J. B. Pieper and J. Goree^{a,b)}

*Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa
52242*

Low-frequency compressional waves were observed in a suspension of strongly-coupled 9.4-um spheres in an rf Kr plasma. Both parts of the complex wavenumber were measured to determine the dispersion relation, which agreed with a theoretical model of damped dust acoustic waves, ignoring strong coupling, but not with a strongly-coupled dust-lattice wave model. The results yield experimental values for the dust plasma frequency, charge, Debye length, and damping rate, and support the applicability of fluid-based dispersion relations to strongly-coupled dusty plasmas, which has been a controversy.

PACS: 52.35.Fp, 52.25.Vy, 52.25.Dg

^{a)} electronic mail: john-goree@uiowa.edu

^{b)} author to whom correspondence should be addressed.

A dusty plasma is a three-component plasma consisting of electrons, ions, and massive solid particles held in suspension. The solid particles usually charge negatively to a large value [1]. The introduction of a third component allows new modes to propagate in the plasma. In particular, the dust acoustic wave (DAW) is analogous to the ion acoustic wave, but occurs at very low frequency [1-7]. Previous laboratory observations of wave motion in suspensions of small particles at frequencies of 10-15 Hz by Barkan et al. were attributed to the DAW [8].

A controversy has arisen whether modes such as the DAW derived theoretically
from continuum-based models are applicable to laboratory dusty plasmas. These
plasmas are generally strongly coupled, i.e., the coupling parameter is large,
*[[Gamma]]* = *q*^{2}/(4*[[pi]][[epsilon]]*0[[Delta]]
*kB*T) >> 1 for particles separated by a typical distance
*[[Delta]]*. This parameter is the ratio of the Coulomb potential and
kinetic energies. Most commonly, plasmas are gaslike and weakly coupled
(*[[Gamma]]* << 1), and continuum models, which are derived from the
BBGKY hierarchy by neglecting Coulomb interactions between discrete particles,
are valid. Dust particles suspended in laboratory plasmas are visibly discrete,
and often strongly coupled (*[[Gamma]]* >> 1). Recently Wang and
Bhattacharjee (WB) expanded the BBGKY hierarchy for *[[Gamma]]* >>
1, and arrived again at the Vlasov equation [9], from which fluid equations are
derived. This result has been controversial because it required the
non-intuitive hypothesis that particles are shielded effectively by other
particles, even when there are fewer than one particle per particle-Debye
sphere. (Although the Debye length *[[lambda]]D* for the electron-ion
plasma component is often larger than *[[Delta]]* in the experiments, the
Debye length *[[lambda]]Dd *of the dust particles themselves is miniscule
due to a high charge and low temperature.) The WB result implies that fluid
models can be applied to strongly-coupled systems, and they should exhibit wave
behavior similar to a weakly coupled plasma.

Here we present measurements of the dispersion relation of driven
compressional waves in a strongly-coupled dusty plasma in the parameter range
*[[lambda]]D*>* *[[Delta]]>>* *[[lambda]]*Dd
*and* [[Gamma]]* >> 1. The DAW dispersion relation derived from a
fluid model shows agreement with the experiment.

We produced a krypton plasma by applying a 13.55 MHz rf voltage to a
horizontal Al electrode with a surface depression 1.5 mm deep and ~6 cm in
diameter. Polymer spheres of diameter 9.4 +/- 0.3 um and density 1.51
g/cm^{3} were shaken into the plasma region above the electrode, where
they were levitated by the force balance between gravity and the electric field
in the electrode's plasma sheath. The depression in the electrode surface acted
as a lateral trap for the particles by producing a horizontal component of the
electric field localized near its edges. The spheres were too massive to
respond to the 13.55 MHz rf. The chamber walls acted as the second electrode,
and at lower pressures (below 220 mtorr) we also used a grounded Al ring
electrode 2.5 cm above the driven electrode. The Kr pressure was regulated at a
small flow rate (1 sccm) that caused no detectable disturbance of the
particles. Experiments were repeated for four pressures. We adjusted the rf
voltage and the number of particles in the cloud to attain the most ordered
particle suspension possible at a given pressure; this is different from phase
transition experiments reported by other authors who did not vary the rf
voltage or particle count while varying the gas pressure [10].

