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We have described the lowest diffusion mode in aFig. 5.5 Motion of a plasma slab in rectilinear and cylindrical geometry, illustrating the difference between a cosine and a Bessel function1545 Diffusion and ResistivityFig. 5.6 The Besselfunction of order zerocylinder. Higher diffusion modes, with more than one maximum in the cylinder,will be given in terms of Bessel functions of higher order, in direct analogy to thecase of slab geometry.5.3Steady State SolutionsIn many experiments, a plasma is maintained in a steady state by continuousionization or injection of plasma to offset the losses. To calculate the density profilein this case, we must add a source term to the equation of continuity:∂n D ∇2 n ¼ QðrÞ∂tð5:35ÞThe sign is chosen so that when Q is positive, it represents a source and contributesto positive ∂n/∂t. In steady state, we set ∂n/∂t ¼ 0 and are left with a Poisson-typeequation for n(r).5.3.1Constant Ionization FunctionIn many weakly ionized gases, ionization is produced by energetic electrons in thetail of the Maxwellian distribution.
In this case, the source term Q is proportional tothe electron density n. Setting Q ¼ Zn, where Z is the “ionization function,” we have∇2 n ¼ ðZ=DÞnð5:36ÞThis is the same equation as that for S, Eq. (5.25). Consequently, the density profileis a cosine or Bessel function, as in the case of a decaying plasma, only in this casethe density remains constant.
The plasma is maintained against diffusion losses bywhatever heat source keeps the electron temperature at its constant value and by asmall influx of neutral atoms to replenish those that are ionized.5.3 Steady State Solutions5.3.2155Plane SourceWe next consider what profile would be obtained in slab geometry if there is alocalized source on the plane x ¼ 0.
Such a source might be, for instance, a slitcollimated beam of ultraviolet light strong enough to ionize the neutral gas. Thesteady state diffusion equation is thend2 nQ¼ δ ð 0Þdx2Dð5:37ÞExcept at x ¼ 0, the density must satisfy ∂2n/∂x2 ¼ 0. This obviously has thesolution (Fig. 5.7)n ¼ n0jxj1Lð5:38ÞThe plasma has a linear profile. The discontinuity in slope at the source is characteristic of δ-function sources.5.3.3Line SourceFinally, we consider a cylindrical plasma with a source located on the axis.
Such asource might, for instance, be a beam of energetic electrons producing ionizationalong the axis. Except at r ¼ 0, the density must satisfy1∂∂nr¼0r ∂r∂rFig. 5.7 The triangulardensity profile resultingfrom a plane source underdiffusionð5:39Þ1565 Diffusion and ResistivityFig. 5.8 The logarithmicdensity profile resultingfrom a line source underdiffusionThe solution that vanishes at r ¼ a isn ¼ n0 lnða=r Þð5:40ÞThe density becomes infinite at r ¼ 0 (Fig. 5.8); it is not possible to determine thedensity near the axis accurately without considering the finite width of the source.5.4RecombinationWhen an ion and an electron collide, particularly at low relative velocity, they havea finite probability of recombining into a neutral atom.
To conserve momentum, athird body must be present. If this third body is an emitted photon, the process iscalled radiative recombination. If it is a particle, the process is called three-bodyrecombination. The loss of plasma by recombination can be represented by anegative source term in the equation of continuity.
It is clear that this term willbe proportional to neni ¼ n2. In the absence of the diffusion terms, the equation ofcontinuity then becomes∂n=∂t ¼ αn2ð5:41ÞThe constant of proportionality α is called the recombination coefficient and hasunits of m3/s. Equation (5.41) is a nonlinear equation for n. This means that thestraightforward method for satisfying initial and boundary conditions by linearsuperposition of solutions is not available. Fortunately, Eq. (5.41) is such a simplenonlinear equation that the solution can be found by inspection. It is11¼þ αtnðr; tÞ n0 ðrÞð5:42Þ5.5 Diffusion Across a Magnetic Field157Fig.
5.9 Density decay curves of a weakly ionized plasma under recombination and diffusion(From S. C. Brown, Basic Data of Plasma Physics, John Wiley and Sons, New York, 1959)where n0(r) is the initial density distribution. It is easily verified that this satisfiesEq.
(5.41). After the density has fallen far below its initial value, it decaysreciprocally with time:n / 1=αtð5:43ÞThis is a fundamentally different behavior from the case of diffusion, in which thetime variation is exponential.Figure 5.9 shows the results of measurements of the density decay in theafterglow of a weakly ionized H plasma. When the density is high, recombination,which is proportional to n2, is dominant, and the density decays reciprocally. Afterthe density has reached a low value, diffusion becomes dominant, and the decay isthenceforth exponential.5.5Diffusion Across a Magnetic FieldThe rate of plasma loss by diffusion can be decreased by a magnetic field; this is theproblem of confinement in controlled fusion research.
