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(4.73) to account for currents due to first-order charged particle motions:c2 ∇ B1 ¼j1 _þ E1ε0ð4:77ÞThe time derivative of this is1 ∂ j1 €c2 ∇ B_ 1 ¼þ E1ε0 ∂tð4:78Þwhile the curl of Eq. (4.72) is∇ ð∇ E1 Þ ¼ ∇ ð∇ E1 Þ ∇2 E1 ¼ ∇ B_ 1ð4:79ÞEliminating ∇ B_ 1 and assuming an exp [i(k · r ωt)] dependence, we havekðk E1 Þ þ k2 E1 ¼iωω2E1jþε0 c 2 1 c 2ð4:80Þ4.12Electromagnetic Waves with B0 ¼ 0107By transverse waves we mean k · E1 ¼ 0, so this becomesω2 c2 k2 E1 ¼ iω j1 =ε0ð4:81Þj1 ¼ n0 eve1ð4:82ÞIf we consider light waves or microwaves, these will be of such high frequency thatthe ions can be considered as fixed.
The current j1 then comes entirely from electronmotion:From the linearized electron equation of motion, we have (for KTe ¼ 0):m∂ve1¼ eE∂teE1ve1 ¼imωð4:83ÞEquation (4.81) now can be writteniωeE1 n0 e2ω2 c2 k2 E1 ¼ n0 e¼E1ε0imω ε0 mð4:84ÞThe expression for ω2p is recognizable on the right-hand side, and the result isω2 ¼ ω2p þ c2 k2ð4:85ÞThis is the dispersion relation for electromagnetic waves propagating in a plasmawith no dc magnetic field. We see that the vacuum relation Eq. (4.76) is modified bya term ω2p reminiscent of plasma oscillations.
The phase velocity of a light wave in aplasma is greater than the velocity of light:v2ϕ ¼ω2pω22> c2¼cþk2k2ð4:86ÞHowever, the group velocity cannot exceed the velocity of light. From Eq. (4.85),we finddωc2¼ vg ¼dkvϕð4:87Þso that vg is less than c whenever vϕ is greater than c. The dispersion relationEq. (4.85) is shown in Fig. 4.26. This diagram resembles that of Fig. 4.5 for plasmawaves, but the dispersion relation is really quite different because the asymptoticvelocity c in Fig. 4.26 is so much larger than the thermal velocity vth in Fig.
4.5.1084 Waves in PlasmasFig. 4.26 Dispersionrelation for electromagneticwaves in a plasma with nodc magnetic fieldMore importantly, there is a difference in damping of the waves. Plasma waves withlarge kvth are highly damped, a result we shall obtain from kinetic theory in Chap. 7.Electromagnetic waves, on the other hand, become ordinary light waves at large kcand are not damped by the presence of the plasma in this limit.A dispersion relation like Eq. (4.85) exhibits a phenomenon called cutoff.
If onesends a microwave beam with a given frequency ω through a plasma, the wavelength 2π/k in the plasma will take on the value prescribed by Eq. (4.85). As theplasma density, and hence ω2p is raised, k2 will necessarily decrease; and thewavelength becomes longer and longer. Finally, a density will be reached suchthat k2 is zero.
For densities larger than this, Eq. (4.85) cannot be satisfied for anyreal k, and the wave cannot propagate. This cutoff condition occurs at a criticaldensity nc such that ω ¼ ωp; namely (from Eq. (4.25))nc ¼ mε0 ω2 =e2ð4:88ÞIf n is too large or ω too small, an electromagnetic wave cannot pass through aplasma. When this happens, Eq. (4.85) tells us that k is imaginary:1=2ck ¼ ω2 ω2p¼ iω2p ω2 1=2ð4:89ÞSince the wave has a spatial dependence exp(ikx), it will be exponentially attenuated if k is imaginary.
The skin depth δ is found as follows:eikx ¼ ejkjx ¼ ex=δδ ¼ jkj1 ¼ cω2p ω21=2ð4:90ÞFor most laboratory plasmas, the cutoff frequency lies in the microwave range.4.134.13Experimental Applications109Experimental ApplicationsThe phenomenon of cutoff suggests an easy way to measure plasma density. Abeam of microwaves generated by a klystron is launched toward the plasma by ahorn antenna (Fig. 4.27). The transmitted beam is collected by another horn and isdetected by a crystal. As the frequency or the plasma density is varied, the detectedsignal will disappear whenever the condition Eq. (4.88) is satisfied somewhere inthe plasma.
This procedure gives the maximum density. It is not a convenient orversatile scheme because the range of frequencies generated by a single microwavegenerator is limited.A widely used method of density measurement relies on the dispersion, orvariation of index of refraction, predicted by Eq. (4.85). The index of refraction n~is defined asen c=vϕ ¼ ck=ωð4:91ÞThis clearly varies with ω, and a plasma is a dispersive medium.
