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It is immersed in a 1-eV atomic hydrogenplasma at density 106 m3. During solar storms the satellite is bombarded byenergetic electrons, which charge it to a potential of 2 kV. Calculate the fluxof energetic ions bombarding each m2 of the panels.8.3 The sheath criterion of Eq. (8.11) was derived for a cold-ion plasma. Supposethe ion distribution had a thermal spread in velocity around an average driftspeed u0. Without mathematics, indicate whether you would expect the value ofu0 to be above or below the Bohm value, and explain why.8.4 An ion velocity analyzer consists of a stainless steel cylinder 5 mm indiameter with one end covered with a fine tungsten mesh grid (grid 1). Behindthis, inside the cylinder, are a series of insulated, parallel grids.
Grid 1 is at“floating” potential—it is not electrically connected. Grid 2 is biased negativeto repel all electrons coming through grid 1, but it transmits ions. Grid 3 is theanalyzer grid, biased so as to decelerate ions accelerated by grid 2. Those ionsable to pass through grid 3 are all collected by a collector plate.
Grid 4 is asuppressor grid that turns back secondary electrons emitted by the collector. Ifthe plasma density is too high, a space charge problem occurs near grid3 because the ion density is so large that a potential hill forms in frontof grid 3 and repels ions which would otherwise reach grid 3. Using theChild–Langmuir law, estimate the maximum meaningful He+ current that canbe measured on a 4-mm-diam collector if grids 2 and 3 are separated by 1 mmand 100 V.8.3Ion Acoustic Shock WavesWhen a jet travels faster than sound, it creates a shock wave. This is a basicallynonlinear phenomenon, since there is no period when the wave is small andgrowing. The jet is faster than the speed of waves in air, so the undisturbedmedium cannot be “warned” by precursor signals before the large shock wavehits it.
In hydrodynamic shock waves, collisions are dominant. Shock waves alsoexist in plasmas, even when there are no collisions. A magnetic shock, the “bowshock,” is generated by the earth as it plows through the interplanetary plasmawhile dragging along a dipole magnetic field. We shall discuss a simpler example:a collisionless, one-dimensional shock wave which develops from a largeamplitude ion wave.8.3 Ion Acoustic Shock Waves8.3.1277The Sagdeev PotentialFigure 8.5 shows the idealized potential profile of an ion acoustic shock wave. Thereason for this shape will be given presently. The wave is traveling to the left with avelocity u0.
If we go to the frame moving with the wave, the function ϕ(x) will beconstant in time, and we will see a stream of plasma impinging on the wave fromthe left with a velocity u0. For simplicity, let Ti be zero, so that all the ions areincident with the same velocity u0, and let the electrons be Maxwellian. Since theshock moves much more slowly than the electron thermal speed, the shift inthe center velocity of the Maxwellian can be neglected. The velocity of the ionsin the shock wave is, from energy conservation,2eϕ 1=22u ¼ u0 Mð8:23ÞIf n0 is the density of the undisturbed plasma, the ion density in the shock isni ¼n0 u02eϕ 1=2¼ n0 1 uMu20ð8:24ÞThe electron density is given by the Boltzmann relation. Poisson’s equation thengives" # d2 ϕeϕ2eϕ 1=2ε0 2 ¼ eðne ni Þ ¼ en0 exp 1dxKT eMu20ð8:25ÞThis is, of course, the same equation (Eq.
(8.6)) as we had for a sheath. A shockwave is no more than a sheath moving through a plasma. We now introduce thedimensionless variablesFig. 8.5 Typical potentialdistribution in an ionacoustic shock wave. Thewave moves to the left, sothat in the wave frame ionsstream into the wave fromthe left with velocity u02788 Nonlinear Effectsχ þeϕKT eξxλDMu0ðKT e =MÞ1=2ð8:26ÞNote that we have changed the sign in the definition of χ so as to keep χ positive inthis problem as well as in the sheath problem.
The quantity M is called the Machnumber of the shock. Equation (8.25) can now be writtend2 χ2χ 1=2dV ðχ Þχ¼e 1 2dχMdξ2ð8:27Þwhich differs from the sheath equation (8.8) only because of the change in sign of χ.The behavior of the solution of Eq. (8.27) was made clear by R. Z. Sagdeev, whoused an analogy to an oscillator in a potential well. The displacement x of anoscillator subjected to a force m · dV(x)/dx is given byd 2 x=dt2 ¼ dV=dxð8:28ÞIf the right-hand side of Eq. (8.27) is defined as dV/dχ, the equation is the sameas that of an oscillator, with the potential χ playing the role of x, and d/dξ replacingd/dt.
The quasipotential V(χ) is sometimes called the Sagdeev potential. Thefunction V( χ) can be found from Eq. (8.27) by integration with the boundarycondition V( χ) ¼ 0 at χ ¼ 0:χV ðχ Þ ¼ 1 e þ M2"2χ1 1 2M1=2 #ð8:29ÞFor M lying in a certain range, this function has the shape shown in Fig. 8.6. If thiswere a real well, a particle entering from the left will go to the right-hand side of theFig. 8.6 The Sagdeev potential V(χ). The upper arrow is the trajectory of a quasiparticledescribing a soliton: it is reflected at the right and returns. The lower arrows show the motion ofa quasiparticle that has lost energy and is trapped in the potential well.
The bouncing back andforth describes the oscillations behind a shock front8.3 Ion Acoustic Shock Waves279Fig. 8.7 The potential in asoliton moving to the leftwell (x > 0), reflect, and return to x ¼ 0, making a single transit. Similarly, aquasiparticle in our analogy will make a single excursion to positive χ and returnto χ ¼ 0, as shown in Fig.
