Dusty Plasmas: ! waves " Ed Thomas " Auburn University - PowerPoint PPT Presentation

Dusty Plasmas: ! waves " Ed Thomas " Auburn University "

References " • Journals " – TPS – IEEE Transactions on Plasma Science " – PoP – Physics of Plasmas " – PRL – Physical Review Letters " – PRE – Physical Review E " – PPCF – Plasma Physics and Controlled Fusion " – PSS – Planetary and Space Science " • Textbooks " – Low Temperature Plasmas: Fundamentals, Technologies, and Techniques - Volume 1 - R. Hippler, H. Kersten, M. Schmidt, K.H. Schoenbach (Eds.) ! Ref: Ch. 6 – Fundamentals of Dusty Plasmas – A. Melzer and J. Goree " – Introduction to Dusty Plasma Physics – P . Shukla and A. Mamun " – Physics and Applications of Complex Plasmas – S. Vladimirov, K. Ostrikov, and A. Samarian " – Plasma Physics – A. Piel "

COLLECTIVE PHENOMENA !

Many forms of dusty plasma instabilities and waves " Dust density waves refers to the general class of low frequency, often self-excited waves in a dusty plasma that are characterized by a modulation of the dust number density. " 1 cm " � ~ 2 mm " Left: DC glow discharge experiment (Auburn) " 1 cm " Above: RF microgravity experiment (Kiel) "

Dust Density Waves vs. Dust Acoustic Waves " � Dust density waves resemble dust acoustic waves, but are also affected by ion drifts. � – A. Piel, et. al., [PRL (2006)] " From: R. L. Merlino, Univ. of Iowa " From: V. Fortov, et al., PoP (2000) " A. Barkan, et al., PoP (1995) "

Dust acoustic waves (DAW) - 1 " • The original derivation of the DAW was given in N. N. Rao, et al., [PSS, (1990)]. " • The dispersion relation is derived starting with the one-dimensional continuity and momentum equations for the dust component of the plasma: " ∂ v d ∂ v d ∂ x = − n d q d ∂ n d ∂ϕ ∂ t + ∂ n d ∂ t + n d v d ∂ x n d v d ( ) = 0 m d ∂ x (continuity) (momentum) " • The system is closed using Poisson � s equation and zero-order quasi-neutrality: " ∂ 2 ϕ ∂ x 2 = − e ( ) n i − n e − Z d n d n i 0 = n e 0 + Z d n d 0 ε 0 (Poisson � s) (quasi-neutrality) "

Dust acoustic waves (DAW) - 2 " • The electrons and ions are assumed to obey a Boltzmann distribution: " $ ' $ ' n i = n i 0 exp − e φ n e = n e 0 exp e φ & ) & ) T i T e % ( % ( • Assuming plane wave solutions, a~a 0 + a 1 e i(kx- � t) the system of equations is solved to obtain the dispersion relation: " 1 For many experiments: " , / 2 $ ' T i ) ε Z 2 T i << T e ( λ Di << λ De ) " . 1 & m d ω 2 = k 2 C DAW 2 . 1 % ( 2 λ D 2 ≈ " ; where, C DAW = ω pd . 1 ( ) 1 + k 2 λ D 2 $ ' 1 + T i Therefore: λ D ~ λ Di " ( ) ) 1 − ε Z . 1 & T e . 1 " % ( - 0 Long wavelength limit: " − 2 and ε = n d − 2 = λ De − 2 + λ Di λ D k � D << 1 " n i

Dust acoustic waves (DAW) - 3 " Comparison of the linearized and complete Rao DAW models. " Linear approximation: " ω = kC D Note the correction at small wavelengths (large k � s). " Model parameters: ! Z = 4600 " r d = 1.5 µm " � = 2.0 g/cm 3 " n i0 = 1 x 10 8 cm -3 " n d0 = 1.35 x 10 4 cm -3 " T i = 0.025 eV " T e = 2.5 eV " " C D = 1.98 cm/s " T i /T e = 0.01 " � = 1.35 x 10 -4 "

