1629373397-425d4de58b7aea127ffc7c337418ea8d (Introduction to Plasma Physics and Controlled Fusion Francis F. Chen), страница 9

A particle with smallv⊥ =vk at the midplane (B ¼ B0) will also escape if the maximum field Bm is not largeenough. For given B0 and Bm, which particles will escape? A particle with v⊥ ¼ v⊥00and vk ¼ vk0 at the midplane will have v⊥ ¼ v⊥ and vk ¼ 0 at its turning point. Letthe field be B0 there. Then the invariance of μ yields1mv2⊥0 =B021202¼ mv ⊥ =BConservation of energy requiresFig. 2.8 A plasma trapped between magnetic mirrors0ð2:43Þ2.3 Nonuniform B Field33022þ v2k0 v20v⊥ ¼ v⊥0ð2:44ÞCombining Eqs. (2.43) and (2.44), we find22B0 v⊥0v⊥0 sin 2 θ¼0 ¼2v20Bv0⊥ð2:45Þwhere θ is the pitch angle of the orbit in the weak-field region.

Particles withsmaller θ will mirror in regions of higher B. If θ is too small, B0 exceeds Bm; andthe particle does not mirror at all. Replacing B0 by Bm in Eq. (2.45), we see that thesmallest θ of a confined particle is given bysin 2 θm ¼ B0 =Bm 1=R mð2:46Þwhere Rm is the mirror ratio. Equation (2.46) defines the boundary of a region invelocity space in the shape of a cone, called a loss cone (Fig. 2.9). Particles lyingwithin the loss cone are not confined. Consequently, a mirror-confined plasma isnever isotropic. Note that the loss cone is independent of q or m.

Without collisions,both ions and electrons are equally well confined. When collisions occur, particlesare lost when they change their pitch angle in a collision and are scattered into theloss cone. Generally, electrons are lost more easily because they have a highercollision frequency.The magnetic mirror was first proposed by Enrico Fermi as a mechanism for theacceleration of cosmic rays. Protons bouncing between magnetic mirrorsapproaching each other at high velocity could gain energy at each bounce(Fig. 2.10).

How such mirrors could arise is another story. A further example ofthe mirror effect is the confinement of particles in the Van Allen belts. Themagnetic field of the earth, being strong at the poles and weak at the equator,forms a natural mirror with rather large Rm.Fig. 2.9 The loss cone342Single-Particle MotionsProblems2.8. Suppose the earth’s magnetic field is 3 105 T at the equator and falls off as1/r3, as for a perfect dipole.

Let there be an isotropic population of 1-eVprotons and 30-keV electrons, each with density n ¼ 107 m3 at r ¼ 5 earthradii in the equatorial plane.(a)(b)(c)(d)Compute the ion and electron ∇B drift velocities.Does an electron drift eastward or westward?How long does it take an electron to encircle the earth?Compute the ring current density in A/m2.Note: The curvature drift is not negligible and will affect the numericalanswer, but neglect it anyway.2.9.

An electron lies at rest in the magnetic field of an infinite straight wirecarrying a current I. At t ¼ 0, the wire is suddenly charged to a positivepotential ϕ without affecting I. The electron gains energy from the electricfield and begins to drift.(a) Draw a diagram showing the orbit of the electron and the relativedirections of I, B, vE, v∇B, and vR.(b) Calculate the magnitudes of these drifts at a radius of 1 cm if I ¼ 500 A,ϕ ¼ 460 V, and the radius of the wire is 1 mm.

Assume that ϕ is held at0 V on the vacuum chamber walls 10 cm away.Hint: A good intuitive picture of the motion is needed in addition to theformulas given in the text.2.10. A 20-keV deuteron in a large mirror fusion device has a pitch angle θ of 45at the midplane, where B ¼ 0.7 T. Compute its Larmor radius.2.11. A plasma with an isotropic velocity distribution is placed in a magneticmirror trap with mirror ratio Rm ¼ 4.

There are no collisions, so the particlesin the loss cone simply escape, and the rest remain trapped. What fraction istrapped?2.12. A cosmic ray proton is trapped between two moving magnetic mirrors withRm ¼ 5 and initially has W ¼ 1 keV and v⊥ ¼ vk at the midplane. Each mirrormoves toward the midplane with a velocity vm ¼ 10 km/s (Fig. 2.10).Fig. 2.10 Acceleration of cosmic rays2.4 Nonuniform E Field35(a) Using the loss cone formula and the invariance of μ, find the energy towhich the proton will be accelerated before it escapes.(b) How long will it take to reach that energy?1.

