1629373397-425d4de58b7aea127ffc7c337418ea8d (Introduction to Plasma Physics and Controlled Fusion Francis F. Chen), страница 8

1.6) has a 1-T magnetic field, and a hydrogen plasma isto be shot out at an E B velocity of 1000 km/s. How much internal electricfield must be present in the plasma?2.4. Show that vE is the same for two ions of equal mass and charge but differentenergies, by using the following physical picture (see Fig. 2.2). Approximatethe right half of the orbit by a semicircle corresponding to the ion energy afteracceleration by the E field, and the left half by a semicircle corresponding tothe energy after deceleration. You may assume that E is weak, so that thefractional change in v⊥ is small.2.5. Suppose electrons obey the Boltzmann relation of Problem 1.5 in a cylindrically symmetric plasma column in which n(r) varies with a scale length λ; thatis, ∂n/∂r ’ n/λ.(a) Using E ¼ ∇ϕ, find the radial electric field for given λ.(b) For electrons, show that finite Larmor radius effects are large if vE is aslarge as vth.

Specifically, show that rL ¼ 2λ if vE ¼ vth.(c) Is (b) also true for ions?Hint: Do not use Poisson’s equation.2.6. Suppose that a so-called Q-machine has a uniform field of 0.2 T and acylindrical plasma with KTe ¼ KTi ¼ 0.2 eV. The density profile is foundexperimentally to be of the formn ¼ n0 exp exp r 2 =a2 1Assume the density obeys the electron Boltzmann relation n ¼ n0exp (eϕ/KTe).(a) Calculate the maximum vE if a ¼ 1 cm.(b) Compare this with vg due to the earth’s gravitational field.(c) To what value can B be lowered before the ions of potassium (A ¼ 39,Z ¼ 1) have a Larmor radius equal to a?262Single-Particle Motions2.7. An unneutralized electron beam has density ne ¼ 1014 m3 and radius a ¼ 1 cmand flows along a 2-T magnetic field.

If B is in the +z direction and E is theelectrostatic field due to the beam’s charge, calculate the magnitude anddirection of the E B drift at r ¼ a (See Fig. P2.7).Fig. P2.72.3Nonuniform B FieldNow that the concept of a guiding center drift is firmly established, we can discussthe motion of particles in inhomogeneous fields—E and B fields which vary inspace or time. For uniform fields we were able to obtain exact expressions for theguiding center drifts. As soon as we introduce inhomogeneity, the problem becomestoo complicated to solve exactly. To get an approximate answer, it is customary toexpand in the small ratio rL/L, where L is the scale length of the inhomogeneity.This type of theory, called orbit theory, can become extremely involved.

We shallexamine only the simplest cases, where only one inhomogeneity occurs at a time.2.3.1∇B⊥B: Grad-B DriftHere the lines of force1 are straight, but their density increases, say, in they direction (Fig. 2.5). We can anticipate the result by using our simple physicalpicture. The gradient in jBj causes the Larmor radius to be larger at the bottom ofthe orbit than at the top, and this should lead to a drift, in opposite directions for ionsand electrons, perpendicular to both B and ∇B. The drift velocity should obviouslybe proportional to rL/L and to v⊥.Consider the Lorentz force F ¼ qv B, averaged over a gyration.

Clearly,Fx ¼ 0; since the particle spends as much time moving up as down. We wishto calculate F y ; in an approximate fashion, by using the undisturbed orbit of theparticle to find the average. The undisturbed orbit is given by Eqs.

(2.4a),1The magnetic field lines are often called “lines of force.” They are not lines of force. Themisnomer is perpetuated here to prepare the student for the treacheries of his profession.2.3 Nonuniform B Field27Fig. 2.5 The drift of a gyrating particle in a nonuniform magnetic field(2.4b), and (2.7) for a uniform B field. Taking the real part of Eqs. (2.4a) and(2.4b), we have∂BF y ¼ qvx Bz ð yÞ ¼ qv⊥ ð cos ωc tÞ B0 r L ð cos ωc tÞ∂yð2:20Þwhere we have made a Taylor expansion of B field about the point x0 ¼ 0, y0 ¼ 0 andhave used Eq. (2.7):B ¼ B0 þ ðr ∇ ÞB þ ð2:21ÞBz ¼ B0 þ yð∂Bz =∂yÞ þ This expansion of course requires rL/L 1, where L is the scale length of ∂Bz/∂y.The first term of Eq.

