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The dependence on m has also been reversed.Keeping in mind that ν is proportional to m1/2, we see that D / m1/2, whileD⊥ / ml/2. In parallel diffusion, electrons move faster than ions because of their5.5 Diffusion Across a Magnetic Field161higher thermal velocity; in perpendicular diffusion, electrons escape more slowlybecause of their smaller Larmor radius.Disregarding numerical factors of order unity, we may write Eq. (5.8) asD ¼ KT=mν v2th τ λ2m =τð5:55ÞThis form, the square of a length over a time, shows that diffusion is a random-walkprocess with a step length λm. Equation (5.54) can be writtenD⊥ ¼2KTνr 2L2 rLνvth 2mω2cτvthð5:56ÞThis shows that perpendicular diffusion is a random-walk process with a step lengthrL, rather than λm.5.5.1Ambipolar Diffusion Across BBecause the diffusion and mobility coefficients are anisotropic in the presence of amagnetic field, the problem of ambipolar diffusion is not as straightforward as in theB ¼ 0 case.

Consider the particle fluxes perpendicular to B (Fig. 5.13). Ordinarily,since Γe⊥ is smaller than Γi⊥, a transverse electric field would be set up so as to aidelectron diffusion and retard ion diffusion. However, this electric field can be shortcircuited by an imbalance of the fluxes along B. That is, the negative chargeresulting from Γe⊥ < Γi⊥ can be dissipated by electrons escaping along the fieldlines. Although the total diffusion must be ambipolar, the perpendicular part of thelosses need not be ambipolar.

The ions can diffuse out primarily radially, while theelectrons diffuse out primarily along B. Whether or not this in fact happens dependson the particular experiment. In short plasma columns with the field lines terminating on conducting plates, one would expect the ambipolar electric field to beshort-circuited out. Each species then diffuses radially at a different rate. In long,thin plasma columns terminated by insulating plates, one would expect the radialdiffusion to be ambipolar because escape along B is arduous.Fig. 5.13 Parallel and perpendicular particle fluxes in a magnetic field1625 Diffusion and ResistivityMathematically, the problem is to solve simultaneously the equations of continuity (5.12) for ions and electrons.

It is not the fluxes Γj, but the divergences ∇ · Γjwhich must be set equal to each other. Separating ∇ · Γj into perpendicular andparallel components, we have∂∂nμi nEz Di∂z∂z ∂∂nμe nEz De∇ Γe ¼ ∇ ⊥ μe⊥ nE⊥ De ⊥ ∇n þ∂z∂z∇ Γi ¼ ∇ ⊥ ðμi⊥ nE⊥ Di⊥ ∇nÞ þð5:57ÞThe equation resulting from setting ∇ · Γi ¼ ∇ · Γe cannot easily be separated intoone-dimensional equations. Furthermore, the answer depends sensitively on theboundary conditions at the ends of the field lines. Unless the plasma is so long thatparallel diffusion can be neglected altogether, there is no simple answer to theproblem of ambipolar diffusion across a magnetic field.5.5.2Experimental ChecksWhether or not a magnetic field reduces transverse diffusion in accordance withEq. (5.51) became the subject of numerous investigations.

The first experimentperformed in a tube long enough that diffusion to the ends could be neglected wasthat of Lehnert and Hoh in Sweden. They used a helium positive column about 1 cmin diameter and 3.5 m long (Fig. 5.14). In such a plasma, the electrons arecontinuously lost by radial diffusion to the walls and are replenished by ionizationof the neutral gas by the electrons in the tail of the velocity distribution.

These fastelectrons, in turn, are replenished by acceleration in the longitudinal electric field.Consequently, one would expect Ez to be roughly proportional to the rate oftransverse diffusion. Two probes set in the wall of the discharge tube were usedto measure Ez as B was varied. The ratio of Ez(B) to Ez(0) is shown as a function of B inFig. 5.15. At low B fields, the experimental points follow closely the predictedcurve, calculated on the basis of Eq. (5.52). At a critical field Bc of about 0.2 T,however, the experimental points departed from theory and, in fact, showed anFig. 5.14 The Lehnert-Hoh experiment to check the effect of a magnetic field on diffusion in aweakly ionized gas5.5 Diffusion Across a Magnetic Field163Fig.