8.7. Such a pulse is called a soliton; it is a potential anddensity disturbance propagating to the left in Fig. 8.7 with velocity u0.Now, if a particle suffers a loss of energy while in the well, it will never return tox ¼ 0 but will oscillate (in time) about some positive value of x.
Similarly, a littledissipation will make the potential of a shock wave oscillate (in space) about somepositive value of ϕ. This is exactly the behavior depicted in Fig. 8.5. Actually,dissipation is not needed for this; reflection of ions from the shock front has thesame effect. To understand this, imagine that the ions have a small thermal spreadin energy and that the height eϕ of the wave front is just large enough to reflectsome of the ions back to the left, while the rest go over the potential hill to the right.The reflected ions cause an increase in ion density in the upstream region to the leftof the shock front (Fig. 8.5). This means that the quantity1χ ¼n00ðξ0ðne ni Þdξ1ð8:30Þis decreased.
Since χ 0 is the analog of dx/dt in the oscillator problem, our virtualoscillator has lost velocity and is trapped in the potential well of Fig. 8.6.8.3.2The Critical Mach NumbersSolutions of either the soliton type or the wave-train type exist only for a range ofM. A lower limit for M is given by the condition that V( χ) be a potential well, ratherthan a hill. Expanding Eq. (8.29) for χ 1 yields1 2χ2 χ 2 =2M 2 > 0M2 > 1ð8:31ÞThis is exactly the same, both physically and mathematically, as the Bohm criterionfor the existence of a sheath (Eq. (8.11)).2808 Nonlinear EffectsAn upper limit to M is imposed by the condition that the function V( χ) of Fig. 8.6must cross the χ axis for χ > 0; otherwise, the virtual particle will not be reflected,and the potential will rise indefinitely.
From Eq. (8.29), we requireeχ 1 < M 2" #2χ 1=21 1 2Mð8:32Þfor some χ > 0. If the lower critical Mach number is surpassed (M > 1), the lefthand side, representing the integral of the electron density from zero to χ, is initiallylarger than the right-hand side, representing the integral of the ion density. As χincreases, the right-hand side can catch up with the left-hand side if M 2 is not toolarge. However, because of the square root, the largest value χ can have is M 2 =2.1This is because eϕ cannot exceed Mu20 ; otherwise, ions would be excluded from2the plasma in the downstream region. Inserting the largest value of χ into Eq. (8.32),we haveexp M 2 =2 1 < M 2orM < 1:6ð8:33ÞThis is the upper critical Mach number. Shock waves in a cold-ion plasma thereforeexist only for 1 < M < 1.6.As in the case of sheaths, the physical situation is best explained by a diagram ofni and ne vs. χ (Fig.
8.8). This diagram differs from Fig. 8.4 because of the change ofsign of ϕ. Since the ions are now decelerated rather than accelerated, ni willapproach infinity at χ ¼ M 2 =2. The lower critical Mach number ensures that theni curve lies below the ne curve at small χ, so that the potential ϕ(x) starts off withthe right sign for its curvature.
When the curve ni1 crosses the ne curve, the solitonϕ(x) (Fig. 8.7) has an inflection point. Finally, when χ is large enough that the areasunder the ni and ne curves are equal, the soliton reaches a peak, and the ni1 and neFig. 8.8 Variation of ionand electron density(logarithmic scale) withnormalized potential χ in asoliton. The ion density isdrawn for two cases: Machnumber greater than andless than 1.68.3 Ion Acoustic Shock Waves281curves are retraced as χ goes back to zero. The equality of the areas ensures that thenet charge in the soliton is zero; therefore, there is no electric field outside. If M islarger than 1.6, we have the curve ni2, in which the area under the curve is too smalleven when χ has reached its maximum value of M 2 =2.8.3.3Wave SteepeningIf one propagates an ion wave in a cold-ion plasma, it will have the phase velocitygiven by Eq.
(4.42), corresponding to M ¼ 1. How, then, can one create shocks withM > 1? One must remember that Eq. (4.42) was a linear result valid only at smallamplitudes. As the amplitude is increased, an ion wave speeds up and also changesfrom a sine wave to a sawtooth shape with a steep leading edge (Fig. 8.9).
Thereason is that the wave electric field has accelerated the ions. In Fig. 8.9, ions at thepeak of the potential distribution have a larger velocity in the direction of vϕ thanthose at the trough, since they have just experienced a period of acceleration as thewave passed by. In linear theory, this difference in velocity is taken into account,but not the displacement resulting from it. In nonlinear theory, it is easy to see thatthe ions at the peak are shifted to the right, while those at the trough are shifted tothe left, thus steepening the wave shape. Since the density perturbation is in phasewith the potential, more ions are accelerated to the right than to the left, and thewave causes a net mass flow in the direction of propagation.
This causes the wavevelocity to exceed the acoustic speed in the undisturbed plasma, so that M is largerthan unity.Fig. 8.9 A large-amplitude ion wave steepens so that the leading edge has a larger slope than thetrailing edge2828.3.48 Nonlinear EffectsExperimental ObservationsIon acoustic shock waves of the form shown in Fig. 8.5 have been generated by R. J.Taylor, D. R. Baker, and H.
Ikezi. To do this, a new plasma source, the DP (doubleplasma) device, was invented. Figure 8.10 shows schematically how it works.Identical plasmas are created in two electrically isolated chambers by dischargesbetween filaments F and the walls W. The plasmas are separated by the negativelybiased grid G, which repels electrons and forms an ion sheath on both sides. Avoltage pulse, usually in the form of a ramp, is applied between the two chambers.This causes the ions in one chamber to stream into the other, exciting a largeamplitude plane wave. The wave is detected by a movable probe or particle velocityanalyzer P.
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