1629373397-425d4de58b7aea127ffc7c337418ea8d (846389), страница 45
Текст из файла (страница 45)
The wave electric field E⊥ can be decomposed intoleft- and right-hand circularly polarized components, as shown in Fig. 7.32. For theleft-hand component, the vector E⊥ at positions A, B, and C along the z axis appearsz direction, a stationaryas shown in Fig. 7.32a. Since the wave propagates in the þ^electron would sample the vectors at C, B, and A in succession and therefore wouldsee a left-rotating E-field. It would not be accelerated because its Larmor gyration isin the right-hand (clockwise) direction. However, if the electron were moving fasterthan the wave in the z^ direction, it would see the vectors at A, B, and C in that orderand hence would be resonantly accelerated if its velocity satisfied the conditionω kzvz ¼ ωc. The right-hand component of E would appear as shown inFig.
7.32b. Now an electron would see a clockwise rotating E-field if it movedFig. 7.32 The mechanism of cyclotron damping2607 Kinetic Theorymore slowly than the wave, so that the vectors at C, B, and A were sampled insuccession. This electron would be accelerated if it met the conditionω kzvz ¼ +ωc. A plane or elliptically polarized wave would, therefore, be cyclotron damped by electrons moving in either direction in the wave frame.7.10.3 Bernstein WavesElectrostatic waves propagating at right angles to B0 at harmonics of the cyclotronfrequency are called Bernstein waves. The dispersion relation can be found byusing the dielectric elements given in Eq. (7.143) in Poisson’s equation ∇ · e · E ¼ 0.If we assume electrostatic perturbations such that E1 ¼ ∇ϕ1, and consider wavesof the form ϕ1 ¼ ϕ1 exp i(k · r ωt), Poisson’s equation can be writtenk2x Exx þ 2kx kz Exz þ k2z Ezz ¼ 0ð7:145ÞNote that we have chosen a coordinate system that has k lying in the x z plane,so that kv ¼ 0.
We next Substitute for exx, exz, and ezz from Eq. (7.143) and expressZ0 (ζ n) in terms of Z(ζ n) with the identity0Z ðζ n Þ ¼ 2½1 þ ζZ ðζ Þð7:146ÞProblems7.12. Prove Eq. (7.146) directly from the integral expressions for Z(ζ) and Z0 (ζ).7.13. The principal part of Z(ζ) for small and large ζ, as used in Eqs.
(7.125) and(7.129), is given by 2 2ζ 1Zðζ Þ ’ 2ζ 1 ζ þ 3 1ζ 1Zðζ Þ ’ ζ 1 1 þ ζ 2 þ 2Prove these by expanding the denominator in the definition (7.118) of Z(ζ).Equation (7.145) becomes1XX ω2pbk2x þ k2z þeζI n ð bÞ0ω2sn¼1"# 1=22n2 k2x Z 2nkx kz ð1 þ ζ n ZÞ 2k2z ζ n ð1 þ ζ n Z Þ ¼ 0bbð7:147Þ7.10Kinetic Effects in a Magnetic Field261The expression in the square brackets can be simplified in a few algebraic steps to2k2z ζ n þ ζ 20 Z ðζ n Þ by using the definitions b ¼ k2x v2th =2ω2c and ζ n ¼ (ω + nωc)/kzvth.Further noting that 2k2x ω2p ζ 0 =ω2 ¼ 2ω2p =v2th k2D for each species, we can writeEq. (7.147) ask2xþk2zþXk2D ebs(1Xn¼1I n ðbÞ½ζ n =ζ 0 þ ζ 0 Z ðζ n Þ)¼0ð7:148ÞThe term ζ n/ζ 0 is 1 nωc/ω. Since In(b) ¼ In(b), the term In(b)nωc/ω sums to zerowhen n goes from 1 to 1; hence, ζ n/ζ 0 can be replaced by 1. Defining k2 k2⊥þk2z ¼ k2x þ k2z we obtain the general dispersion relation for Bernstein waves:X k2D b1þe2ks(1Xn¼1I n ðbÞ½1 þ ζ 0 Z ðζ n Þ)¼ 0:ð7:149Þ(A) Electron Bernstein Waves.
Let us first consider high-frequency waves inwhich the ions do not move. These waves are not sensitive to small deviationsfrom perpendicular propagation, and we may set kz ¼ 0, so that ζ n ! 1. There is,therefore, no cyclotron damping; the gaps in the spectrum that we shall find are notcaused by such damping. For large ζ n, we may replace Z(ζ n) by l/ζ n, according tothe integral of Eq. (7.129). The n ¼ 0 term in the second sum of Eq.
(7.149) thencancels out, and we can divide the sum into two sums, as follows:k2 þXk2D ebs"1Xn¼1I n ðbÞð1 ζ 0 =ζ n Þ þ1Xn¼1#I n ðbÞð1 ζ 0 =ζ n Þ ¼ 0; ð7:150Þor2k þXsk2D eb1XωωI n ð bÞ 2 ¼0ω þ nωc ω nωcn¼1ð7:151ÞThe bracket collapses to a single term upon combining over a commondenominator:1¼1XX k22n2 ω2cD bI n ð bÞ 2e2ω n2 ω2cs kn¼1ð7:152ÞUsing the definitions of kD and b, one obtains the well-known kz ¼ 0 dispersionrelation1¼1X ω2p 2XI n ð bÞeb22bωcsn¼1 ðω=nωc Þ 1ð7:153Þ2627 Kinetic TheoryFig. 7.33 The function α(ω, b) for electron Bernstein waves [From I.
