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(4.30), with precision. As an example of more modern experimental864 Waves in PlasmasFig. 4.8 Schematic of an experiment to measure plasma waves. [From P. J. Barrett, H. G. Jones,and R. N. Franklin, Plasma Physics 10, 911 (1968).]technique, we show the results of Barrett, Jones, and Franklin. Figure 4.8 is aschematic of their apparatus. The cylindrical column of quiescent plasma is produced in a Q-machine by thermal ionization of Cs atoms on hot tungsten plates (notshown). A strong magnetic field restricts electrons to motions along the column.The waves are excited by a wire probe driven by an oscillator and are detected by asecond, movable probe.
A metal shield surrounding the plasma prevents communication between the probes by ordinary microwave (electromagnetic wave) propagation, since the shield constitutes a waveguide beyond cutoff for the frequencyused. The traveling waveforms are traced by interferometry: the transmitted andreceived signals are detected by a crystal which gives a large dc output when thesignals are in phase and zero output when they are 90 out of phase. The resultingsignal is shown in Fig.
4.9 as a function of position along the column. Synchronousdetection is used to suppress the noise level. The excitation signal is chopped at500 kHz, and the received signal should also be modulated at 500 kHz. By detectingonly the 500-kHz component of the received signal, noise at other frequencies iseliminated. The traces of Fig.
4.9 give a measurement of k. When the oscillatorfrequency ω is varied, a plot of the dispersion curve (ω/ωp)2 vs. ka is obtained,where a is the radius of the column (Fig. 4.10). The various curves are labeledaccording to the value of ωpa/vth. For vth ¼ 0, we have the curve labeled 1, whichcorresponds to the dispersion relation ω ¼ ωp. For finite vth, the curves correspond4.4 Electron Plasma Waves87Fig. 4.9 Spatial variation of the perturbed density in a plasma wave, as indicated by an interferometer, which multiplies the instantaneous density signals from two probes and takes the timeaverage.
The interferometer is tuned to the wave frequency, which varies with the density. Theapparent damping at low densities is caused by noise in the plasma. [From Barrett, Jones, andFranklin, loc. cit.]to that of Fig. 4.5. There is good agreement between the experimental points and thetheoretical curves. The decrease of ω at small ka is the finite-geometry effect shownin Fig. 4.4. In this particular experiment, that effect can be explained another way.To satisfy the boundary condition imposed by the conducting shield, namely thatE ¼ 0 on the conductor, the plasma waves must travel at an angle to the magneticfield. Destructive interference between waves traveling with an outward radialcomponent of k and those traveling inward enables the boundary condition to besatisfied.
However, waves traveling at an angle to B have crests and troughsseparated by a distance larger than λ/2 (Fig. 4.11). Since the electrons can moveonly along B (if B is very large), they are subject to less acceleration, and thefrequency is lowered below ωp.Problems4.6 Electron plasma waves are propagated in a uniform plasma with KTe ¼ 100 eV,n ¼ 1016 m3, and B ¼ 0. If the frequency f is 1.1 GHz, what is the wavelengthin cm?884 Waves in PlasmasFig. 4.10 Comparison of the measured and calculated dispersion curves for electron plasmawaves in a cylinder of radius a.
[From Barrett, Jones, and Franklin, loc. cit.]Fig. 4.11 Wavefronts traveling at an angle to the magnetic field are separated, in the fielddirection, by a distance larger than the wavelength λ4.7 (a) Compute the effect of collisional damping on the propagation of Langmuirwaves (plasma oscillations), by adding a term mnνv to the electronequation of motion and rederiving the dispersion relation for Te ¼ 0. Heren is the electron collision frequency with ions and neutral atoms.(b) Write an explicit expression for Im (ω) and show that its sign indicates thatthe wave is damped in time.4.5 Sound Waves4.589Sound WavesAs an introduction to ion waves, let us briefly review the theory of sound waves inordinary air.
Neglecting viscosity, we can write the Navier–Stokes equation (3.48),which describes these waves, asρ∂vγpþ ðv ∇ Þv ¼ ∇ p ¼ ∇ρ∂tρð4:32ÞThe equation of continuity is∂ρþ ∇ ðρvÞ ¼ 0∂tð4:33ÞLinearizing about a stationary equilibrium with uniform p0 and ρ0, we haveγ p0ikρ1ρ0ð4:34Þiωρ1 þ ρ0 ik v1 ¼ 0ð4:35Þiωρ0 v1 ¼ where we have again taken a wave dependence of the formexp½iðk r ωtÞFor a plane wave with k ¼ k^x and v ¼ v^x ; we find, upon eliminating ρ1,γ p0 ρ0 ikv1ikρ0iωγpω2 v1 ¼ k2 0 v1ρ0iωρ0 v1 ¼ orω¼kγ p0 1=2γKT 1=2 cs¼Mρ0ð4:36ÞThis is the expression for the velocity cs of sound waves in a neutral gas.
