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78, 2128 (1997))The average interparticle distance a should vary as n1/3, but the coefficient dependson the shape of the enclosing volume. The value of a can be approximated by theWigner-Seitz radius:a ¼ ð3=4πnÞ1=3 0:62n1=3 :ð9:7ÞWhen Γ, as defined by Eqs. (9.6) and (9.7), becomes large, the plasma first turnsinto a liquid as fluid elements become highly correlated. At Γ > 174, the plasmabecomes a solid crystal such as the one shown in Fig.
9.4. To achieve such a largevalue of Γ, the plasma is cooled to a temperature of 10 mK (10 milliKelviin, or0.01 K) by laser cooling. In this process, a laser is tuned to a frequency just below thatof a transition in the atom. A laser photon is absorbed only by atoms moving rapidlytoward the laser with a blue Doppler shift. The atom’s momentum is lowered by thatof the photon. The photon is later re-emitted in a random direction at a lowerfrequency, so that the fastest atoms are removed, and the plasma is cooled.The Penning trap is a useful device in which unusual plasmas have been created.For instance, a positronium plasma has been formed, consisting of positrons andelectrons, with no ions. Also possible is the creation of anti-hydrogen, whose atomsconsist of negative antiprotons and positrons.9.3 Pair-ion Plasmas337Fig.
9.4 A plasma crystal(from C.F. Driscoll et al.,Physica C:Superconductivity369, Nos. 1–4, 21 (2002).)This is produced in aPenning trap such as the onein Fig. 9.29.3Pair-ion PlasmasOne would expect that a plasma with equal masses would behave quite differentlyfrom a normal plasma with slow ions immersed in a sea of fast electrons. Though itis possible to make a positronium plasma, the recombination rate is so fast that thereis no time to do experiments. Fullerenes, stable molecules of 60 carbon atomsarranged in a hollow sphere, move and recombine slowly because of the large mass.It is thus possible to produce a long-lived pair-ion plasma with C60. In Fig.
9.5,neutral C60 is injected from an oven. A ring of fast electrons up to 150 eV driventhrough the C60 forms Cþ60 ions by ionization and C60 ions by attachment. Amagnetic field confines the electrons but allows the heavy ions to diffuse into thecenter to form a fullerene pair-ion plasma. The plasma then passes through a holeinto a chamber for experimentation.Consider a singly charged pair-ion plasma with a common mass M and temperature KT.
The equations of motion for the ions of charge + or areMn0dv¼ n0 EdtAssuming waves of the form exp½iðkzð9:8Þ∇ p:ωtÞ and E ¼∇ϕ ¼ikϕ, we haveiωMn0 v ¼ ikn0 eϕ 3KTikn ;keϕn þ c2sv ¼, where c2s 3KT=M:ωMn0ð9:9Þð9:10Þ3389 Special PlasmasFig. 9.5 Schematic of a fullerene plasma source. Adapted from W. Oohara and R. Hatakeyama,Phys. Plasmas 14, 055704 (2007)From the equation of continuity ∂n=∂t þ ∇ ðnvÞ ¼ 0, we haven k¼ v :n0 ωð9:11Þiωv ¼ ikðe=MÞϕ ikðk=ωÞc2s v ,k2 2k eϕv 1:c¼ωMω2 sð9:12Þiωn þ ikn0 v ¼ 0,Using this in Eq. (9.9), we findThere are two solutions, depending on whether ϕ is zero. If ϕ is zero, the + andions move together without creating an E-field, and we have an ordinary soundwave:ω2 =k2 ¼ c2s :ð9:13ÞIf ϕ is not zero, Eq. (9.12) says that v+ ¼ v .