Dust acoustic waves (DAW) - 4 " In most laboratory experiments, the effect of neutral drag is critical. " 2 λ Di " ω 2 + i βω = k 2 ω pd 2 ≈ k 2 C DAW 2 This modifies the dispersion relation ( 1 + k 2 λ Di 2 ) ( 1 + k 2 λ D 2 ) with the introduction of a damping parameter, β . " real% imaginary% β = 0.5 ω pd " β = 0.1 ω pd " β = 0 "

Dust density waves (DDW) - 1 " • In the most general description, the continuity and momentum equations are solved for all three plasma species ( � = e, i, d). " • We allow for: " o a pressure term " o an electric field, E , which – in zero-order - gives rise to drifts " o collisions with background neutrals " ∂ n α ∂ t + ∂ ∂ x n α u α ( ) = 0 $ ∂ u α ∂ u α ' ∂ n α m α n α ∂ t + u α ) + k B T α ∂ x − n α q α E = − m α n α ν α n u α & ∂ x % ( We solve for the zeroth- and first-order terms assuming plane waves: ~e i(kx- � t) "

Dust density waves (DDW) - 2 " • The resulting fluid dispersion relation contains the effects of ion drift, thermal effects, and collisions. " 2 2 2 ω pi ω pe ω pd 1 = 2 + 2 + ) − k 2 V ti ) − k 2 V te ) − k 2 V td 2 ( ( ( Ω i Ω i + i ν in Ω e Ω e + i ν en Ω d Ω d + i ν dn 1 1 ( 2 + 2 ( + Where : Ω α = ω − ku α 0 , ω p α = n α q α , V t α = k B T α 2 * - * - * - ε 0 m α m α ) , ) , A number of authors have studied various forms of the dispersion relation: " " Kaw and Singh, PRL (1997), Mamun and Shukla, PoP (2000), " Merlino and D � Angelo, PoP (2005), Piel, et al., PRL (2007), " Williams and Thomas, PoP (2008) "

Dust density waves (DDW) - 3 " • Comparison of the Rao results with the full fluid dispersion relation. " • The fluid dispersion contains the effects of the ion flow on the waves. " Typical experiments: " " � ~ 40 - 100 rad/s " k ~ 2 – 6 mm -1 " f ~ 6 – 16 Hz " � ~ 1 to 3 mm "

Experiments on DDWs " • Experiments on DDWs have been ongoing since the earliest days of dusty plasma research. " • DDWs have been studied in RF and DC glow discharge plasmas, in Q-machine plasmas, in hot filament discharge plasmas, and under microgravity conditions. " • Two basic classes of experiments are performed: " – Experiments on self-excited DDWs " – Experiments on driven DDWs "

DAW/DDW basic properties - 1 " • Early experiments on DDW/DAW focused on characterizing the basic properties of self-excited waves. " • The first experimental result was reported by Barkan, et al., PoP , 1995. " Measurement of the The displacement of single wavefront displacement of a wave is recorded using a video camera and a front giving a velocity of: He-Ne laser as the light source. " C D ~ 9 cm/s. "

DAW/DDW basic properties - 2 " • In another early experiment, measurements of the frequency of DDWs were performed. " • A photodiode records the fluctuations in the scattered light intensity of a He-Ne laser that illuminated the dust cloud. " Dominant peak @ ! f ~ 5.1 Hz " Prabhakara and Tanna, PoP , 3 , 3176 (1996) "

DDW as a diagnostic for charge " 1 * - # & T i 2 ; ε = n d ω ε Z 2 In the long λ limit, phase velocity of k = , / % ( m d n i DAW/DDW is: " $ ' + . Use the phase velocity of the DDW, measured dust number density, and ion number density to estimate grain charge: q d = - Z d e ! r d = 0.8 µ m m d = 6 x 10 -16 kg ε = 2 to 5 x 10 -4 T i (est.) = 0.03 eV v phase ~ 12 cm/s slope = 1/v phase ! Z d ~ 1300 C. Thompson, et al., PoP (1997) "