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(Hint: From the equation of motion,derive an expression for vx/vy in terms of ω/ωc.)4.9 Find the dispersion relation for electrostatic electron waves propagating at anarbitrary angle θ relative to B0. Hint: Choose the x axis so that k and E lie in thex–z plane (Fig. P4.9).
Then1004 Waves in PlasmasFig. 4.21 The Trivelpiece–Gould dispersion curves forelectrostatic electron wavesin a conducting cylinderfilled with a uniform plasmaand a coaxial magneticfield. [From A. W.Trivelpiece and R. W.Gould, J. Appl. Phys. 30,1784 (1959).]Ex ¼ E1 sin θ,Ez ¼ E1 cos θ,Ey ¼ 0and similarly for k.
Solve the equations of motion and continuity and Poisson’sequation in the usual way with n0 uniform and v0 ¼ E0 ¼ 0.Fig. P4.94.9 Electrostatic Electron Oscillations Perpendicular to B101Fig. 4.22 Experimentalverification of theTrivelpiece –Gould curves,showing the existence ofbackward waves; that is,waves whose groupvelocity, as indicated by theslope of the dispersioncurve, is opposite indirection to the phasevelocity. [From Trivelpieceand Gould, loc.
cit.](a) Show that the answer isω2 ω2 ω2h þ ω2c ω2p cos 2 θ ¼ 0(b) Write out the two solutions of this quadratic for ω2, and show that in thelimits θ ! 0 and θ ! π/2, our previous results are recovered. Show that inthese limits, one of the two solutions is a spurious root with no physicalmeaning.(c) By completing the square, show that the above equation is the equation ofan ellipse:ð y 1Þ 2 x 2þ 2¼1a12wherex cos θ, y 2ω2 =ω2h , and a ω2h =2ωc ω p :1024 Waves in Plasmas(d) Plot the ellipse for ωp/ωc ¼ 1, 2, and 1.(e) Show that if ωc > ωp, the lower root for ω is always less than ωp for anyθ > 0 and the upper root always lies between ωc and ωh; and that ifωp > ωc, the lower root lies below ωc while the upper root is between ωpand ωh.4.10Electrostatic Ion Waves Perpendicular to BWe next consider what happens to the ion acoustic wave when k is perpendicular toB0. It is tempting to set k · B0 exactly equal to zero, but this would lead to a result(Sect.
4.11) which, though mathematically correct, does not describe what usuallyhappens in real plasmas. Instead, we shall let k be almost perpendicular to B0; whatwe mean by “almost” will be made clear later. We shall assume the usual infiniteplasma in equilibrium, with n0 and B0 constant and uniform and v0 ¼ E0 ¼ 0. Forsimplicity, we shall take Ti ¼ 0; we shall not miss any important effects because weknow that acoustic waves still exist if Ti ¼ 0. We also assume electrostatic waveswith k E ¼ 0, so that E ¼ ∇ϕ.
The geometry is shown in Fig. 4.23. The angle½π θ is taken to be so small that we may take E ¼ E1 ^x and∇ ¼ ik^x as far as theion motion is concerned. For the electrons, however, it makes a great deal ofdifference whether ½π θ is zero, or small but finite. The electrons have suchsmall Larmor radii that they cannot move in the x direction to preserve chargeneutrality; all that the E field does is make them drift back and forth in they direction. If θ is not exactly π/2, however, the electrons can move along thedashed line (along B0) in Fig. 4.23 to carry charge from negative to positive regionsin the wave and carry out Debye shielding.
The ions cannot do this effectivelybecause their inertia prevents them from moving such long distances in a waveperiod; this is why we can neglect kz for ions. The critical angle χ ¼ ½π θ isproportional to the ratio of ion to electron parallel velocities: χ ’ (m/M )1/2Fig. 4.23 Geometry of anelectrostatic ion cyclotronwave propagating nearly atright angles to B04.10Electrostatic Ion Waves Perpendicular to B103(in radians). For angles χ larger than this, the following treatment is valid. Forangles χ smaller than this, the treatment of Sect. 4.11 is valid.After this lengthy introduction, we proceed to the brief derivation of the result.For the ion equation of motion, we haveM∂vi1¼ e∇ϕ1 þ evi1 B0∂tð4:61ÞAssuming plane waves propagating in the x direction and separating into components, we haveiωMvix ¼ eikϕ1 þ eviy B0iωMviy ¼ evix B0ð4:62ÞSolving as before, we findvix ¼1ekΩ2ϕ1 1 2cMωωð4:63Þwhere Ωc ¼ eB0/M is the ion cyclotron frequency.
The ion equation of continuityyields, as usual,ni1 ¼ n0kvixωð4:64ÞAssuming the electrons can move along B0 because of the finiteness of the angle χ,we can use the Boltzmann relation for electrons. In linearized form, this isne1 eϕ1¼KT en0ð4:65ÞThe plasma approximation ni ¼ ne now closes the system of equations.