(2.20) averages to zero in a gyration, and the average of cos2ωct is ½, so that12F y ¼ qv⊥ r L ð∂B=∂yÞð2:22ÞThe guiding center drift velocity is thenvgc ¼1 F B 1 Fyv⊥ r L 1 ∂Bx^¼x^ ¼ q B2q jBjB 2 ∂yð2:23Þwhere we have used Eq. (2.17). Since the choice of the y axis was arbitrary, this canbe generalized to12v∇B ¼ v⊥ r LB ∇BB2ð2:24ÞThis has all the dependences we expected from the physical picture; only the factor½ (arising from the averaging) was not predicted.

Note that the stands for the signof the charge, and lightface B stands for jBj. The quantity v∇B is called the grad-Bdrift; it is in opposite directions for ions and electrons and causes a currenttransverse to B. An exact calculation of v∇B would require using the exact orbit,including the drift, in the averaging process.282.3.22Single-Particle MotionsCurved B: Curvature DriftHere we assume the lines of force to be curved with a constant radius of curvatureRc, and we take jBj to be constant (Fig.

2.6). Such a field does not obey Maxwell’sequations in a vacuum, so in practice the grad-B drift will always be added to theeffect derived here. A guiding center drift arises from the centrifugal force felt bythe particles as they move along the field lines in their thermal motion. If v2k denotesthe average square of the component of random velocity along B, the averagecentrifugal force isFcf ¼mv2kRcr^ ¼ mv2kRcR2cð2:25ÞAccording to Eq.

(2.17), this gives rise to a drift2vR ¼1 Fc f B mvk Rc B¼ 2q B2R2cqBð2:26ÞThe drift vR is called the curvature drift.We must now compute the grad-B drift which accompanies this when thedecrease of jBj with radius is taken into account. In a vacuum, we have ∇ B ¼ 0.In the cylindrical coordinates of Fig. 2.6, ∇ B has only a z component, since B hasonly a θ component and ∇B only an r component. We then haveFig. 2.6 A curvedmagnetic field2.3 Nonuniform B Field29ð∇ B Þz ¼1∂ðrBθ Þ ¼ 0r ∂rBθ /1rð2:27ÞThusjBj /∇ j BjRc¼ 2j BjRc1Rcð2:28ÞUsing Eq.

(2.24), we havev∇B ¼ 122v⊥ r LRc1 v Rc B1 m 2 Rc Bv¼B j Bj 2 ¼ ⊥22 ωc R 2 B2 q ⊥ R2 B 2BRcccð2:29ÞAdding this to vR, we have the total drift in a curved vacuum field:vR þ v∇B ¼m Rc B 2 1 2 vk þ v⊥2q R2c B2ð2:30ÞIt is unfortunate that these drifts add. This means that if one bends a magneticfield into a torus for the purpose of confining a thermonuclear plasma, the particleswill drift out of the torus no matter how one juggles the temperatures and magneticfields.For a Maxwellian distribution, Eqs. (1.7) and (1.10) indicate that v2k and 12v2⊥ areeach equal to KT/m, since v⊥ involves two degrees of freedom. Equations (2.3) and(1.6) then allow us to write the average curved-field drift asvRþ ∇B ¼ v2thrLy^ ¼ vth y^Rc ω cRcð2:30aÞwhere ^y here is the direction of Rc B.

This shows that vRþ∇B depends on thecharge of the species but not on its mass.2.3.3∇BjjB: Magnetic MirrorsNow we consider a magnetic field which is pointed primarily in the z direction andwhose magnitude varies in the z direction. Let the field be axisymmetric, withBθ ¼ 0 and ∂/∂θ ¼ 0.

