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In particular, ω does not depend on k, so the groupvelocity dω/dk is zero. The disturbance does not propagate. How this can happen canbe made clear with a mechanical analogy (Fig. 4.3). Imagine a number of heavy ballssuspended by springs equally spaced in a line. If all the springs are identical, eachball will oscillate vertically with the same frequency.
If the balls are started in theproper phases relative to one another, they can be made to form a wave propagatingin either direction. The frequency will be fixed by the springs, but the wavelengthcan be chosen arbitrarily. The two undisturbed balls at the ends will not be affected,and the initial disturbance does not propagate. Either traveling waves or standingwaves can be created, as in the case of a stretched rope.
Waves on a rope, however,must propagate because each segment is connected to neighboring segments.Fig. 4.3 Synthesis of a wave from an assembly of independent oscillators824 Waves in PlasmasFig. 4.4 Plasma oscillations propagate in a finite medium because of fringing fieldsThis analogy is not quite accurate, because plasma oscillations have motions inthe direction of k rather than transverse to k.
However, as long as electrons do notcollide with ions or with each other, they can still be pictured as independentoscillators moving horizontally (in Fig. 4.3). But what about the electric field?Won’t that extend past the region of initial disturbance and set neighboring layers ofplasma into oscillation? In our simple example, it will not, because the electric fielddue to equal numbers of positive and negative infinite plane charge sheets is zero. Inany finite system, however, plasma oscillations will propagate. In Fig. 4.4, thepositive and negative (shaded) regions of a plane plasma oscillation are confined ina cylindrical tube.
The fringing electric field causes a coupling of the disturbance toadjacent layers, and the oscillation does not stay localized.Problems4.2 The plasma density in the lower ionosphere has been measured during satellitere-entry to be about 1018 m3 at 50 km altitude, 1017 at 70 km, ad 1014 at 85 km.What are the plasma frequencies there?4.3 Calculate the plasma frequency with the ion motions included, thus justifyingour assumption that the ions are essentially fixed. (Hint: include the term n1i inPoisson’s equation and use the ion equations of motion and continuity.)4.4 For a simple plasma oscillation with fixed ions and a space-time behavior exp[i(kx ωt)], calculate the phase δ for ϕ1, E1, and v1 if the phase of n1, is zero.Illustrate the relative phases by drawing sine waves representing n1, ϕ1, E1, andv1 (a) as a function of x at t ¼ 0, (b) as a function of t at x ¼ 0 for ω/k > 0, and(c) as a function of t at x ¼ 0 for ω/k < 0.
Note that the time patterns can beobtained by translating the x patterns in the proper direction, as if the wavewere passing by a fixed observer.4.5 By writing the linearized Poisson’s equation used in the derivation of simpleplasma oscillations in the form∇ ðεEÞ ¼ 0derive an expression for the dielectric constant ε applicable to high-frequencylongitudinal motions.4.4 Electron Plasma Waves4.483Electron Plasma WavesThere is another effect that can cause plasma oscillations to propagate, and that isthermal motion. Electrons streaming into adjacent layers of plasma with theirthermal velocities will carry information about what is happening in the oscillatingregion.
The plasma oscillation can then properly be called a plasma wave. We caneasily treat this effect by adding a term ∇pe to the equation of motion Eq. (4.12).In the one-dimensional problem, γ will be three, according to Eq. (3.53). Hence,∇ pe ¼ 3KT e ∇ne ¼ 3KT e ∇ ðn0 þ n1 Þ ¼ 3KT e∂n1^x∂xand the linearized equation of motion ismn0∂v1∂n1¼ en0 E1 3KT e∂t∂xð4:28ÞNote that in linearizing we have neglected the terms n1 ∂v1/∂t and n1E1 as wellas the (v1 · ∇)v1 term. With Eq.
