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

= −ndqdmd

∂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 Poisson s equation and zero-order quasi-neutrality:"

€

∂ 2ϕ

∂x2= −

eε0

ni − ne − Zdnd( )

€

ni0 = ne0 + Zdnd0

(continuity) (momentum)"

(Poisson s) (quasi-neutrality)"

Dust acoustic waves (DAW) - 2"

ω2 =k 2CDAW

2

1+k 2λD2( )

; where, CDAW =ωpd2 λD

2 ≈

Timd

$

%&

'

()εZ 2

1+ TiTe

$

%&

'

() 1−εZ( )

,

-

.

.

.

.

.

/

0

11111

12

λD−2 = λDe

−2 + λDi−2 and ε=

ndni

•  The electrons and ions are assumed to obey a Boltzmann distribution:"

€

ni = ni 0 exp −eφTi

$

% &

'

( ) ne = ne0 exp

eφTe

$

% &

'

( )

For many experiments:"Ti

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 k s)."

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."

ω2 + iβω =k 2ωpd

2 λDi2

1+k 2λDi2( )

≈k 2CDAW

2

1+k 2λD2( )

6.6 Waves in Dusty Plasmas 187

6.6.1 Waves in Weakly-Coupled Plasmas: Dust-Acoustic Wave (DAW)

We start with the discussion of waves in weakly coupled dusty plasmas where we like topresent an example of an electrostatic wave, the dust acoustic wave [105, 106]. The DAW is avery low-frequent wave with wave frequencies of the order of the dust plasma frequency !pdwhich is much less than the ion plasma and electron plasma frequency (!pi,!pe)

!pd =

sZ2de

2nd0≤0md

ø !pi,!pe , (6.34)

where nd0 is the equilibrium (undisturbed) dust density. The DAW is a complete analog to thecommon ion-acoustic wave, where the dust particles take the role of the ions and the ions andelectrons take the role of the electrons in the ion-acoustic wave. Thus, the DAW is driven bythe electron and ion pressure and the inertia is provided by the massive dust particles.

The dispersion relation of the DAW is given by, see e.g. [107],

!2 + iØ! =!2pdq

2∏2Di1 + q2∏2Di

(6.35)

under the (generally justified) assumption of cold dust Td = 0 and cold ions Ti ø Te (The fulldispersion is given e.g. in [107]). Here, q is the wave vector and ∏Di is the ion Debye length.In contrast to the common ion-acoustic wave the governing screening length is here the ionDebye length. Second, damping of the wave by friction with the neutral gas is included interms of the friction coefficient Ø.

The calculated dispersion relation of the DAW is shown in Fig. 6.14a. For large wavenumbers q∏Di ¿ 1 the wave is not propagating and oscillates at the dust plasma frequency!pd. For long wavelengths q∏Di ø 1 the wave is acoustic ! = qCDAW with the dust-acoustic

Figure 6.14: a) Dispersion relation of the dust-acoustic wave without damping. The solid line is thefull dispersion relation, the dotted line indicates the acoustic limit with the dust-acoustic velocity. b)Dispersion relation with small friction constant Ø = 0.1!pd and c) with large friction constant Ø =0.5!pd. Here, the solid line refers to the real part of the wave vector and the dashed line to the imaginarypart. Note, that in b) and c) the axes have been exchanged with respect to a).

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:"o  a pressure term"o  an electric field, E, which – in zero-order - gives rise to drifts"o  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( ) − k2Vti2+

ωpe2

Ωe Ωe + iν en( ) − k2Vte2+

ωpd2

Ωd Ωd + iν dn( ) − k2Vtd2

Where : Ωα = ω − kuα 0, ωpα =nα qα

2

ε0mα

(

) * *

+

, - -

12, Vtα =

kBTαmα

(

) *

+

, -

12

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 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"

€

ωk

=Timd

#

$ %

&

' ( εZ2

*

+ ,

-

. /

12; ε = nd

niIn 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