Consider a weakly ionizedplasma in a magnetic field (Fig. 5.10). Charged particles will move along B bydiffusion and mobility according to Eq. (5.10), since B does not affect motion in theparallel direction. Thus we have, for each species,1585 Diffusion and ResistivityFig.
5.10 A chargedparticle in a magnetic fieldwill gyrate about the sameline of force until it makes acollisionFig. 5.11 Particle drifts ina cylindrically symmetricplasma column do not leadto lossesΓ z ¼ μnEz D∂n∂zð5:44ÞIf there were no collisions, particles would not diffuse at all in the perpendiculardirection—they would continue to gyrate about the same-line of force. There are, ofcourse, particle drifts across B because of electric fields or gradients in B, but thesecan be arranged to be parallel to the walls. For instance, in a perfectly symmetriecylinder (Fig.
5.11), the gradients are all in the radial direction, so that the guidingcenter drifts are in the azimuthal direction. The drifts would then be harmless.When there are collisions, particles migrate across B to the walls along thegradients. They do this by a random-walk process (Fig. 5.12).
When an ion, say,collides with a neutral atom, the ion leaves the collision traveling in a differentdirection. It continues to gyrate about the magnetic field in the same direction, butits phase of gyration is changed discontinuously. (The Larmor radius may alsochange, but let us suppose that the ion does not gain or lose energy on the average.)The guiding center, therefore, shifts position in a collision and undergoes arandom walk. The particles will diffuse in the direction opposite ∇n.
The steplength in the random walk is no longer λm, as in magnetic-field-free diffusion, buthas instead the magnitude of the Larmor radius rL. Diffusion across B can thereforebe slowed down by decreasing rL; that is, by increasing B.5.5 Diffusion Across a Magnetic Field159Fig. 5.12 Diffusion ofgyrating particles bycollisions with neutralatomsTo see how this comes about, we write the perpendicular component of the fluidequation of motion for either species as follows:mndv⊥¼ enðE þ v⊥ BÞ KT∇n mnνv ¼ 0dtð5:45ÞWe have again assumed that the plasma is isothermal and that ν is large enough forthe dv⊥/dt term to be negligible. The x and y components are∂n env y B∂x∂n envx Bmnνv y ¼ enE y KT∂ymnνvx ¼ enEx KTð5:46ÞUsing the definitions of μ and D, we haveD ∂n ωc vyn ∂x νD ∂n ωc vxv y ¼ μE y n ∂y νvx ¼ μEx ð5:47ÞSubstituting for vx, we may solve for vy:D ∂nExKT 1 ∂n ω2c τ2 ω2c τ2v y 1 þ ω2c τ2 ¼ μE y n ∂yeB n ∂xBð5:48Þwhere τ ¼ ν1.
Similarly, vx is given byEyD ∂nKT 1 ∂nvx 1 þ ω2c τ2 ¼ μEx þ ω2c τ2 ω2c τ2n ∂xeB n ∂yBð5:49Þ1605 Diffusion and ResistivityThe last two terms of these equations contain the E B and diamagnetic drifts:EyBKT 1 ∂n¼eB n ∂yvEx ¼vDxExBKT 1 ∂n¼eB n ∂xvEy ¼ vDyð5:50ÞThe first two terms can be simplified by defining the perpendicular mobility anddiffusion coefficients:μ⊥ ¼μ1 þ ω2c τ2D⊥ ¼D1 þ ω2c τ2ð5:51ÞWith the help of Eqs. (5.50) and (5.51), we can write Eqs. (5.48) and (5.49) asv⊥ ¼ μ⊥ E D⊥∇nvE þ vDþn1 þ ν2 =ω2cð5:52ÞFrom this, it is evident that the perpendicular velocity of either species iscomposed of two parts.
First, there are usual vE and vD drifts perpendicular to thegradients in potential and density. Thesedrifts are slowed down by collisions withneutrals; the drag factor 1 þ ν2 =ω2c becomes unity when ν ! 0. Second, there arethe mobility and diffusion drifts parallel to the gradients in potential and density.These drifts have the same form as in the B ¼ 0 case, but the coefficients μ and D arereduced by the factor 1 þ ω2c τ2 :The product ωcτ is an important quantity in magnetic confinement.
When ω2c τ2 1; the magnetic field has little effect on diffusion. When ω2c τ2 1; the magneticfield significantly retards the rate of diffusion across B. The following alternativeforms for ωcτ can easily be verified:ωc τ ¼ ωc =ν ¼ μB ffi λm =r Lð5:53ÞIn the limit ω2c τ2 1; we haveD⊥ ¼KT 1KTν¼22mν ωc τmω2cð5:54ÞComparing with Eq. (5.8), we see that the role of the collision frequency ν has beenreversed. In diffusion parallel to B, D is proportional to ν1, since collisions retardthe motion. In diffusion perpendicular to B, D⊥ is proportional to ν, since collisionsare needed for cross-field migration.
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