(4.20), Eq. (4.28) becomesimωn0 v1 ¼ en0 E1 3KT e ikn1ð4:29ÞE1 and n1 are still given by Eqs. (4.23) and (4.22), and we have en0 ikv1þ 3KT e ikimωn0 v1 ¼ en0ikε0iω 2n0 e3KT e 2k v1þω2 v 1 ¼mε0 m3ω2 ¼ ω2p þ k2 v2th2ð4:30Þwhere v2th 2KT e =m: The frequency now depends on k, and the group velocityis finite:32ω dω ¼ v2th 2k dk2vg ¼dω 3 k 23 v2th¼vth ¼dk 2 ω2 vϕð4:31ÞThat vg is always less than c can easily be seen from a graph of Eq. (4.30). Figure 4.5is a plot of the dispersion relation ω(k) as given by Eq. (4.30).
At any point P onthis curve, the slope of a line drawn from the origin gives the phase velocity ω/k.844 Waves in PlasmasFig. 4.5 Dispersion relation for electron plasma waves (Bohm–Gross waves)The slope of the curve at P gives the group velocity. This is clearly always less than(3/2)1/2 vth, which, in our nonrelativistic theory, is much less than c. Note that atlarge k (small λ), information travels essentially at the thermal velocity.
At smallk (large λ), information travels more slowly than vth even though vϕ is greater thanvth. This is because the density gradient is small at large λ, and thermal motionscarry very little net momentum into adjacent layers.The existence of plasma oscillations has been known since the days of Langmuirin the 1920s. It was not until 1949 that Bohm and Gross worked out a detailedtheory telling how the waves would propagate and how they could be excited.A simple way to excite plasma waves would be to apply an oscillating potential to agrid or a series of grids in a plasma; however, oscillators in the GHz range were notgenerally available in those days. Instead, one had to use an electron beam to exciteplasma waves. If the electrons in the beam were bunched so that they passed by anyfixed point at a frequency fp, they would generate an electric field at that frequencyand excite plasma oscillations. It is not necessary to form the electron bunchesbeforehand; once the plasma oscillations arise, they will bunch the electrons, andthe oscillations will grow by a positive feedback mechanism.
An experiment to testthis theory was first performed by Looney and Brown in 1954. Their apparatus wasentirely contained in a glass bulb about 10 cm in diameter (Fig. 4.6). A plasmafilling the bulb was formed by an electrical discharge between the cathodes K andan anode ring A under a low pressure (3 103 Torr) of mercury vapor. An electronbeam was created in a side arm containing a negatively biased filament. The emittedelectrons were accelerated to 200 V and shot into the plasma through a small hole.A thin, movable probe wire connected to a radio receiver was used to pick up theoscillations. Figure 4.7 shows their experimental results for f2 vs.
discharge current,which is generally proportional to density. The points show a linear dependence, inrough agreement with Eq. (4.26). Deviations from the straight line could beattributed to the k2v2th term in Eq. (4.30). However, not all frequencies wereobserved; k had to be such that an integral number of half wavelengths fit along4.4 Electron Plasma Waves85Fig. 4.6 Schematic of the Looney–Brown experiment on plasma oscillationsFig. 4.7 Square of theobserved frequencyvs.
plasma density, whichis generally proportionalto the discharge current.The inset shows theobserved spatial distributionof oscillation intensity,indicating the existenceof a different standing wavepattern for each of thegroups of experimentalpoints. [From D. H.
Looneyand S. C. Brown, Phys. Rev.93, 965 (1954).]the plasma column. The standing wave patterns are shown at the left of Fig. 4.7. Thepredicted traveling plasma waves could not be seen in this experiment, probablybecause the beam was so thin that thermal motions carried electrons out of thebeam, thus dissipating the oscillation energy. The electron bunching was accomplished not in the plasma but in the oscillating sheaths at the ends of the plasmacolumn. In this early experiment, one learned that reproducing the conditionsassumed in the uniform plasma theory requires considerable skill.A number of recent experiments have verified the Bohm–Gross dispersionrelation, Eq.
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