To detect the waves, we illuminated particles by a horizontal sheet of laser light and imaged them with a video camera looking through the ring's opening down at the lower electrode. Details of the apparatus are presented elsewhere [11]. In the vertical direction, the particle cloud was three layers thick in the region where we made our measurements, and it had a simple-hexagonal structure [11]. Only one of the three layers was imaged at a time.

Analysis [12] of the equilibrium (without driven waves) revealed that the
particles were arranged in an ordered structure. The structure was generally
liquidlike, with six-fold particle coordination and short-range translational
and orientational order. The orientational correlation length *[[xi]]*6
increased from about 2*[[Delta]]* at the lowest pressures to
14*[[Delta]]* at 300 mtorr, while the translational correlation length
*[[xi]]* was almost constant at 2-3 *[[Delta]]* over the pressure
range.

To characterize the dust velocity distribution *fd*(v*)* at
equilibrium, we imaged particles in a vertical (x-y) plane using a
long-distance microscope. We found *fd*(v*) *was anisotropic and
Maxwellian, with a higher kinetic temperature (Table I) in the *xy* plane,
*kB*T*d ^{xy}*= [

To drive an electrostatic wave, we used the scheme of Zuzic et al. [13]. An electrically isolated W wire of diameter 0.25 mm was stretched horizontally above the depressed electrode surface, at a height of 3 to 5 mm, which was adjusted in each experiment to be near the equilibrium particle height. To excite longitudinal waves in the particle cloud, the wire was driven by a sinusoidal voltage source of about 30 V p-p.

To avoid disturbing the plasma, the sinusoid was given a dc offset of -40 V relative to the chamber walls. Without the dc offset, a large electron current would be drawn during the positive half-cycle of the ac voltage, whereas with the negative offset, a much smaller ion current was drawn that varied little over the ac cycle. As a test, a Langmuir probe, which was dc-coupled to an oscilloscope, was inserted into the plasma a few mm from the driven wire. We found the probe's floating potential was modulated at the wire's frequency with an amplitude two decades smaller than at 13.55 MHz.

We recorded the particle motions on video tape. The driving frequency was
varied in steps from 1 to 10 Hz. Later, we digitized 1 sec (30 frame) segments
of the recording [14]. Image-processing software yielded the coordinates of all
the particles in an image. A given particle was tracked from one frame to the
next through the 30-frame sequence by searching a frame in a rectangular area
centered on a particle's position in the previous frame. Each particle's motion
was separated into the time-averaged position [*x* and the deviation
*dx*(*tk*). The deviation was then averaged over the ignorable
coordinate *y*, parallel to the wire and wavefronts, by binning particles
in each frame *tk* based on their average distance [*x* from the
wire. Finally, the averaged time-dependent deviations in each bin were assumed
to have the form *dx*(*t*) = *A* cos([[omega]]*t* - j)
where [[omega]] is the driving frequency. The amplitude *A* and phase
j were extracted by calculating the Fourier sum

(1)

where *n* is the number *of complete half-cycles contained in the data
set, N* is the nearest integer to
*n[[pi]]*/*[[omega]]*[[Delta]]*t*, and [[Delta]]*t* =
(1/30) s. The sinusoid constructed from *A* and j fit the data very
accurately.

The charging time was six orders of magnitude faster than the wave oscillation, so that the charge on the particles might vary synchronously with fluctuations in the electron and ion densities due to a large amplitude wave [15]. This and other nonlinear effects were not observed. In tests of linearity, a full spectral analysis of the motion showed that harmonics of the drive frequency were typically 30 dB or more below the fundamental response, and the amplitude was found to scale proportionally with the peak-to-peak wire voltage.

To obtain the experimental dispersion relation *k* vs *[[omega]]* as
shown in Fig. 2, we found both parts of the wavenumber *k* =* kr*
+* i ki* by fitting the amplitude and phase as functions of *x* to
decaying-exponential and linear functions, respectively.