Since the lines of force converge and diverge, there isnecessarily a component Br (Fig. 2.7). We wish to show that this gives rise to aforce which can trap a particle in a magnetic field.302Single-Particle MotionsFig. 2.7 Drift of a particle in a magnetic mirror fieldWe can obtain Br from ∇ · B ¼ 0:1∂∂Bz¼0ðrBr Þ þr ∂r∂zð2:31ÞIf ∂Bz/∂z is given at r ¼ 0 and does not vary much with r, we have approximatelyðrh i∂Bz1zrBr ¼ rdr ’ r 2 ∂B∂z r¼02∂z0h i1zBr ¼ r ∂B∂z2ð2:32Þr¼0The variation of jBj with r causes a grad-B drift of guiding centers about the axis ofsymmetry, but there is no radial grad-B drift, because ∂B/∂θ ¼ 0.

The componentsof the Lorentz force areFr = q (vq Bz − vz Bq )Fq = q (−vr Bz + vz Br )ð2:33ÞFz = q (vr Bq − vq Br)Two terms vanish if Bθ ¼ 0, and terms 1 and 2 give rise to the usual Larmorgyration. Term 3 vanishes on the axis; when it does not vanish, this azimuthalforce causes a drift in the radial direction. This drift merely makes the guidingcenters follow the lines of force. Term 4 is the one we are interested in. UsingEq. (2.32), we obtain12Fz ¼ qvθ r ð∂Bz =∂z Þð2:34Þ2.3 Nonuniform B Field31We must now average over one gyration.

For simplicity, consider a particle whoseguiding center lies on the axis. Then vθ is a constant during a gyration; depending onthe sign of q, vθ is v⊥. Since r ¼ rL, the average force is12Fz ¼ qv⊥ r L22∂Bz1 v ∂Bz1 mv⊥ ∂Bz¼ q ⊥¼2 ωc ∂z2 B ∂z∂zð2:35ÞWe define the magnetic moment of the gyrating particle to be12μ mv2⊥ =Bð2:36ÞFz ¼ μð∂Bz =∂zÞð2:37Þso thatThis is a specific example of the force on a diamagnetic particle, which in generalcan be writtenFk ¼ μ∂B=∂s¼ μ ∇ k Bð2:38Þwhere ds is a line element along B.

Note that the definition (2.36) is the same as theusual definition for the magnetic moment of a current loop with area A and currentI: μ ¼ IA. In the case of a singly charged ion, I is generated by a charge e comingaround ωc/2π times a second: I ¼ eωc/2π. The area A is πr 2L ¼ πv2⊥ =ω2c . Thusμ¼πv2⊥ eωc 1 v2⊥ e 1 mv2⊥¼:¼2 ωc2 Bω2c 2πAs the particle moves into regions of stronger or weaker B, its Larmor radiuschanges, but μ remains invariant. To prove this, consider the component of theequation of motion along B:mdvk∂B¼ μ∂sdtð2:39ÞMultiplying by vk on the left and its equivalent ds/dt on the right, we havemvkdvkd 1 2 ∂B dsdB¼ μ¼mvk ¼ μdt 2∂s dtdtdtð2:40ÞHere dB/dt is the variation of B as seen by the particle; B itself is constant.The particle’s energy must be conserved, so we have322Single-Particle Motions dd 111mv2k þ mv2⊥ ¼mv2k þ μB ¼ 02dt 2dt 2ð2:41ÞWith Eq.

(2.40) this becomesμdB dþ ðμBÞ ¼ 0dt dtso thatdμ=dt ¼ 0ð2:42ÞThe invariance of μ is the basis for one of the primary schemes for plasmaconfinement: the magnetic mirror. As a particle moves from a weak-field region toa strong-field region in the course of its thermal motion, it sees an increasing B, andtherefore its v⊥ must increase in order to keep μ constant. Since its total energy mustremain constant, vk must necessarily decrease.

If B is high enough in the “throat” ofthe mirror, vk eventually becomes zero; and the particle is “reflected” back to theweak-field region. It is, of course, the force Fk which causes the reflection. Thenonuniform field of a simple pair of coils forms two magnetic mirrors betweenwhich a plasma can be trapped (Fig. 2.8). This effect works on both ions andelectrons.The trapping is not perfect, however. For instance, a particle with v⊥ ¼ 0 willhave no magnetic moment and will not feel any force along B.