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(9.19) in Eq. (9.23) gives2∇ ϕ¼ne0 e2ni0 e2þϕε0 KT e ε0 KT ieZ dnd1 ¼ k2De þ k2Di ϕε0eZ dnd1 ;ε0ð9:25Þwhere k2D j n j0 e2 =ε0 KT j , so that, with k2D k2De þ k2Di ; Eq. (9.25) simplifies to∇ 2 ϕ ¼ k2D ϕðeZ d =ε0 Þnd1ð9:26ÞRemembering that nd1 is the density fluctuation of a dust wave, we can let nd1 and ϕtake the usual form exp½iðkz ωtÞ. Equations (9.24) and (9.26) then become,respectively,ω2andThus,eZ d nd0 2k2 v2thd nd1 ¼k ϕMdeZ dnd1 :k2 þ k2D ϕ ¼ε0k2 þ k2D ¼nd0 e2 Z2dk2:ε0 Md ω2 k2 v2thdð9:27Þð9:28Þð9:29ÞThe coefficient on the r.h.s can be called the square of the “dust plasma frequency”ωpd in analogy with Eq.
(4.25). The dispersion relation for these low-frequency dustacoustic waves is then3449 Special Plasmasω2pdk21 þ D2kω2k2 v2thd¼ 0,ω pdnd0 e2 Z2dε0 M d1=2;ð9:30Þwhere kD refers to the electron-ion plasma. In laboratory plasmas, typicallyKTe KTi, so that k2D k2Di ¼ λDi2 ¼ ni0 e2 =ε0 KT i andω2k2 v2thd ¼k2 ω2pdk2 þ k2Di¼k2 λ2Di ω2pd1 þ k2 λ2Di:ð9:31ÞWe can definecd ω pd λDind0 2 KT i 1=2Z¼ni0 d Mdð9:32Þas the dust acoustic speed, in analogy with cs ¼ ω p λD .
For cold dust, we can neglectvthd, and then have for the phase velocityωcd1=2 :k1 þ k2 λ2Dið9:33ÞFor k2 λ2Di << 1, the waves are non-dispersive sound-like waves with ω/k cd.1=2These dust plasma waves, with cd / 1=Md , propagate very slowly. For example,in a typical laboratory dusty plasma with dust particles of 1 μm size, KTe ¼ 2.5 eV,KTi ¼ 0.025 eV, Zd 2000, Md 10 15 kg, and nd0/ni0 ~ 10 5, the dust acousticspeed cd is ~1 cm/s.
Dust acoustic waves can be seen by illuminating the dust with athin sheet of laser light as shown in Fig. 9.7. Figure 9.10 shows a frame from a videoof a dust acoustic wave.9.4.2Dust Ion-acoustic WavesDust has a physically reasonable effect on the normal ion waves of Sect. 4.6. Theion equations of motion and continuity are:∂vi¼∂te∇ϕMv2thi∇ni1ni0∂ni1þ ni0 ∇ vi ¼ 0:∂tInserting Eq.
(9.34) into the time derivative of Eq. (9.35) givesð9:34Þð9:35Þ9.4 Dusty Plasmas345Fig. 9.10 A single-framevideo image of a dustacoustic wave propagatingfrom right to left. The brightvertical features are thewave crests imaged fromscattered laser light(courtesy of R. Merlino)2∂∂t2!ev2thi ∇2 ni1 ¼ ni0 ∇2 ϕ:MAgain let the waves have the usual form exp½iðkzω2ð9:36ÞωtÞ, so thateni0 2k ϕ:k2 v2thi ni1 ¼Mð9:37ÞSince the phase velocity ω/k will be scaled to KTe, as in ion sound waves, andKTi KTe, the vthi2 term can be neglected, so thatω2 ni1 ¼eni0 2q nd0 2k ϕ, and, similarly, ω2 nd1 ¼ dk ϕ:MMdUsing Eq. (9.19) for ne1 and Eq.
(9.38) for ni1 and nd1 in Eq. (9.23) yieldsk2 2k2 222k ϕ ¼ kDe ϕω ϕω ϕ ,ω2 piω2 pdω2pi ω2pdk2¼þ 21 þ Deω2ωk2ð9:38Þð9:39ÞThe phase velocity is then given byω2pi λ2Deω2 ω2pi þ ω2pd¼:k2k2 þ k2De1 þ k2 λ2Deð9:40ÞSince Zdnd0 is at most ni0, the ratio ωpd2/ωpi2 ¼ (nd0/ni0)(Zd2M/Md) is less than ZdM/Md. We can then neglect the ωpd2 term and have done so. Equation (9.40) justifiesthe definition of cd as a phase velocity in Eq. (9.32).3469 Special PlasmasIf n0e is not so small that λD is as large as the wavelength of the wave, we canneglect the k2λDe2 term in the denominator of Eq.
(9.40). Also, qd is usually muchlarger than e, so ne can be much smaller than ni. The numerator is thenω2pi λ2De ¼n0i e2 ε0 KT e n0i 2¼ cs >> c2s :n0eε0 M n0e e2Equation (9.40), finally, gives the phase velocityω¼kn0in0e1=2cs :ð9:41ÞThis is the dispersion relation for dust-modified ion acoustic waves, or dust ionacoustic waves. The dust density does not appear explicitly here, but it determinesthe ni0/ne0 ratio via the quasineutrality condition ni0 ¼ ne0 + jZdjnd0 (Eq. (9.18)) fornegatively charged dust.