Thenn keϕ=M:¼ v ¼ k2 2n0 ωωk2 c2sPoisson’s equation then yieldsε0 k2 ϕ ¼ eðnþn Þ ¼ en0 k22k2 eϕ=Mω2 k2 c2sð9:14Þ9.4 Dusty Plasmas33921¼2Ω pen0 2e=M;¼ 2222ε0 ωk cs ωk2 c2sω2 ¼ k2 c2s þ 2Ω2pð9:15Þwhere Ωp is the ion plasma frequency (ne2/ε0M )1/2. Thus, a pair-ion plasmasupports an ion acoustic wave and an ion plasma wave. The latter (Eq. (9.15)) isthe analog of the Bohm-Gross wave of Eq. (4.30), but with a factor two because ofthe two ion species. These two waves are connected by an intermediate-frequencywave which can be derived only with kinetic theory.9.4Dusty PlasmasIn Chap.
5 we considered three-component plasmas consisting ions, electrons, andneutral atoms in partially ionized plasmas. In general, there can be contaminants ofmacroscopic size, “dust”, made of other atomic species. In outer space, comet tailsare dusty plasmas, as are some nebulas, such as the Orion nebula, and planetaryrings, such as the one on Saturn. On earth, dusty plasmas can exist in flames; rocketexhausts; thermonuclear explosions; atmospheric-pressure plasmas (Sect. 9.6); and,importantly, in plasma processing (see Chap. 10). We shall find that dust has twomain effects. First, it introduces low-frequency waves in the motions of the chargeddust.
Second, it changes the quasineutrality condition so that ne is no longer equal toZni, thus modifying the normal waves in the plasma.Dust grains have sizes from tens of nanometers to hundreds of microns. Sinceelectrons impinge on them much more often than ions do, the grains will have anegative surface potential Vs. Consider a spherical grain of radius a and chargeq < 0. The capacitance of the sphere (with distant walls) isC ¼ 4πε0 a:ð9:16Þq ¼ CV s ¼ 4πε0 aV s :ð9:17ÞThe surface charge q is thenThe value of either q or Vs depends on the Debye length λD in the backgroundplasma.If λD is a, the grain is a small, isolated particle like a spherical Langmuirprobe (Sect.
8.2.5). An ion will be attracted to the grain and will either strike it ororbit around it, depending on its orbital angular momentum around the grain.Enough electrons will be collected to ensure that the net current is zero. Therequired value of Vs is the “floating potential” of probe theory, which we neednot discuss here. On the other hand, if λD is a, the grains are a third chargedcomponent of the plasma along with the ions and electrons. For instance, in a Cs340800PROBE CURRENT ( mA )Fig. 9.6 Langmuir probetraces of a Q-machineplasma with (bottom trace)and without (top trace) dust.(A. Barkan, N. D’Angelo,and R.
L. Merlino, Phys.Rev. Lett. 73, 3093 (1994))9 Special Plasmas600400DUSTOFF2000-100-105-50PROBE POTENTIAL (V)10plasma with KT ¼ 0.21 eV, λD is 34 μm at n ¼ 1010 cm 3 while a 1 μm, so that λD a is satisfied.Assume the latter condition, and let the dust grains have a charge Zd < 0 and adensity nd. Charge neutrality requiresni ¼ neZ d nd :ð9:18ÞThus, the “electron” density is lowered by the presence of dust. This has beenobserved by Barkan et al.
in a potassium Q-machine (Fig. 4.14). The dust,consisting of aluminum silicate particles of 5-μm average radius, was droppedinto the plasma through a mesh on a rotating cylinder surrounding the plasma.The dust density was about 5 104 cm 3. Figure 9.6 shows Langmuir probe tracesof the plasma with and without the dust. In the presence of dust, the electronsaturation current is seen to be lowered by the slow velocities of the heavy dust.Studies of how dust is charged include many minor effects too detailed to bedescribed here; for instance, secondary electron emission upon ion or electronimpact, photoemission of electrons (often the dominant mechanism in cosmicplasmas), and field emission.