The imaginary part of *k* arises from neutral gas drag. The drag also
cooled the particle thermal motion so the particles would be strongly coupled.
Although *ki* was often larger than *kr*, the particle motion was
always unambiguously wavelike, so our analysis was able to yield accurate
measurements of *k*. The frequency is purely real since the mode was
driven.

Analytical theory for the low-frequency compressional dispersion relation can
be done easily in two extreme limits: a damped dust lattice wave (DLW) [15] and
a damped DAW. In the limit of [[Delta]] >> [[lambda]]D, particles in one
dimension interact only with their two nearest neighbors through the shielded
Coulomb potential. Expanding this potential about the equilibrium particle
positions, particles in one-dimension behave like a simple lattice of
identical masses connected by springs. This yields a DLW dispersion relation
where *kr* > *ki* for all *[[omega]]*, and *kr*
increases linearly with [[omega]] in the high-frequency limit. This is not seen
in the experimental results, where *kr* rolls over at high frequencies.
This rollover indicates a transition from a damped propagating wave to an
evanescent wave, which does not occur in the DLW model.

A damped DAW, on the other hand, successfully fits the experimental
measurements. This model assumes a continuum limit [[Delta]] >>
[[lambda]]D, and ignores particle discreteness. We follow Rao et al. [2], but
add a drag term - *m*v*[[nu]]* in the fluid momentum equation for
dust, yielding the dispersion relation:

(2)

This assumes charge neutrality, no dust motion in the equilibrium state,
Boltzmann electron and ion responses, and a plane wave exp(*ikx -
i[[omega]]t*). Here [[nu]] is the damping rate, and *kD* =
1/*[[lambda]]D* = (*[[lambda]]De*^{-2} +
*[[lambda]]Di*^{-2})^{1/2} is the total inverse Debye
length, with *[[lambda]]De*and *[[lambda]]Di*the electron and ion
Debye lengths
[*[[epsilon]]*0k*B*T*e,i*/(*ne,i*e^{2})]^{1/2.
The dust Debye length is assumed to be negligible and does not enter into our
expressions. The dust plasma frequency is [[omega]]pd =
[Z2e2nd0/([[epsilon]]0m)]1/2,
where Z, m and nd0 are the charge, mass and number density of the
dust. In the absence of damping, the wavenumber would be purely real, resonant
(k ->[[infinity]] ), and purely imaginary (evanescent) for
[[omega]] < [[omega]]pd, [[omega]] = [[omega]]pd
[[omega]] >[[omega]]pd, respectively. With strong
damping, the dispersion does not have this discontinuity at the plasma
frequency.}

The real and imaginary parts of Eq. (2) were fitted to the experimental data.
These fits are shown along with the data in Fig. 2. The free parameters
*kD*, *[[omega]]pd*, and *[[nu]]* had the same trial values in
the simultaneous real and imaginary fits. The results are given in Table I,
including *[[nu]]* calculated for Epstein drag with diffuse scattering of
gas molecules from spherical particles [16, 17] , the particle charge *Z,*
and coupling parameter *[[Gamma]]*. The last two quantities were
calculated from *[[omega]]pd* using the measured dust kinetic temperature
*Td *and particle separation *[[Delta]]*, and the value of the
three-dimensional dust density *nd0* obtained from the measured
two-dimensional density and spacing between horizontal particle layers. The
agreement between the measured damping rate and the theoretical value is within
40% for the three highest pressures, but less satisfactory for the lowest.

The data of Fig. 2 are plotted in Fig. 3 in dimensionless form, *k/kD*
as a function of *[[omega]]/[[omega]]pd*. The data nearly coincide in a
similarity curve, as would be expected if the dimensionless parameter *[[nu]]
/ [[omega]]pd* were constant, which is nearly so.

The model presented in Eq. (2) is for a homogeneous dusty plasma. In the experiment the particles were in a slab only three layers thick. Although the particle motion in both cases is one-dimensional, the wave electric fields should have vertical structure in the experiment and not in the model. Nevertheless, the model shows good agreement with the experiment. Fully three-dimensional theories, modeling the particles as a slab of continuum or discrete particles, require numerical techniques, which are now being developed.