Treat the mirrors as flat pistons and show that the velocity gained ateach bounce is 2vm.2. Compute the number of bounces necessary.3. Compute the time T it takes to traverse L that many times. Factor-oftwo accuracy will suffice.2.4Nonuniform E FieldNow we let the magnetic field be uniform and the electric field be nonuniform. Forsimplicity, we assume E to be in the x direction and to vary sinusoidally in thex direction (Fig. 2.11):E E0 ð cos kxÞ^xð2:47ÞThis field distribution has a wavelength λ ¼ 2π/k and is the result of a sinusoidaldistribution of charges, which we need not specify.

In practice, such a chargedistribution can arise in a plasma during a wave motion. The equation of motion ismðdv=dtÞ ¼ q½EðxÞþv Bð2:48Þwhose transverse components arev_ x ¼qBqv y þ Ex ð x Þmmv_ y ¼ €vx ¼ ω2c vx ωcqBvxmE_ xBFig. 2.11 Drift of a gyrating particle in a nonuniform electric fieldð2:49Þð2:50Þ362€v y ¼ ω2c v y ω2cSingle-Particle MotionsEx ð x ÞBð2:51ÞHere Ex(x) is the electric field at the position of the particle. To evaluate this, weneed to know the particle’s orbit, which we are trying to solve for in the first place.If the electric field is weak, we may, as an approximation, use the undisturbed orbitto evaluate Ex(x).

The orbit in the absence of the E field was given in Eq. (2.7):x ¼ x0 þ r L sin ωc tð2:52ÞFrom Eqs. (2.51) and (2.47), we now have€v y ¼ ω2c v y ω2cE0cos kðx0 þ r L sin wc tÞBð2:53ÞAnticipating the result, we look for a solution which is the sum of a gyration at ωcand a steady drift vE. Since we are interested in finding an expression for vE, we takeout the gyratory motion by averaging over a cycle. Equation (2.50) then givesvx ¼ 0. In Eq. (2.53), the oscillating term €v y clearly averages to zero, and we have€v y ¼ 0 ¼ ω2c v y ω2cE0cos kðx0 þ r L sin ωc tÞBð2:54ÞExpanding the cosine, we havecos kðx0 þ r L sin ωc tÞ ¼ cos ðkx0 Þ cos ðkr L sin ωc tÞ sin ðkx0 Þ sin ðkr L sin ωc tÞð2:55ÞIt will suffice to treat the small Larmor radius case, krL 1.

The Taylor expansions12cos ε ¼ 1 ε 2 þ ð2:56Þsin ε ¼ ε þ allow us to write1cos k ðx0 þ r L sin ωc tÞ ð cos kx0 Þ 1 k2 r 2L sin 2 ωc t ð sin kx0 Þkr L sin ωc t2The last term vanishes upon averaging over time, and Eq. (2.54) givesvy ¼ E0Ex ð x 0 Þ 111 k2 r 2Lð cos kx0 Þ 1 k2 r 2L ¼ 44BBð2:57Þ2.5 Time-Varying E Field37Thus the usual E B drift is modified by the inhomogeneity to readvE ¼E B1 2 2kr1L4B2ð2:58ÞThe physical reason for this is easy to see.

An ion with its guiding center at amaximum of E actually spends a good deal of its time in regions of weaker E. Itsaverage drift, therefore, is less than E/B evaluated at the guiding center. In a linearlyvarying E field, the ion would be in a stronger field on one side of the orbit and in afield weaker by the same amount on the other side; the correction to vE then cancelsout.

From this it is clear that the correction term depends on the second derivative ofE. For the sinusoidal distribution we assumed, the second derivative is alwaysnegative with respect to E. For an arbitrary variation of E, we need only replace ikby ∇ and write Eq. (2.58) asE B1vE ¼ 1 þ r 2L ∇ 24B2ð2:59ÞThe second term is called the finite-Larmor-radius effect.

What is the significance ofthis correction? Since rL is much larger for ions than for electrons, vE is no longerindependent of species. If a density clump occurs in a plasma, an electric field cancause the ions and electrons to separate, generating another electric field. If there is afeedback mechanism that causes the second electric field to enhance the first one,E grows indefinitely, and the plasma is unstable. Such an instability, called a driftinstability, will be discussed in a later chapter. The grad-B drift, of course, is also afinite-Larmor-radius effect and also causes charges to separate.

According toEq. (2.24), however, v∇B is proportional to krL, whereas the correction term inEq. (2.58) is proportional to k2r2L . The nonuniform-E-field effect, therefore, isimportant at relatively large k, or small scale lengths of the inhomogeneity. Forthis reason, drift instabilities belong to a more general class called microinstabilities.2.5Time-Varying E FieldLet us now take E and B to be uniform in space but varying in time. First, considerthe case in which E alone varies sinusoidally in time, and let it lie along the x axis:E ¼ E0 eiωt x^ð2:60ÞSince E_ x ¼ iωEx , we can write Eq.