A microwaveinterferometer employing the same physical principles as the Michelson interferometer is used to measure density (Fig. 4.28). The signal from a klystron is split intoFig. 4.27 Microwave measurement of plasma density by the cutoff of the transmitted signalFig. 4.28 A microwave interferometer for plasma density measurement1104 Waves in PlasmasFig. 4.29 The observed signal from an interferometer (right) as plasma density is increased, andthe corresponding wave patterns in the plasma (left)Fig. 4.30 A plasma lenshas unusual opticalproperties, since theindex of refraction isless than unitytwo paths. Part of the signal goes to the detector through the “reference leg.” Theother part is sent through the plasma with horn antennas.
The detector responds tothe mean square of the sum of the amplitudes of the two received signals. Thesesignals are adjusted to be equal in amplitude and 180 out of phase in the absence ofplasma by the attenuator and phase shifter, so that the detector output is zero. Whenthe plasma is turned on, the phase of the signal in the plasma leg is changed as thewavelength increases (Fig. 4.29). The detector then gives a finite output signal. Asthe density increases, the detector output goes through a maximum and a minimumevery time the phase shift changes by 360 .
The average density in the plasma isfound from the number of such fringe shifts. Actually, one usually uses a highenough frequency that the fringe shift is kept small. Then the density is linearlyproportional to the fringe shift (Problem 4.13). The sensitivity of this technique atlow densities is limited to the stability of the reference leg against vibrations andthermal expansion. Corrections must also be made for attenuation due to collisionsand for diffraction and refraction by the finite-sized plasma.The fact that the index of refraction is less than unity for a plasma has someinteresting consequences.
A convex plasma lens (Fig. 4.30) is divergent rather4.13Experimental Applications111Fig. 4.31 A plasma confined in a long, linear solenoid will trap the CO2 laser light used to heat itonly if the plasma has a density minimum on axis. The vacuum chamber has been omitted forclaritythan convergent.
This effect is important in the laser-solenoid proposal for a linearfusion reactor. A plasma hundreds of meters long is confined by a strong magneticfield and heated by absorption of CO2 laser radiation (Fig. 4.31). If the plasma has anormal density profile (maximum on the axis), it behaves like a negative lens andcauses the laser beam to diverge into the walls. If an inverted density profile(minimum on the axis) can be created, however, the lens effect becomes converging; and the radiation is focused and trapped by the plasma. The inverted profile canbe produced by squeezing the plasma with a pulsed coil surrounding it, or it can beproduced by the laser beam itself.
As the beam heats the plasma, the latter expands,decreasing the density at the center of the beam. The CO2 laser operates atλ ¼ 10.6 μm, corresponding to a frequencyf ¼c3 108¼¼ 2:8 1013 Hzλ 10:6 106The critical density is, from Eq. (4.88),nc ¼ mε0 ð2π f Þ2 =e2 ¼ 1025 m3However, because of the long path lengths involved, the refraction effects areimportant even at densities of 1022 m3. The focusing effect of a hollow plasmahas been shown experimentally.Perhaps the best known effect of the plasma cutoff is the application to shortwave radio communication.
When a radio wave reaches an altitude in the ionosphere where the plasma density is sufficiently high, the wave is reflected(Fig. 4.32), making it possible to send signals around the earth. If we take themaximum density to be 1012 m3, the critical frequency is of the order of 10 MHz(cf. Eq. (4.26)). To communicate with space vehicles, it is necessary to usefrequencies above this in order to penetrate the ionosphere. However, duringreentry of a space vehicle, a plasma is generated by the intense heat of friction.This causes a plasma cutoff, resulting in a communications blackout during reentry(Fig. 4.32).1124 Waves in PlasmasFig.
4.32 Exaggeratedview of the earth’sionosphere, illustrating theeffect of plasma on radiocommunicationsProblems4.10. A space capsule making a reentry into the earth’s atmosphere suffers acommunications blackout because a plasma is generated by the shock wavein front of the capsule.
If the radio operates at a frequency of 300 MHz, whatis the minimum plasma density during the blackout?4.11. Hannes Alfvén, the first plasma physicist to be awarded the Nobel prize, hassuggested that perhaps the primordial universe was symmetric betweenmatter and antimatter. Suppose the universe was at one time a uniformmixture of protons, antiprotons, electrons, and positrons, each species havinga density n0.(a) Work out the dispersion relation for high-frequency electromagneticwaves in this plasma. You may neglect collisions, annihilations, andthermal effects.(b) Work out the dispersion relation for ion waves, using Poisson’s equation.
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