The charge ranged from -3400 to -7200 *e*, corresponding to a surface
potential of -1.05 to -2.20 V with respect to the local dc plasma potential.
This is comparable to the charge measured by Trottenberg et al. [18], who used
spheres identical to ours and a similar apparatus. Their experiment differed in
using He gas, a higher rf power, and exciting the particle cloud in the
vertical direction. The two charge measurement methods differ as follows:
vertical excitation is a disturbance of the cloud's levitation equilibrium and
the analysis requires assumptions about the equilibrium. Horizontal excitation
launches a wave, and the analysis requires choosing a theoretical dispersion
relation; additionally it yields *[[lambda]]D* as an output.

The coupling parameters we computed satisfy *[[Gamma]]* >> 1,
indicating the particles were strongly coupled. This conclusion is consistent
with the long correlation lengths measured from the static structure. The sharp
decline of *[[Gamma]]* below 220 mtorr is mostly due to an increase in
the dust temperature as the pressure decreases. This transition from low to
high *Td* was seen by Thomas and Morfill [10] at about the same Kr
pressure.

It is striking that a plasma that is strongly coupled according to static
measurements obeys a wave dispersion relation that is suitable for a continuum
rather than for discrete particles that interact only with their nearest
neighbors. Whether this should be attributed to the condition
*[[lambda]]D* > *[[Delta]]* in our experiment, or to the
WB hypothesis of shielding by the dust particles themselves in the range
*[[lambda]]Dd* < *[[Delta]]*, remains to be determined.

At the present time, our method may be the only practical way of measuring
both *[[lambda]]D *and *[[omega]]pd*(i.e., charge). This new
capability is needed, but it is tempered by a lack of verification by other
means. Calculations of the shielding length and charge based on theoretical
models are not practical, since these quantities depend on the ion flow
conditions around a grain and plasma non-neutrality in the electrode sheath,
which cannot be measured readily.

We have excited low-frequency longitudinal waves in a dusty plasma. They are dispersive and become evanescent at frequencies above the dust plasma frequency. The dispersion cannot be explained by a model involving nearest-neighbor particle interactions alone; the motion also involves the shielding electron and ion components. The measured dispersion relation is fit well by a damped DAW model, based on a fluid model, even though the particles are conspicuously discrete and strongly coupled. We used the model to extract useful information about the system including the particle charge, Debye length, and damping rate.

We thank N. Otani for helpful discussions. This work was supported by NASA and the National Science Foundation.

**References**

[1] C. K. Goertz, Rev. Geophys. **27**, 271 (1989).

[2] N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. **38**, 543
(1990).

[3] N. D'Angelo, Planet. Space Sci. **38**, 1143 (1990).

[4] F. Melandsø, Phys. Scri. **45**, 515 (1992).

**[5] F. Melandsø, T. K. Aslaksen, and O. Havnes, Planet. Space Sci.
41**, 321 (1993).

**[6] M. Rosenberg, Planet. Space Sci. 41**, 229 (1993).

**[7] H. R. Prabhakar and V. L. Tanna, Phys. Plasmas, in press.**

**[8] A. Barkan, R. L. Merlino, and N. D'Angelo, Phys. Plasmas 2**, 3563
(1995).

[9] X. Wang and A. Bhattacharjee, Phys. Plasmas **3**, 1189 (1996).

[10] H. M. Thomas and G. E. Morfill, Nature **379**, 806 (1996).

[11] J. B. Pieper, J. Goree and R. A. Quinn, J. Vac. Sci Technol. A **14**,
519 (1996).

[12] R. A. Quinn *et al.*, Phys. Rev. E **53**, R2049 (1996).

[13] M. Zuzic, H. Thomas and G. E. Morfill, J. Vac. Sci. Technol. A **14**,
496 (1996).

[14] Video data from this experiment are available from the server http://dusty.physics.uiowa.edu

[15] F. Melandsø, Wave Propagation in Dust Plasma Crystals, submitted to Phys. Plasmas.

[16] P. Epstein, Phys. Rev. **23**, 710 (1924).

[17] M. J. Baines, I. P. Williams, and A. S. Asebiomo, Mon. Not. R. Astron.
Soc. **130**, 63 (1965).