Thewaves are pressure waves propagating from one layer to the next by collisionsamong the air molecules. In a plasma with no neutrals and few collisions,an analogous phenomenon occurs. This is called an ion acoustic wave, or, simply,an ion wave.904.64 Waves in PlasmasIon WavesIn the absence of collisions, ordinary sound waves would not occur. Ions can stilltransmit vibrations to each other because of their charge, however; and acoustic wavescan occur through the intermediary of an electric field. Since the motion of massiveions will be involved, these will be low-frequency oscillations, and we can use theplasma approximation of Sect.
3.6. We therefore assume ni ¼ ne ¼ n and do not usePoisson’s equation. The ion fluid equation in the absence of a magnetic field isMn∂viþ ðvi ∇Þvi ¼ enE ∇ p ¼ en∇ϕγ i KTi ∇n∂tð4:37ÞWe have assumed E ¼ ∇ ϕ and used the equation of state. Linearizing andassuming plane waves, we haveiωMn0 vi1 ¼ en0 ikϕ1 γ i KTi ikn1ð4:38ÞAs for the electrons, we may assume m ¼ 0 and apply the argument of Sect. 3.5,regarding motions along B, to the present case of B ¼ 0.
The balance of forces onelectrons, therefore, requireseϕ1ne ¼ n ¼ n0 expKT e¼ n0eϕ1 þ 1 þ KT eThe perturbation in density of electrons, and, therefore, of ions, is thenn1 ¼ n0eϕ1KT eð4:39ÞHere the n0 of Boltzmann’s relation also stands for the density in the equilibriumplasma, in which we can choose ϕ0 ¼ 0 because we have assumed E0 ¼ 0. Inlinearizing Eq. (4.39), we have dropped the higher-order terms in the Taylorexpansion of the exponential.The only other equation needed is the linearized ion equation of continuity.From Eq.
(4.22), we haveiωn1 ¼ n0 ikvi1ð4:40ÞIn Eq. (4.38), we may substitute for ϕ1 and n1 in terms of vi1 from Eqs. (4.39) and(4.40) and obtainKT en0 ikvi1þ γi KTi ikiωMn0 vi1 ¼ en0 iken0iωKT e γi KTiþω2 ¼ k 2MM4.6 Ion Waves91Fig. 4.12 Dispersionrelation for ion acousticwaves in the limit of smallDebye lengthω¼kKT e þ γi KTi 1=2 vsMð4:41ÞThis is the dispersion relation for ion acoustic waves; υ s is the sound speed in aplasma. Since the ions suffer one-dimensional compressions in the plane waves wehave assumed, we may set γ i ¼ 3 here. The electrons move so fast relative to thesewaves that they have time to equalize their temperature everywhere; therefore, theelectrons are isothermal, and γ e ¼ 1.
Otherwise, a factor γ e would appear in front ofKTe in Eq. (4.41).The dispersion curve for ion waves (Fig. 4.12) has a fundamentally differentcharacter from that for electron waves (Fig. 4.5). Plasma oscillations are basicallyconstant-frequency waves, with a correction due to thermal motions.
Ion waves arebasically constant-velocity waves and exist only when there are thermal motions.For ion waves, the group velocity is equal to the phase velocity. The reasons for thisdifference can be seen from the following description of the physical mechanismsinvolved. In electron plasma oscillations, the other species (namely, ions) remainsessentially fixed. In ion acoustic waves, the other species (namely, electrons) is farfrom fixed; in fact, electrons are pulled along with the ions and tend to shield outelectric fields arising from the bunching of ions. However, this shielding is notperfect because, as we saw in Sect.
1.4, potentials of the order of KTe/e can leak outbecause of electron thermal motions. What happens is as follows. The ions formregions of compression and rarefaction, just as in an ordinary sound wave. Thecompressed regions tend to expand into the rarefactions, for two reasons. First, theion thermal motions spread out the ions; this effect gives rise to the second term inthe square root of Eq. (4.41). Second, the ion bunches are positively charged andtend to disperse because of the resulting electric field. This field is largely shieldedout by electrons, and only a fraction, proportional to KTe, is available to act on theion bunches. This effect gives rise to the first term in the square root of Eq.
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