B. Bernstein, Phys. Rev.109, 10 (1958)]We now specialize to the case of electron oscillations with kz ¼ 0. Dropping the sumover species, we obtain from Eq. (7.152)1Xk2⊥eb I n ðbÞ 22¼2ωn αðω; bÞcω2 nω2ck2Dn¼1ð7:154ÞThe function α(ω, b) for one value of b is shown in Fig. 7.33.
The possible values ofω are found by drawing a horizontal line at αðω; bÞ ¼ k2⊥ =k2D > 0: It is then clearthat possible values of ω lie just above each cyclotron harmonic, and that there is aforbidden gap just below each harmonic.To obtain the fluid limit, we replace In(b) by its small-b value (b/2)n/n! inEq. (7.153). Only the n ¼ 1 term remains in the limit b ! 0, and we obtain1ω2p 2 b ω2ω2p1¼ 2¼1ωc b 2 ω2cω2 ω2cð7:155Þor ω2 ¼ ω2p þ ω2c ¼ ω2h , which is the upper hybrid oscillation.
As k⊥ ! 0, thisfrequency must be one of the roots. If ωh falls between two high harmonics of ωc,the shape of the ω k curves changes near ω ¼ ωh to allow this to occur. The ω kcurves are computed by multiplying Eq. (7.154) by 2ω2p /ω2c to obtaink2⊥ r 2L ¼ 4ω2p αðω; bÞ. The resulting curves for ω/ωc vs. k⊥rL are shown inFig. 7.34 for various values of ω2p /ω2c . Note that for each such value, the curveschange in character above the corresponding hybrid frequency for that case. At theextreme left of the diagram, where the phase velocity approaches the speed of light7.10Kinetic Effects in a Magnetic Field263Fig.
7.34 ElectronBernstein wave dispersionrelation [Adapted fromF. W. Crawford, J. Appl.Phys. 36, 2930 (1965)]waves in the plasma, these curves must be modified by including electromagneticcorrections.Electron Bernstein modes have been detected in the laboratory, but inexplicablylarge spontaneous oscillations at high harmonics of ωc have also been seen in gasdischarges. The story is too long to tell here.(B) Ion Bernstein Waves.
In the case of waves at ion cyclotron harmonics, one hasto distinguish between pure ion Bernstein waves, for which kz ¼ 0, and neutralizedion Bernstein waves, for which kz has a small but finite value. The difference, as wehave seen earlier for lower hybrid oscillations, is that finite kz allows electrons toflow along B0 to cancel charge separations. Though the kz ¼ 0 case has already beentreated in Eq.
(7.153), the distinction between the two cases will be clearer if we goback a step to Eqs. (7.148) and (7.149). Separating out the n ¼ 0 term and usingEq. (7.146), we havek2⊥ þ k2z þXshi XX1 0k2D eb I 0 ðbÞ Z ðζ 0 Þ þk2D ebI n ðbÞ½1 þ ζ 0 Z ðζ n Þ ¼ 02sn6¼0ð7:156ÞThe dividing line between pure and neutralized ion Bernstein waves lies in theelectron n ¼ 0 term. If ζ 0e 1 for the electrons, we can use Eq.
(7.129) to write0Z ðζ 0e Þ ’ 1=ζ 20e . Since ω/kz vthe in this case, electrons cannot flow rapidly2647 Kinetic Theoryenough along B0 to cancel charge. If ζ 0e 1, we can use Eq. (7.126) to write Z0(ζ 0e) ’ 2. In this case we have ω/kz vthe, and the electrons have time to followthe Boltzmann relation (3.73).Taking first the ζ 0e 1 case, we note that ζ 0i 1 is necessarily true also, so thatthe n ¼ 0 term in Eq. (7.156) becomes"ω2p Ω2pk2z 2 þ 2 eb I 0 ðbÞωω#Here we have taken be ! 0 and omitted the subscript from bi. The n 6¼ 0 terms inEq.
(7.156) are treated as before, so that the electron part is given by Eq. (7.155),and the ion part by the ion term in Eq. (7.153). The pure ion Bernstein wavedispersion relation then becomesk2z"#ω2p Ω2p b1 2 2 e I 0 ð bÞωω"#221XωΩ2IðbÞnppþ k2⊥ 1 2eb¼0ω ω2c Ω2c b n¼1 ðω=nΩc 1Þ2ð7:157ÞSince ζ 0e 1 implies small k2z , the first term is usually negligible.
To examine thefluid limit, we can set the second bracket to zero, separate out the n ¼ 1 term, anduse the small-b expansion of In(b), obtaining11 n2 Ω2 ðb=2Þn1Xω2pΩ2p p¼0ω2 ω2c ω2 Ω2c n¼2 n! ω2 n2 Ω2cð7:158ÞThe sum vanishes for b ¼ 0, and the remaining terms are equal to the quantity S ofAppendix B. The condition S ¼ 0 yields the upper and lower hybrid frequencies [seethe equation following Eq. (4.70)]. Thus, for k⊥ ! 0, the low-frequency rootapproaches ωl.
For finite b, one of the terms in the sum can balance the electronterm if ω ’ nΩc, so there are roots near the ion cyclotron harmonics. The dispersioncurves ω/Ωc vs. k⊥rLi resemble the electron curves in Fig. 7.34. The lowest tworoots for the ion case are shown in Fig. 7.35, together with experimental measurements verifying the dispersion relation.The lower branches of the Bernstein wave dispersion relation exhibit the backward-wave phenomenon, in which the ω k curve has a negative slope, indicatingthat the group velocity is opposite in direction to the phase velocity.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
