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(4.41).The ions overshoot because of their inertia, and the compressions and rarefactionsare regenerated to form a wave.The second effect mentioned above leads to a curious phenomenon. When KTigoes to zero, ion waves still exist. This does not happen in a neutral gas (Eq. (4.36)).The acoustic velocity is then given by924 Waves in Plasmasvs ¼ ðKTe =MÞ1=2ð4:42ÞThis is often observed in laboratory plasmas, in which the condition Ti Te is acommon occurrence.
The sound speed υs depends on electron temperature (becausethe electric field is proportional to it) and on ion mass (because the fluid’s inertia isproportional to it).4.7Validity of the Plasma ApproximationIn deriving the velocity of ion waves, we used the neutrality condition ni ¼ ne whileallowing E to be finite. To see what error was engendered in the process, we nowallow ni to differ from ne and use the linearized Poisson equation:ε0 ∇ E1 ¼ ε0 k2 ϕ1 ¼ eðni1 ne1 Þð4:43ÞThe electron density is given by the linearized Boltzmann relation Eq. (4.39):ne1 ¼eϕ1n0KT eð4:44ÞInserting this into Eq. (4.43), we haven0 e 2ε0 ϕ1 k 2 þ¼ eni1ε0 KT eð4:45Þε0 ϕ1 k2 λ2D þ 1 ¼ eni1 λ2DThe ion density is given by the linearized ion continuity equation (4.40):ni1 ¼kn0 vi1ωð4:46ÞInserting Eqs.
(4.45) and (4.46) into the ion equation of motion Eq. (4.38), we findeno ik eλ2Dkn0 vi1þγKTikii22ε 0 1 þ k λDω2k2 n0 e2 ε10 λDω2 ¼þγKTiiM 1 þ k2 λ2DiωMn0 vi1 ¼ω¼kKT e1γ i KT i 1=2þM 1 þ k2 λ2DMð4:47Þð4:48Þ4.8 Comparison of Ion and Electron Waves93This is the same as we obtained previously (Eq. (4.41)) except for the factor 1 þ k2λ2D : Our assumption ni ¼ ne has given rise to an error of order k2 λ2D ¼ ð2πλD =λÞ2 :Since λD is very small in most experiments, the plasma approximation is valideverywhere except in a thin layer, called a sheath (Chap.
8), a few λD’s in thickness,next to a wall.4.8Comparison of Ion and Electron WavesIf we consider these short-wavelength waves by taking k2 λ2D 1; Eq. (4.47)becomesω2 ¼ k 2n0 e 2n0 e 2 Ω2p¼ε0 Mk2 ε0 Mð4:49ÞWe have, for simplicity, also taken the limit Ti ! 0. Here Ωp is the ion plasmafrequency. For high frequencies (short wavelengths) the ion acoustic wave turns intoa constant-frequency wave. There is thus a complementary behavior between electron plasma waves and ion acoustic waves: the former are basically constant frequency, but become constant velocity at large k; the latter are basically constantvelocity, but become constant frequency at large k. This comparison is showngraphically in Fig. 4.13.Experimental verification of the existence of ion waves was first accomplishedby Wong, Motley, and D’Angelo.
Figure 4.14 shows their apparatus, which wasagain a Q-machine. (It is no accident that we have referred to Q-machines so often;careful experimental checks of plasma theory were possible only after schemes tomake quiescent plasmas were discovered.) Waves were launched and detected bygrids inserted into the plasma. Figure 4.15 shows oscilloscope traces of thetransmitted and received signals. From the phase shift, one can find the phasevelocity (same as group velocity in this case).
These phase shifts are plotted asFig. 4.13 Comparison of the dispersion curves for electron plasma waves and ion acoustic waves944 Waves in PlasmasFig. 4.14 Q-machine experiment to detect ion waves. [From N. Rynn and N. D’Angelo, Rev. Sci.Instrum. 31, 1326 (1960).]Fig. 4.15 Oscillograms ofsignals from the driver andreceiver grids, separated bya distance d, showing thedelay indicative of atraveling wave. [From A. Y.Wong, R. W. Motley, andN. D’Angelo, Phys.