With thehelp of Eqs. (4.64) and (4.65), we can write Eq. (4.63) asΩ2ek KT e n0 kvix1 2c vix ¼Mω en0 ωωω2 Ω2c ¼ k2KT eMð4:66ÞSince we have taken KTi ¼ 0, we can write this asω2 ¼ Ω2c þ k2 v2sThis is the dispersion relation for electrostatic ion cyclotron waves.ð4:67Þ1044 Waves in PlasmasFig. 4.24 Schematic of a Q-machine experiment on electrostatic ion cyclotron waves.[After R. W. Motley and N. D’Angelo, Phys. Fluids 6, 296 (1963).]Fig. 4.25 Measurements offrequency of electrostaticion cyclotron wavesvs.
magnetic field.[From Motley andD’Angelo, loc. cit.]The physical explanation of these waves is very similar to that in Fig. 4.19 forupper hybrid waves. The ions undergo an acoustic-type oscillation, but the Lorentzforce constitutes a new restoring force giving rise to the Ω2c term in Eq. (4.67). Theacoustic dispersion relation ω2 ¼ k2 v2s is valid if the electrons provide Debyeshielding. In this case, they do so by flowing long distances along B0.Electrostatic ion cyclotron waves were first observed by Motley and D’Angelo,again in a Q-machine (Fig. 4.24). The waves propagated radially outward across themagnetic field and were excited by a current drawn along the axis to a smallauxiliary electrode.
The reason for excitation is rather complicated and will notbe given here. Figure 4.25 gives their results for the wave frequency vs. magneticfield. In this experiment, the k2v2s term was small compared to the Ω2c term, and themeasured frequencies lay only slightly above Ωc.4.11The Lower Hybrid FrequencyWe now consider what happens when θ is exactly π/2, and the electrons are notallowed to preserve charge neutrality by flowing along the lines of force. Instead ofobeying Boltzmann’s relation, they will obey the full equation of motion,4.11The Lower Hybrid Frequency105Eq. (3.62).
If we keep the electron mass finite, this equation is nontrivial even if weassume Te ¼ 0 and drop the ∇ pe term: hence, we shall do so in the interest ofsimplicity. The ion equation of motion is unchanged from Eq. (4.63):1ekΩ2cϕ 1 2vix ¼Mω 1ωð4:68ÞBy changing e to e, M to m, and Ωc to ωc in Eq. (4.68), we can write down theresult of solving Eq. (3.62) for electrons, with Te ¼ 0:vex ¼ 1ekω2ϕ1 1 c2mωωð4:69ÞThe equations of continuity givekni1 ¼ n0 vi1ωkne1 ¼ n0 ve1ωð4:70ÞThe plasma approximation ni ¼ ne then requires vi1 ¼ ve1. Setting Eqs. (4.68) and(4.69) equal to each other, we haveΩ2cω2cM 1 2 ¼ m 1 2ωωω2 ðM þ mÞ ¼ mω2c þ MΩ2c ¼ e2 B2ω2 ¼11þm Me 2 B2¼ Ω c ωcMmω ¼ ðΩc ωc Þ1=2 ωlð4:71ÞThis is called the lower hybrid frequency. These oscillations can be observed only ifθ is very close to π/2.
If we had used Poisson’s equation instead of the plasmaapproximation, we would have obtained111¼þ 22ω c Ωc Ω pωlð4:71aÞIn very low-density plasmas the latter term actually dominates because the plasmaapproximation is not valid when the Debye length is not negligibly small.1064.124 Waves in PlasmasElectromagnetic Waves with B0 ¼ 0Next in the order of complexity come waves with B1 6¼ 0. These are transverseelectromagnetic waves—light waves or radio waves traveling through a plasma.We begin with a brief review of light waves in a vacuum. The relevant Maxwellequations are∇ E1 ¼ B_ 1ð4:72Þc2 ∇ B1 ¼ E_ 1ð4:73Þsince in a vacuum j ¼ 0 and ε0μ0 ¼ c2. Taking the curl of Eq.
(4.73) and substituting into the time derivative of Eq. (4.72), we have€1c2 ∇ ð∇ B1 Þ ¼ ∇ E_ 1 ¼ Bð4:74ÞAgain assuming planes waves varying as exp [i(kx ωt)], we haveω2 B1 ¼ c2 k ðk B1 Þ ¼ c2 kðk B1 Þ k2 B1ð4:75Þω2 ¼ k 2 c 2ð4:76ÞSince k · B1 ¼ i∇ · B1 ¼ 0 by another of Maxwell’s equations, the result isand c is the phase velocity ω/k of light waves.In a plasma with B0 ¼ 0, Eq. (4.72) is unchanged, but we must add a term j1/E0 toEq.
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