1629373397-425d4de58b7aea127ffc7c337418ea8d (846389), страница 43
Текст из файла (страница 43)
If nb is large, there will be a two-streaminstability. The critical nb at which instability sets in can be found by settingthe slope of the total distribution function equal to zero. To keep the algebrasimple, we can find an approximate answer as follows.7.8 Experimental Verification247Fig. 7.29 Experimentalmeasurement of thedispersion relation forplasma waves in planegeometry [From H. Derflerand T.
Simonen, J. Appl.Phys. 38, 5018 (1967)]Fig. P7.3 Unperturbed distribution functions fp(vx) and fb(vx) for the plasma and beam electrons,respectively, in a beam–plasma interaction2487 Kinetic Theory(a) Write expressions for fp(v) and fb(v), using the abbreviations v ¼ vx,a2 ¼ 2KTp/m, b2 ¼ 2KTb/m.(b) Assume that the phase velocity vϕ will be the value of v at which fb(v) has0the largest positive slope. Find vϕ and f b(vϕ).000(c) Find f p(vϕ) and set f p vϕ þ f b vϕ ¼ 0.(d) For V b, show that the critical beam density is given approximately bynbTb Vexp V 2 =a2¼ ð2eÞ1=2npTp a7.4 To model a warm plasma, assume that the ion and electron distributionfunctions are given by^f e0 ðvÞ ¼ ae 1π v2 þ a2e^f i0 ðvÞ ¼ ai 1π v2 þ a2i(a) Derive the exact dispersion relation in the Vlasov formalism assuming anelectrostatic perturbation.(b) Obtain an approximate expression for the dispersion relation if ω Ωp.Under what conditions are the waves weakly damped? Explain physicallywhy ω ’ Ωp for very large k.7.5 Consider an unmagnetized plasma with a fixed, neutralizing ion background.The one-dimensional electron velocity distribution is given byf 0e ðvÞ ¼ g0 ðvÞ þ h0 ðvÞwhereae 1, h0 ðvÞ ¼ nb δðv v0 Þπ v2 þ a2e:n0 ¼ np þ nb and nb npg0 ðvÞ ¼ np(a) Derive the dispersion relation for high-frequency electrostaticperturbations.(b) In the limit ω/k ae show that a solution exists in which Im(ω) > 0 (i.e.,growing oscillations).7.6 Consider the one-dimensional distribution function f ðvÞ ¼ A v < vm f ðvÞ ¼ 0 v vm7.9 Ion Landau Damping249(a) Calculate the value of the constant A in terms of the plasma density n0.(b) Use the Vlasov and Poisson equations to derive an integral expression forelectrostatic electron plasma waves.(c) Evaluate the integral and obtain a dispersion relation ω(k), keeping termsto third order in the small quantity kvm/ω.7.9Ion Landau DampingElectrons are not the only possible resonant particles.
If a wave has a slow enoughphase velocity to match the thermal velocity of ions, ion Landau damping canoccur. The ion acoustic wave, for instance, is greatly affected by Landau damping.Recall from Eq. (4.41) that the dispersion relation for ion waves isω¼ vs ¼kKT e þ γi KT iM1=2ð7:112ÞIf Te Ti, the phase velocity lies in the region where f0i(v) has a negative slope, asshown in Fig. 7.30a. Consequently, ion waves are heavily Landau-damped ifTe Ti.
Ion waves are observable only if Te Ti [Fig. 7.30b], so that the phasevelocity lies far in the tail of the ion velocity distribution. A clever way to introduceLandau damping in a controlled manner was employed by Alexeff, Jones, andMontgomery. A weakly damped ion wave was created in a heavy-ion plasma (suchas xenon) with Te Ti. A small amount of a light atom (helium) was then added.Since the helium had about the same temperature as the xenon but had muchsmaller mass, its distribution function was much broader, as shown by the dashedcurve in Fig.
7.30b. The resonant helium ions then caused the wave to damp. Ionacoustic waves generated in instabilities are often not seen because of the severedamping by light-ion impurities.Fig. 7.30 Explanation of Landau damping of ion acoustic waves. For Te Ti, the phase velocitylies well within the ion distribution; for Te Ti, there are very few ions at the phase velocity.Addition of a light ion species (dashed curve) increases Landau damping2507.9.17 Kinetic TheoryThe Plasma Dispersion FunctionTo introduce some of the standard terminology of kinetic theory, we now calculatethe ion Landau damping of ion acoustic waves in the absence of magnetic fields.Ions and electrons follow the Vlasov equation (7.23) and have perturbations ofthe form of Eq. (7.46) indicating plane waves propagating in the x direction.
Thesolution for f1 is given by Eq. (7.48) with appropriate modifications:f1j ¼ iqj E ∂ f 0 j =∂v jm j ω kv jð7:113Þwhere E and vj stand for E1x, vxj; and the jth species has charge qj, mass mj, andparticle velocity vj. The density perturbation of the jth species is given byn1j ¼ð11 iqj Ef 1j vj dvj ¼ mjð11∂ f 0j =∂vjdvjω kvjð7:114ÞLet the equilibrium distributions f0j be one-dimensional Maxwellians:f 0j ¼n0 j v2j =v2th jevth j π 1=21=2vth j 2KT j =mjð7:115ÞIntroducing the dummy integration variable s ¼ vj/vthj, we can write n1j asn1 jiqj En0j 1¼kmj v2th j π 1=2ð 1 ðd=dsÞ es21sζjdsð7:116Þwhereζ j ω=kvth jð7:117ÞWe now define the plasma dispersion function Z(ζ):Z ðζ Þ ¼1π 1=2ð112esds I m ðζ Þ > 0sζð7:118ÞThis is a contour integral, as explained in Sect.