Rev.133, A436 (1964).]4.8 Comparison of Ion and Electron Waves95Fig. 4.16 Experimental measurements of delay vs. probe separation at various frequencies of thewave exciter. The slope of the lines gives the phase velocity. [From Wong, Motley, and D’Angelo,loc. cit.]Fig. 4.17 Measured phase velocity of ion waves in potassium and cesium plasmas as a function offrequency.
The different sets of points correspond to different plasma densities. [From Wong,Motley, and D’Angelo, loc. cit.]functions of distance in Fig. 4.16 for a plasma density of 3 1017 m3. The slopesof such lines give the phase velocities plotted in Fig. 4.17 for the two masses andvarious plasma densities n0. The constancy of υ s with ω and n0 is demonstratedexperimentally, and the two sets of points for K and Cs plasmas show the properdependence on M.964.94 Waves in PlasmasElectrostatic Electron Oscillations Perpendicular to BUp to now, we have assumed B ¼ 0. When a magnetic field exists, many more typesof waves are possible. We shall examine only the simplest cases, starting with highfrequency, electrostatic, electron oscillations propagating at right angles to themagnetic field. First, we should define the terms perpendicular, parallel, longitudinal, transverse, electrostatic, and electromagnetic.
Parallel and perpendicular willbe used to denote the direction of k relative to the undisturbed magnetic field B0.Longitudinal and transverse refer to the direction of k relative to the oscillatingelectric field E1. If the oscillating magnetic field B1 is zero, the wave is electrostatic; otherwise, it is electromagnetic. The last two sets of terms are related byMaxwell’s equation∇ E1 ¼ B_ 1ð4:50Þk E1 ¼ ωB1ð4:51ÞorIf a wave is longitudinal, k E1 vanishes, and the wave is also electrostatic. If thewave is transverse, B1 is finite, and the wave is electromagnetic. It is of coursepossible for k to be at an arbitrary angle to B0 or E1; then one would have a mixtureof the principal modes presented here.Coming back to the electron oscillations perpendicular to B0, we shall assumethat the ions are too massive to move at the frequencies involved and form a fixed,uniform background of positive charge.
We shall also neglect thermal motions andset KTe ¼ 0. The equilibrium plasma, as usual, has constant and uniform n0 and B0and zero E0 and v0. The motion of electrons is then governed by the followinglinearized equations:m∂ve1¼ eðE1 þ ve1 B0 Þ∂tð4:52Þ∂ne1þ n0 ∇ ve 1 ¼ 0∂tð4:53Þε0 ∇ E1 ¼ ene1ð4:54ÞWe shall consider only longitudinal waves with k║E1.
Without loss of generality,we can choose the x axis to lie along k and E1, and the z axis to lie along B0(Fig. 4.18). Thus ky ¼ kz ¼ Ey ¼ Ez ¼ 0, k ¼ k^x , and E ¼ E^x : Dropping the subscripts 1 and e and separating Eq. (4.52) into components, we haveiωmvx ¼ eE ev y B0ð4:55Þ4.9 Electrostatic Electron Oscillations Perpendicular to B97Fig. 4.18 Geometry of alongitudinal plane wavepropagating at rightangles to B0iωmv y ¼ þevx B0ð4:56Þ iωmvz ¼ 0Solving for υy in Eq. (4.56) and substituting into Eq.
(4.55), we haveiωmvx ¼ eE þ eB0vx ¼ieB0vxmωeE=imω1 ω2c =ω2ð4:57ÞNote that υx becomes infinite at cyclotron resonance, ω ¼ ωc. This is to be expected,since the electric field changes sign with υ x and continuously accelerates theelectrons. [Note that the fluid and single-particle equations are identical when the(v · ∇)v and ∇p terms are both neglected, so that all the particles move together.]From the linearized form of Eq. (4.53), we haven1 ¼kn 0 vxωð4:58ÞLinearizing Eq.
