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The result is the same as in aglancing collision, in which the trajectories are hardly disturbed. The worst that canhappen is a 90 collision, in which the velocities are changed 90 in direction. Theorbits after collision will then be the dashed circles, and the guiding centers will haveshifted.
However, it is clear that the “center of mass” of the two guiding centersremains stationary. For this reason, collisions between like particles give rise to verylittle diffusion. This situation is to be contrasted with the case of ions colliding withneutral atoms. In that case, the final velocity of the neutral is of no concern, and theion random-walks away from its initial position.
In the case of ion–ion collisions,however, there is a detailed balance in each collision; for each ion that movesoutward, there is another that moves inward as a result of the collision.When two particles of opposite charge collide, however, the situation is entirelydifferent (Fig. 5.17). The worst case is now the 180 collision, in which the particlesemerge with their velocities reversed. Since they must continue to gyrate about thelines of force in the proper sense, both guiding centers will move in the samedirection.
Unlike-particle collisions give rise to diffusion. The physical picture issomewhat different for ions and electrons because of the disparity in mass. Theelectrons bounce off the nearly stationary ions and random-walk in the usualfashion. The ions are slightly jostled in each collision and move about as a resultof frequent bombardment by electrons. Nonetheless, because of the conservation ofmomentum in each collision, the rates of diffusion are the same for ions andelectrons, as we shall show.1665 Diffusion and ResistivityFig.
5.17 Shift of guidingcenters of two oppositelycharged particles making a180 collision5.6.1Plasma ResistivityThe fluid equations of motion including the effects of charged-particle collisionsmay be written as follows (cf. Eq. (3.47)):dvi¼ enðE þ vi BÞ ∇ pi ∇ πi þ Piedtdve¼ enðE þ ve BÞ ∇ pe ∇ πe þ PeimndtMnð5:58ÞThe terms Pie and Pei represent, respectively, the momentum gain of the ion fluidcaused by collisions with electrons, and vice versa. The stress tensor Pj has beensplit into the isotropic part pj and the anisotropic viscosity tensor πj.
Like-particlecollisions, which give rise to stresses within each fluid individually, are contained inπj. Since these collisions do not give rise to much diffusion, we shall ignore theterms ∇ · πj. As for the terms Pei and Pie, which represent the friction between thetwo fluids, the conservation of momentum requiresPie ¼ Peið5:59Þ5.6 Collisions in Fully Ionized Plasmas167We can write Pei in terms of the collision frequency in the usual manner:Pei ¼ mnðvi ve Þνeið5:60Þand similarly for Pie. Since the collisions are Coulomb collisions, one would expectPei to be proportional to the Coulomb force, which is proportional to e2 (for singlycharged ions).
Furthermore, Pei must be proportional to the density of electrons neand to the density of scattering centers ni, which, of course, is equal to ne. Finally,Pei should be proportional to the relative velocity of the two fluids. On physicalgrounds, then, we can write Pei asPei ¼ ηe2 n2 ðvi ve Þð5:61Þwhere η is a constant of proportionality. Comparing this with Eq.
(5.60), we see thatνei ¼ne2ηmð5:62ÞThe constant η is the specific resistivity of the plasma; that this jibes with the usualmeaning of resistivity will become clear shortly.5.6.2Mechanics of Coulomb CollisionsWhen an electron collides with a neutral atom, no force is felt until the electron isclose to the atom on the scale of atomic dimensions; the collisions are like billiardball collisions. When an electron collides with an ion, the electron is graduallydeflected by the long-range Coulomb field of the ion.
Nonetheless, one can derivean effective cross section for this kind of collision. It will suffice for our purposes togive an order-of-magnitude estimate of the cross section. In Fig. 5.18, an electron ofvelocity v approaches a fixed ion of charge e. In the absence of Coulomb forces, theelectron would have a distance of closest approach r0, called the impact parameter.In the presence of a Coulomb attraction, the electron will be deflected by an angle χ,which is related to r0.
The Coulomb force isF¼e24πε0 r 2ð5:63ÞThis force is felt during the time the electron is in the vicinity of the ion; this time isroughlyT r 0 =vð5:64Þ1685 Diffusion and ResistivityFig. 5.18 Orbit of an electron making a Coulomb collision with an ionThe change in the electron’s momentum is therefore approximatelyΔðmvÞ ¼ jFTj e24πε0 r 0 vð5:65ÞWe wish to estimate the cross section for large-angle collisions, in which χ 90 .For a 90 collision, the change in mv is of the order of mv itself.