In terms of the dust fraction δ nd0/ni0, we can writeEq. (9.41) asω2¼k2Thus, the phase velocity isω¼k1ni0!ni0c2s2 Zd nd0 cs ¼ 1 δZ d :cs 1=2 ,δZ d δ ¼ nd0 =ni0 :ð9:42ÞThis shows that dust increases the velocity of ion acoustic waves, with the consequence that Landau damping of those waves is decreased.9.5Helicon PlasmasSo far, we have mainly treated plasmas consisting of charged species: negativeones, such as electrons and charged dust; and positive ones, such as ions andpositrons. Such fully ionized plasmas, however, have to be specially prepared inthe laboratory in fusion devices, Q-machines, and such. Most laboratory plasmasare partially ionized and include neutral atoms.
Collisions with neutrals wereconsidered in Chap. 5 on diffusion. Plasmas that have practical applications, suchas semiconductor etching and magnetic sputtering, are partially ionized. Thegranddaddy of three-component plasmas is the helicon plasma, which includesnot only neutrals but also a magnetic field. Though helicons are complicated,they have been studied exhaustibly worldwide and are well understood.9.5 Helicon Plasmas347Helicon plasmas are ionized by helicon waves, which are basically whistlerwaves (R waves) confined to a cylinder.
Their frequencies generally lie between ωcand the lower hybrid frequency ωl (Eq. (4.71)). To satisfy the boundary conditionson the cylinder, a second wave has to be generated near the boundary. This secondwave is an electron cyclotron wave traveling obliquely to the B-field; it is called theTrivelpiece-Gould (TG) mode (Fig. 4.21).
Let β be the total k such thatβ2 ¼ k2⊥ þ k2z . The R wave dispersion relation for propagation at an angle θ toB is (Problem 9.2)c2 β 2¼1ω21ω2p =ω2ðωc =ωÞ cos θ!ωc >>ωω2p:ωωc cos θð9:45ÞDefining k kz ¼ β cos θ, we haveβ¼ω2pω2 1ω ω2p 1 ωn0 e2 m¼ ε0 μ0¼22c β ωωc cos θ c ωc k kε0 m eB0Thus,β¼ω n0 eμ0:k B0ð9:46ÞThis is the basic equation for helicon waves. It shows that the density increaseslinearly with B0, and the electron mass m has cancelled out to this order. Thefrequency is much lower than the electron frequencies ωc and ωp. When terms inm are kept, a second wave, the TG mode, is obtained.
The relation between thehelicon (H) and TG waves has been clarified in papers by D. Arnush, who alsowrote a computer program HELIC for the properties of these waves. Damping byelectron-neutral collisions is important in the exact theory. The relation between theH and TG waves is shown in Fig. 9.11. There, k (the wave number parallel to B0) isplotted against β, the cylindrical wave number in the radial direction, for azimuthalwave number m ¼ 1. Above a minimum, there are two solutions for k for givenn and B0, the one with large β being the TG mode.Helicon discharges have been studied experimentally in many machines withlong, uniform magnetic fields of order 0.1 T.
The first such machine, built byR.W. Boswell in Australia, reached a density of almost 1020 m 3 on axis. Differenttypes of antennas have been used, the most efficient being helical ones matching thehelicity of m ¼ +1 waves obeying Eq. (9.46). For reasons not well understood,m ¼ 1 waves rotating in the opposite direction do not propagate as well.
Thecoupling of radiofrequency (RF) energy from the antenna to the plasma has beenfound to involve parametric instabilities. Instabilities such as the drift-wave instability (Fig. 6.14) have been studied in helicon discharges. Magnetic confinement ofelectrons (but not of argon ions, which have Larmor orbits larger than the dischargeradius), combined with efficient antenna coupling, enables helicons to achieve3489 Special PlasmasFig. 9.11 The dispersion relation for m ¼ 1 helicon and TG waves at one density and fourmagnetic fields (F.F. Chen, Plasma Sources Sci.
Technol. 24, 014001 (2015))Fig. 9.12 A large helicondischarge inside its magnetcoils (R.T.S. Chen,R.A. Breun, S. Gross,N. Hershkowitz, M.J. Hsieh,and J. Jacobs, PlasmaSources Sci. Technol. 4, 337(1995))higher densities than in other RF plasmas at the same power. However, the expenseof a DC B-field has so far prevented helicons from being accepted by industry. Toovercome this, arrays of short helicon discharges with permanent magnets havebeen proposed for producing large, uniform, high-density plasmas for plasmaprocessing (see Sect. 10.3).Figure 9.12 shows a large helicon discharge.
Note that At high B-field and highpower, the plasma can shrink into a dense blue core which is almost fully ionized.9.6 Plasmas in Space3491435G, 4E11Jz, Bz (arb. units)12108Jz dataBz datan(r)6420-6-4-20r (cm)246Fig. 9.13 Radial profiles of n(r), Bz(r), and Jz(r) in a helicon discharge at B ¼ 3.5 mT andn ¼ 4 1017 m 3 (D.D. Blackwell, T.G. Madziwa, D. Arnush, and F.F.
Chen, Phys. Rev. Lett.88, 145002 (2002))Helicon devices proposed for semiconductor etching and spacecraft propulsion willbe shown in Chap. 10.The phase velocity of helicon waves along B is usually comparable to thevelocities of “primary” electrons. . . those that do the ionization. This fact gaverise to a hypothesis that helicons accelerate the ionizing electrons by inverseLandau damping. However, measurements of electron velocity distributionsshowed no such population of fast electrons. It turns out that most of the RF energygoes into the TG mode near the radial boundary, where the antenna is; and only asmall amount goes directly into the helicon mode, which peaks near the axis.
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