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"

Left: DC glow discharge experiment (Auburn) " Above: RF microgravity experiment (Kiel)"

1 cm" ~ 2 mm"

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"

• A. Barkan, et al., PoP (1995)"

From:

• V. Fortov, et al., PoP (2000)"

Dust acoustic waves (DAW) - 1"

∂nd ∂t + ∂ ∂x ndvd

( ) = 0

nd ∂vd ∂t + ndvd ∂vd ∂x = − ndqd md ∂ϕ ∂x

• 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:"

• The system is closed using Poissons equation and zero-order quasi-neutrality:"

∂ 2ϕ ∂x2 = − e ε0 ni − ne − Zdnd

( )

ni0 = ne0 + Zdnd0 (continuity) (momentum)" (Poissons) (quasi-neutrality)"

Dust acoustic waves (DAW) - 2"

ω 2 = k 2CDAW

2

1+k 2λD

2

( )

; where, CDAW = ωpd

2 λD 2 ≈

Ti md $ % & ' ( )εZ 2 1+ Ti Te $ % & ' ( ) 1−εZ

( )

,

• .

. . . . / 1 1 1 1 1

1 2

λD

−2 = λDe −2 + λDi −2 and ε=nd

ni

• The electrons and ions are assumed to obey a Boltzmann

distribution:"

ni = ni 0 exp − eφ Ti $ % & ' ( ) ne = ne0 exp eφ Te $ % & ' ( )

For many experiments:" Ti << Te (λDi << λDe)" " Therefore: λD ~ λDi " " Long wavelength limit:" kD << 1"

• Assuming plane wave solutions, a~a0 + a1 ei(kx-t) the system of

equations is solved to obtain the dispersion relation:"

Dust acoustic waves (DAW) - 3"

Model parameters:!

Z = 4600" rd = 1.5 µm" = 2.0 g/cm3" ni0 = 1 x 108 cm-3" nd0 = 1.35 x 104 cm-3" Ti = 0.025 eV" Te = 2.5 eV" " CD = 1.98 cm/s" Ti/Te = 0.01" = 1.35 x 10-4"

Linear approximation: ω = kCD

Comparison of the linearized and complete Rao DAW models." " Note the correction at small wavelengths (large ks)."

Dust acoustic waves (DAW) - 4"

In most laboratory experiments, the effect of neutral drag is critical." " This modifies the dispersion relation with the introduction of a damping parameter, β."

ω 2 + iβω = k 2ωpd

2 λDi 2

1+k 2λDi

2

( )

≈ k 2CDAW

2

1+k 2λD

2

( )

β = 0.1ωpd" β = 0.5ωpd" real% imaginary% β = 0"

Dust density waves (DDW) - 1"

∂nα ∂t + ∂ ∂x nα uα

( ) = 0

mα nα ∂uα ∂t +uα ∂uα ∂x $ % & ' ( ) + kBTα ∂nα ∂x − nα qα E = −mα nαναnuα

• In the most general description, the continuity and momentum equations are solved for

all three plasma species ( = e, i, d)."

• We allow for:"
• a pressure term"
• an electric field, E, which – in zero-order - gives rise to drifts"
• collisions with background neutrals"

We solve for the zeroth- and first-order terms assuming plane waves: ~ei(kx-t)"

Dust density waves (DDW) - 2"

• The resulting fluid dispersion relation contains the effects of ion

drift, thermal effects, and collisions."

1= ωpi

2

Ωi Ωi + iν in

( ) − k2Vti

2 +

ωpe

2

Ωe Ωe + iν en

( ) − k2Vte

2 +

ωpd

2

Ωd Ωd + iν dn

( ) − k2Vtd

2

Where : Ωα = ω − kuα 0, ωpα = nα qα

2

ε0mα ( ) * * + ,

• 1

2

, Vtα = kBTα mα ( ) * + ,

• 1

2

A number of authors have studied various forms of the dispersion relation:" "

Kaw and Singh, PRL (1997), Mamun and Shukla, PoP (2000), " Merlino and DAngelo, 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 displacement of a wave front giving a velocity of: CD ~ 9 cm/s." The displacement of single wavefront is recorded using a video camera and a He-Ne laser as the light source. "

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"

ω k = Ti md # $ % & ' ( εZ2 * + ,

• .

/

1 2

; ε = nd ni

In the long λ limit, phase velocity of DAW/DDW is:" Use the phase velocity of the DDW, measured dust number density, and ion number density to estimate grain charge: qd = -Zde!

slope = 1/vphase

• C. Thompson, et al., PoP (1997)"

rd = 0.8 µm md = 6 x 10-16 kg ε = 2 to 5 x 10-4 Ti (est.) = 0.03 eV vphase ~ 12 cm/s ! Zd ~ 1300