ThusΔðmvÞ ffi mv ffi e2 =4πε0 r 0 v,r 0 ¼ e2 =4πε0 m v2ð5:66ÞThe cross section is thenσ ¼ πr 20 ¼ e4 =16πε20 m2 v4ð5:67ÞThe collision frequency is, therefore,νei ¼ nσv ¼ ne4 =16πε20 m2 v3ð5:68Þand the resistivity isη¼me2ν¼eine216πε20 mv3ð5:69ÞFor a Maxwellian distribution of electrons, we may replace v2 by KTe/m for ourorder-of-magnitude estimate:ηπe2 m1=2ð4πε0 Þ2 ðKT e Þ3=2ð5:70Þ5.6 Collisions in Fully Ionized Plasmas169Equation (5.70) is the resistivity based on large-angle collisions alone. Inpractice, because of the long range of the Coulomb force, small-angle collisionsare much more frequent, and the cumulative effect of many small-angle deflectionsturns out to be larger than the effect of large-angle collisions.
It was shown bySpitzer that Eq. (5.70) should be multiplied by a factor ln Λ:ηπe2 m1=2ð4πε0 Þ2 ðKT e Þ3=2ð5:71ÞlnΛwhereΛ ¼ λD =r 0 ¼ 12πnλ3Dð5:72ÞThis factor represents the maximum impact parameter, in units of r0 as given byEq. (5.66), averaged over a Maxwellian distribution. The maximum impact parameter is taken to be λD because Debye shielding suppresses the Coulomb field atlarger distances. Although Λ depends on n and KTe, its logarithm is insensitive tothe exact values of the plasma parameters. Typical values of ln Λ are given below.KTe (eV)0.22100104103n (m3)10151017101910211027ln Λ9.110.213.716.06.8(Q-machine)(lab plasma)(typical torus)(fusion reactor)(laser plasma)It is evident that ln Λ varies only a factor of two as the plasma parameters rangeover many orders of magnitude.
For most purposes, it will be sufficiently accurateto let ln Λ ¼ 10 regardless of the type of plasma involved.5.6.3Physical Meaning of ηLet us suppose that an electric field E exists in a plasma and that the current that itdrives is all carried by the electrons, which are much more mobile than the ions. LetB ¼ 0 and KTe ¼ 0, so that ∇ · Pe ¼ 0. Then, in steady state, the electron equation ofmotion (5.58) reduces toen E ¼ Peið5:73ÞSince j ¼ en(vi ve), Eq. (5.61) can be writtenPei ¼ ηen jð5:74Þ1705 Diffusion and Resistivityso that Eq. (5.73) becomesE ¼ ηjð5:75ÞThis is simply Ohm’s law, and the constant η is just the specific resistivity.
Theexpression for η in a plasma, as given by Eq. (5.71) or Eq. (5.69), has severalfeatures which should be pointed out.(a) In Eq. (5.71), we see that η is independent of density (except for the weakdependence in ln Λ). This is a rather surprising result, since it means that if afield E is applied to a plasma, the current j, as given by Eq.
(5.75), isindependent of the number of charge carriers. The reason is that althoughj increases with ne, the frictional drag against the ions increases with ni. Sincene ¼ ni these two effects cancel. This cancellation can be seen in Eqs. (5.68)and (5.69). The collision frequency νei is indeed proportional to n, but thefactor n cancels out in η.
A fully ionized plasma behaves quite differently froma weakly ionized one in this respect. In a weakly ionized plasma, we havej ¼ neve, ve ¼ μeE, so that j ¼ neμeE. Since μe depends only on the densityof neutrals, the current is proportional to the plasma density n.(b) Equation (5.71) shows that η is proportional to (KTe)3/2. As a plasma isheated, the Coulomb cross section decreases, and the resistivity drops ratherrapidly with increasing temperature. Plasmas at thermonuclear temperatures(tens of keV) are essentially collisionless; this is the reason so much theoretical research is done on collisionless plasmas. Of course, there must always besome collisions; otherwise, there wouldn’t be any fusion reactions either.
Aneasy way to heat a plasma is simply to pass a current through it. The I2R(or j2η) losses then turn up as an increase in electron temperature. This is calledohmic heating. The (KTe)3/2 dependence of η, however, does not allow thismethod to be used up to thermonuclear temperatures. The plasma becomessuch a good conductor at temperatures above 1 keV that ohmic heating is avery slow process in that range.(c) Equation (5.68) shows that νei varies as v3.
The fast electrons in the tail of thevelocity distribution make very few collisions. The current is therefore carriedmainly by these electrons rather than by the bulk of the electrons in the mainbody of the distribution. The strong dependence on v has another interestingconsequence. If an electric field is suddenly applied to a plasma, a phenomenon known as electron runaway can occur. A few electrons which happen tobe moving fast in the direction of E when the field is applied will havegained so much energy before encountering an ion that they can make only aglancing collision. This allows them to pick up more energy from the electricfield and decrease their collision cross section even further. If E is largeenough, the cross section falls so fast that these runaway electrons nevermake a collision.
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