily be solved by the method of separation ofvariables.
We letnðr; tÞ ¼ T ðtÞSðrÞð5:20Þwhereupon Eq. (5.17), with the subscript on Da understood, becomesSdT¼ DT ∇2 Sdt1 dT D 2¼ ∇ ST dtSð5:21Þð5:22ÞSince the left side is a function of time alone and the right side a function of spacealone, they must both be equal to the same constant, which we shall call 1/τ. Thefunction T then obeys the equationdTT¼dtτð5:23ÞT ¼ T 0 et=τð5:24Þwith the solutionThe spatial part S obeys the equation∇2 S ¼ 1SDτð5:25Þ5.2 Decay of a Plasma by Diffusion151Fig. 5.3 Density of aplasma at various timesas it decays by diffusionto the wallsIn slab geometry, this becomesd2 S1¼ S2dxDτð5:26Þwith the solutionS ¼ A cosxðDτÞ1=2þ B sinxðDτÞ1=2ð5:27ÞWe would expect the density to be nearly zero at the walls (Fig. 5.3) and to have oneor more peaks in between. The simplest solution is that with a single maximum.
Bysymmetry, we can reject the odd (sine) term in Eq. (5.27). The boundary conditionsS ¼ 0 at x ¼ L then requiresLðDτÞ1=2¼π2or 22L 1τ¼π Dð5:28ÞCombining Eqs. (5.20), (5.24), (5.27), and (5.28), we haven ¼ n0 et=τ cosπx2Lð5:29Þ1525 Diffusion and ResistivityFig. 5.4 Decay of aninitially nonuniformplasma, showing the rapiddisappearance of the higherorder diffusion modesThis is called the lowest diffusion mode. The density distribution is a cosine, and thepeak density decays exponentially with time. The time constant τ increases withL and varies inversely with D, as one would expect.There are, of course, higher diffusion modes with more than one peak. Supposethe initial density distribution is as shown by the top curve in Fig.
5.4. Such anarbitrary distribution can be expanded in a Fourier series:n ¼ n0Xl!l þ 12 πx Xmπxþbm sinal cosLLmð5:30ÞWe have chosen the indices so that the boundary condition at x ¼ L is automatically satisfied. To treat the time dependence, we can try a solution of the formn ¼ n0Xlal et=τl!l þ 12 πx Xmπxt=τmþcosbm esinLLmð5:31ÞSubstituting this into the diffusion equation (5.17), we see that each cosine termyields a relation of the form 11 π 2 ¼ D l þτl2 Lð5:32Þand similarly for the sine terms. Thus the decay time constant for the lth mode isgiven byτl ¼"Ll þ 12 π#21Dð5:33ÞThe fine-grained structure of the density distribution, corresponding to largel numbers, decays faster, with a smaller time constant τl.
The plasma decay will5.2 Decay of a Plasma by Diffusion153proceed as indicated in Fig. 5.4. First, the fine structure will be washed out bydiffusion. Then the lowest diffusion mode, the simple cosine distribution ofFig. 5.3, will be reached. Finally, the peak density continues to decay while theplasma density profile retains the same shape.5.2.3Diffusion in a CylinderThe spatial part of the diffusion equation, Eq. (5.25), reads, in cylindrical geometry,d2 S 1 dS1þS¼0þdr 2 r dr Dτð5:34ÞThis differs from Eq.
(5.26) by the addition of the middle term, which merelyaccounts for the change in coordinates. The need for the extra term is illustratedsimply in Fig. 5.5. If a slice of plasma in (a) is moved toward larger x without beingallowed to expand, the density would remain constant. On the other hand, if a shellof plasma in (b) is moved toward larger r with the shell thickness kept constant, thedensity would necessarily decrease as 1/r. Consequently, one would expect thesolution to Eq. (5.34) to be like a damped cosine (Fig. 5.6). This function is called aBessel function of order zero, and Eq. (5.34) is called Bessel’s equation (of orderzero). Instead of the symbol cos, it is given the symbol J0. The function J0(r/[Dτ]1/2)is a solution to Eq.
(5.34), just as cos [x/(Dτ)1/2] is a solution to Eq. (5.26). Both coskx and J0(kr) are expressible in terms of infinite series and may be found inmathematical tables. Unfortunately, Bessel functions are not yet found in mosthand calculators.To satisfy the boundary condition n ¼ 0 at r ¼ a, we must set a/(Dτ)l/2 equal tothe first zero of J0; namely, 2.4. This yields the decay time constant τ. The plasmaagain decays exponentially, since the temporal part of the diffusion equation,Eq. (5.23), is unchanged.
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