(4.54) and using the last two results, we haveikε0 E ¼ e1keEω2n01 c2ω imωωω2pω21 c2 E ¼ 2 Eωωð4:59ÞThe dispersion relation is thereforeω2 ¼ ω2p þ ω2c ω2hð4:60Þ984 Waves in PlasmasFig. 4.19 Motion ofelectrons in an upper hybridoscillationThe frequency ωh is called the upper hybrid frequency. Electrostatic electron wavesacross B have this frequency, while those along B are the usual plasma oscillationswith ω ¼ ωp.
The group velocity is again zero as long as thermal motions areneglected.A physical picture of this oscillation is given in Fig. 4.19. Electrons in the planewave form regions of compression and rarefaction, as in a plasma oscillation.However, there is now a B field perpendicular to the motion, and the Lorentzforce turns the trajectories into ellipses. There are two restoring forces acting onthe electrons: the electrostatic field and the Lorentz force. The increased restoringforce makes the frequency larger than that of a plasma oscillation.
As the magneticfield goes to zero, ωc goes to zero in Eq. (4.60), and one recovers a plasmaoscillation. As the plasma density goes to zero, ωp goes to zero, and one has asimple Larmor gyration, since the electrostatic forces vanish with density.The existence of the upper hybrid frequency has been verified experimentally bymicrowave transmission across a magnetic field. As the plasma density is varied,the transmission through the plasma takes a dip at the density that makes ωh equal tothe applied frequency. This is because the upper hybrid oscillations are excited, andenergy is absorbed from the beam. From Eq.
(4.60), we find a linear relationshipbetween ω2c /ω2 and the density:ω2pω2cne2¼1¼1ε0 mω2ω2ω2This linear relation is followed by the experimental points on Fig. 4.20, whereω2c /ω2 is plotted against the discharge current, which is proportional to n.If we now consider propagation at an angle θ to B, we will get two possiblewaves. One is like the plasma oscillation, and the other is like the upper hybridoscillation, but both will be modified by the angle of propagation. The details of this4.9 Electrostatic Electron Oscillations Perpendicular to B99Fig. 4.20 Results of anexperiment to detect theexistence of the upperhybrid frequency bymapping the conditionsfor maximum absorption(minimum transmission)of microwave energy sentacross a magnetic field.The field at which thisoccurs (expressed as ω2c /ω2)is plotted against dischargecurrent (proportional toplasma density).
[FromR. S. Harp, Proceedingsof the Seventh InternationalConference on Phenomenain Ionized Gases, Belgrade,1965, II, 294 (1966).]are left as an exercise (Problem 4.8). Figure 4.21 shows schematically the ω–kzdiagram for these two waves for fixed kx, where kx/kz ¼ tan θ. Because of thesymmetry of Eq. (4.60), the case ωc > ωp is the same as the case ωp > ωc with thesubscripts interchanged. For large kz, the wave travels parallel to B0. One wave isthe plasma oscillation at ω ¼ ωp; the other wave, at ω ¼ ωc, is a spurious root atkz ! 1.
For small kz, we have the situation of k ⊥ B0 discussed in this section. Thelower branch vanishes, while the upper branch approaches the hybrid oscillation atω ¼ ωh. These curves were first calculated by Trivelpiece and Gould, who alsoverified them experimentally (Fig. 4.22). The Trivelpiece–Gould experiment wasdone in a cylindrical plasma column; it can be shown that varying kz in this case isequivalent to propagating plane waves at various angles to B0.Problems4.8 For the upper hybrid oscillation, show that the elliptical orbits (Fig. 4.19) arealways elongated in the direction of k.
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