5.15 The normalizedlongitudinal electric fieldmeasured as a function ofB at two different pressures.Theoretical curves areshown for comparison(From F. C. Hoh andB. Lehnert, Phys. Fluids 3,600 (1960))increase of diffusion with B. The critical field Bc increased with pressure, suggestingthat a critical value of ωcτ was involved and that something went wrong with the“classical” theory of diffusion when ωcτ was too large.The trouble was soon found by Kadomtsev and Nedospasov in theU.S.S.R. These theorists discovered that an instability should develop at highmagnetic fields; that is, a plasma wave would be excited by the Ez field, and thatthis wave would cause enhanced radial losses.

The theory correctly predicted thevalue of Bc. The wave, in the form of a helical distortion of the plasma column,was later seen directly in an experiment by Allen, Paulikas, and Pyle at Berkeley.This helical instability of the positive column was the first instance in which“anomalous diffusion” across magnetic fields was definitively explained, but theexplanation was applicable only to weakly ionized gases. In the fully ionizedplasmas of fusion research, anomalous diffusion proved to be a much tougherproblem to solve.Problems5.1.

The electron–neutral collision cross section for 2-eV electrons in He is about6πa20 , where a0 ¼ 0.53 108 cm is the radius of the first Bohr orbit of thehydrogen atom. A positive column with no magnetic field has p ¼ 1 Torr of He(at room temperature) and KTe ¼ 2 eV.1645 Diffusion and Resistivity(a) Compute the electron diffusion coefficient in m2/s, assuming that σvaveraged over the velocity distribution is equal to σv for 2-eV electrons.(b) If the current density along the column is 2 kA/m2 and the plasma densityis 1016 m3, what is the electric field along the column?5.2. A weakly ionized plasma slab in plane geometry has a density distributionnðxÞ ¼ n0 cos ðπx=2LÞLxLThe plasma decays by both diffusion and recombination.

If L ¼ 0.03 m,D ¼ 0.4 m2/s, and α ¼ 1015 m3/s, at what density will the rate of loss bydiffusion be equal to the rate of loss by recombination?5.3. A weakly ionized plasma is created in a cubical aluminum box of length L oneach side. It decays by ambipolar diffusion.(a) Write an expression for the density distribution in the lowest diffusionmode.(b) Define what you mean by the decay time constant and compute it ifDa ¼ 103 m2/s.5.4. A long, cylindrical positive column has B ¼ 0.2 T, KTi ¼ 0.1 eV, and otherparameters the same as in Problem 5.1. The density profile isnðr Þ ¼ n0 J 0 r=½Dτ1=2with the boundary condition n ¼ 0 at r ¼ a ¼ l cm.

Note: J0(z) ¼ 0 at z ¼ 2.4.(a) Show that the ambipolar diffusion coefficient to be used above can beapproximated by D⊥e.(b) Neglecting recombination and losses from the ends of the column, compute the confinement time τ.5.5. For the density profile of Fig. 5.7, derive an expression for the peak density n0in terms of the source strength Q in ion-electron pairs per m2.5.6. You do a recombination experiment in a weakly ionized gas in which the mainloss mechanism is recombination. You create a plasma of density 1020 m3 bya sudden burst of ultraviolet radiation and observe that the density decays tohalf its initial value in 10 ms.

What is the value of the recombination coefficient α? Give units.5.6Collisions in Fully Ionized PlasmasWhen the plasma is composed of ions and electrons alone, all collisions are Coulombcollisions between charged particles. However, there is a distinct difference between(a) collisions between like particles (ion–ion or electron–electron collisions) and5.6 Collisions in Fully Ionized Plasmas165Fig.

5.16 Shift of guidingcenters of two like particlesmaking a 90 collision(b) collisions between unlike particles (ion–electron or electron–ion collisions).Consider two identical particles colliding (Fig. 5.16). If it is a head-on collision, theparticles emerge with their velocities reversed; they simply interchange their orbits,and the two guiding centers remain in the same places.

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