It is not surprising that dust particles can arrangethemselves in crystal arrays. Figure 9.7 shows how a picture of such a crystal can beobtained. A plasma is formed by RF applied to the bottom electrode, and dust isintroduced by a shaker that is not shown; The dust is illuminated by a laser beamspread into a plane by a cylindrical lens, and a camera records the dust through ahole in the upper, grounded electrode.
The dust is suspended above the RF9.4 Dusty Plasmas341Fig. 9.7 Schematic of asetup to photograph themotion of dust grains in anRF plasma. The laser beamis spread into a sheet by alens, shown rotated 90 . Thedust dispenser is not shown.(Adapted from G. E. Morfilland H. Thomas, J.
Vac. Sci.Technol. A 14, 490 (1996))Fig. 9.8 Picture of a dustcrystal taken by Morfillet al. (loc. cit.) with theapparatus shown in Fig. 9.7.The grains are trapped in thesheath on the lowerelectrodeelectrode, which is negative relative to the plasma. A still picture of a dust crystalarray is shown in Fig. 9.8.By adding straight barriers to make a channel on the lower electrode, lineararrays of dust particles can be produced, as shown in Fig. 9.9.
These can be pushedby the laser from the right. Only the first particle is pushed, and the others maintainthe crystal spacing.When the dust density is sufficiently high, the charged dust in a plasma can beconsidered as an additional fluid component exhibiting collective effects. Thepresence of charged dust in a plasma modifies all of the wave modes in Chap. 4,even with a DC magnetic field, and introduces new “dust waves” involving themotions of the charged dust.
The dispersion relations for the plasma waves aremodified through the quasi-neutrality condition (Eq. (9.18)), an example of which isthe “dust ion-acoustic wave”, which is analyzed below. We begin by consideringthe dust acoustic wave, which is a very low frequency, longitudinal, compressionalwave involving the dynamics of the dust particles in a plasma.3429 Special PlasmasFig.
9.9 A linear array ofdust particles pushed by alaser from the right. Theframes are 100 ms apart.(A. Homann et al., Phys.Rev. E 56, 7138 (1997);P.K. Shukla and A. A.Mamun, Intro. to DustyPlasma Physics (IOP Press,Bristol, UK, 2002.)9.4.1Dust Acoustic WavesThe dust acoustic wave is a very low-frequency, longitudinal compressional waveinvolving the motions of the dust particles. Because the heavy dust moves moreslowly than the ions and electrons, the latter have time to relax into Maxwelliandistributions:ne ¼ ne0 expðeϕ=KT e Þ ¼ ne0 ð1 þ eϕ=KT e þ . .
.Þ ,ne1 ¼ ne0 ðeϕ=KT e Þni ¼ ni0 expð eϕ=KT i Þ ¼ ni0 ð1ni1 ¼ ni0 ð eϕ=KT i Þeϕ=KT i þ . . .Þ ,ð9:19ÞUsing the subscript d for dust, we can write the 1-D dust equations of motion andcontinuity as:∂vd¼∂tqd∇ϕMdv2thd∇nd1nd0ð9:20Þ9.4 Dusty Plasmas343∂nd1þ nd0 ∇ vd ¼ 0;∂twhere v2thd 3KT d =Md ;ð9:21Þð9:22Þand the subscript 1 has been dropped from vd1 since vd0 ¼ 0. Poisson’s equation is∇2 ϕ ¼1ðene1ε0eni1qd nd1 Þ:ð9:23ÞThe dust charge is taken to be constant. To express nd1 in terms of ϕ, take the timederivative of Eq. (9.21) and use Eq. (9.20):2∂ nd1¼∂t2nd0 ∇ ¼ nd02∂∂t2or∂vd¼∂tnd0 ∇ qd∇ϕMdv2thd∇nd1nd0qd 2∇ ϕ þ v2thd ∇ 2 nd1Md!q nd0 222∇ ϕ:vthd ∇ nd1 ¼ dMdð9:24ÞUsing Eq.
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