7.4, and analytic continuation to thelower half plane must be used if Im(ζ) < 0. Z(ζ) is a complex function of a complexargument (since ω or k usually has an imaginary part). In cases where Z(ζ) cannotbe approximated by an asymptotic formula, one can use the tables of Fried andConte or a standard computer subroutine.7.9 Ion Landau Damping251To express n1j in terms of Z(ζ), we take the derivative with respect to ζ:10Z ðζ Þ ¼π 1=2ð12es1ðs ζ Þ2dsIntegration by parts yields"#1ð 1 ðd=dsÞ es2s21 e10dsZ ðζ Þ ¼ 1=2þ 1=2sζsζππ11The first term vanishes, as it must for any well-behaved distribution function.Equation (7.116) can now be writtenn1 j ¼iq j En0 j 0 Z ζjkm j v2th jð7:119ÞPoisson’s equation isε0 ∇ E ¼ ikε0 E¼ Σ q j n1 jjð7:120ÞCombining the last two equations, separating out the electron term explicitly, anddefining1=2Ω p j n0 j Z2j e2 =ε0 Mjð7:121ÞWe obtain the dispersion relationk2 ¼ω2p0v2theZ ðζ e Þ þX Ω2p jjv2th j0Z ζjð7:122ÞElectron plasma waves can be obtained by setting Ωpj ¼ 0 (infinitely massive ions).Definingk2D ¼ 2ω2p =v2the ¼ λ2D ,ð7:123Þ1 0k2 =k2D ¼ Z ðζ e Þ2ð7:124Þwe then obtainwhich is the same as Eq.
(7.54) when f0e is Maxwellian.2527.9.27 Kinetic TheoryIon Waves and Their DampingTo obtain ion waves, go back to Eq. (7.122) and use the fact that their phase velocityω/k is much smaller than vthe; hence ζ e is small, and we can expand Z(ζ e) in apower series:pffiffiffi 22Z ðζ e Þ ¼ i π eζe 2ζ e 1 ζ 2e þ ð7:125Þpffiffiffi20Z ðζ e Þ ¼ 2i π ζ e eζe 2 þ ’ 2ð7:126Þ3The imaginary term comes from the residue at a pole lying near the real s axis[of Eq. (7.56)] and represents electron Landau damping. For ζ e 1, the derivativeof Eq.
(7.125) givesElectron Landau damping can usually be neglected in ion waves because the slopeof fe(v) is small near its peak. Replacing Z0 (ζ e) by 2 in Eq. (7.122) gives the ionwave dispersion relationλ2DX Ω2p jjv2th j0Z ζ j ¼ 1 þ k2 λ2D ’ 1ð7:127ÞThe term k2λ2D represents the deviation from quasineutrality.We now specialize to the case of a single ion species. Since n0e ¼ Zin0i, thecoefficient in Eq. (7.127) isλ2DΩ2pv2thi¼ε0 KT e n0i Z 2 e2 M1 ZT e¼n0e e2 ε0 M 2kT i 2 T iFor k2 λ2D 1, the dispersion relation becomesωZkvthi0¼2T i 2¼ZT e θθZT eTið7:128ÞSolving this equation is a nontrivial problem. Suppose we take real k and complex ωto study damping in time.
Then the real and imaginary parts of ω must be adjustedso that Im(Z0 ) ¼ 0 and Re(Z0 ) ¼ 2Ti/ZTe. There are in general many possible roots ωthat satisfy this, all of them having Im ω < 0. The least damped, dominant root is theone having the smallest jIm ωj. Damping in space is usually treated by taking ω realand k complex. Again we get a series of roots k with Im k > 0, representing spatialdamping. However, the dominant root does not correspond to the same value of ζ ias in the complex ω case.
It turns out that the spatial problem has to be treated with7.9 Ion Landau Damping253special attention to the excitation mechanism at the boundaries and with morecareful treatment of the electron term Z0 (ζ e).To obtain an analytic result, we consider the limit ζ i 1, corresponding to largetemperature ratio θ ZTe/Ti. The asymptotic expression for Z0 (ζ i) ispffiffiffi203 4Z ðζ i Þ ¼ 2i π ζ i eζi þ ζ 2i þ ζi þ 2ð7:129ÞIf the damping is small, we can neglect the Landau term in the first approximation.Equation (7.128) becomes1311þ 222ζ iζi!¼2θSince θ is assumed large, ζ 2i is large; and we can approximate ζ 2i by θ/2 in thesecond term. Thus1321þ¼ ,2θθζiζ 2i ¼3 θþ2 2ð7:130Þorω2 2KT i 3 ZT eZKT e þ 3KT iþ¼¼22 2T iMMkð7:131ÞThis is the ion wave dispersion relation (4.41) with γi ¼ 3, generalized toarbitrary Z.We now substitute Eqs.
(7.129) and (7.130) into Eq. (7.128) retaining theLandau term:pffiffiffi21321þ 2i π ζ i eζi ¼2θθζipffiffiffi132ζ 2i1þiπθζe1þ¼iθθζ 2ipffiffiffi3þθ 2 11 þ i π θζ i eζiζ 2i ¼2Expanding the square root, we haveζi ’3þθ21=2 21 pffiffiffi1 i π θζ i eζi2ð7:132Þ2547 Kinetic TheoryThe approximate damping rate is found by using Eq. (7.130) in the imaginary term:Im ζ iIm ω π 1=2¼¼θð3 þ θÞ1=2 eð3þθÞ=2Re ω8Re ζ ið7:133Þwhere θ ¼ ZTe/Ti and Re ω is given by Eq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