[18] T. Trottenberg, A. Melzer, and A. Piel, Plasma Sources Sci. Technol.
**4**, 450 (1995).

TABLE I. Experimental parameters at each Kr gas pressure. The rf electrode
voltage *V*rf,p-p was measured with an oscilloscope probe. Dust thermal
energies *kB*T*d ^{xy,z}* were calculated from measurements of
the random particle velocities. The average particle separation

+"'hParameter

g'"Gas pressure

55 mtorr 100 mtorr 220 mtorr 300 mtorr

&:hMeasured

&g:ø

*&[[arrowhorizex]]:*

&:[[rho]]

&[[theta]]:(c)

9[[Xi]][[eta]]Vrf,p-p (V)

9gXø36

9æX31

9Xr90

9qX"108

Wuhk*B*T*d ^{xy}*(eV)

Wguø1.1 +/- 0.2

Wæu3.8 +/- 0.3

Wur0.024 +/- 0.002

Wqu"0.028 +/- 0.002

têh*kB*T*d ^{z}*(eV)

tgêø0.039 +/- 0.005

tæê2.8 +/- 0.3

têr0.0078 +/- 0.0009

tqê"0.0036 +/- 0.0004

è!=h*[[Delta]]* (mm)

èg!=ø0.58 +/- 0.09

èæ!=0.48 +/- 0.05

è!=r0.29 +/- 0.02

èq!="0.28 +/- 0.02

Ä[[dieresis]]æh__Fit parameters__

*€<-[[gamma]][[arrowhorizex]][[carriagereturn]]*

Ä[[dieresis]]ææ

Ä[[dieresis]]ær

Ä[[dieresis]]qæ"

ÄÄ[[Omega]]'h*[[omega]]pd*/ 2[[pi]] (Hz)

*€€[[arrowvertex]][[gamma]][[product]][[carriagereturn]]*.2
+/- 0.3

ÄÄ[[Omega]]æ'4.6 +/- 0.2

ÄÄ[[Omega]]'r9.0 +/- 0.3

ÄÄ[[Omega]]q'"22 +/- 5

ÄÄ`[[apple]]h[[lambda]]*D* (mm)

*€€(TM)[[gamma]][[apple]][[carriagereturn]]*3.85 +/-
0.15

ÄÄ`æ[[apple]]2.56 +/- 0.06

ÄÄ`[[apple]]r1.19 +/- 0.03

ÄÄ`q[[apple]]"0.46 +/- 0.06

ÄÄÔ

h*[[nu]]*fit / 2[[pi]] (Hz)

*€€ï[[gamma]]*

*[[carriagereturn]]*3.9 +/- 0.9

ÄÄÔæ

5.2 +/- .7

ÄÄÔ

r9.0 +/- 0.7

ÄÄÔq

"15.9 +/- 3.4

ÄÄ)h*[[xi]]*6 (mm)

*€€[[gamma]])[[carriagereturn]]*1.50 +/-
0.08

ÄÄæ)0.98 +/- 0.02

ÄÄ)r2.20 +/- 0.04

ÄÄq)"3.80 +/- 0.06

ÄÄ(:h__Computed__

*€€([[gamma]]:[[carriagereturn]]*

ÄÄ(æ:

ÄÄ(:r

ÄÄ(q:"

ÄÄ9Kh*[[nu]]*calc /2[[pi]] (Hz)

__ÄÄ9gKø__2.1

ÄÄ9æK3.8

ÄÄ9Kr8.5

ÄÄ9qK"11.5

ÄJgh*Z* (10^{3})

*€J[[gamma]][[gamma]][[carriagereturn]]*-3.8 +/-
0.4

ÄJæg-3.8 +/- 0.2

ÄJgr-3.4 +/- 0.1

ÄJqg"-7.2 +/- 1.6

fÉh[[Gamma]]

[[phi]][[gamma]]ƒ[[carriagereturn]]32 +/- 8

fæÉ11.4 +/- 1.4

fÉr2400 +/- 220

fqÉ"10000 +/- 4600

Ç+h*[[Delta]]*/*[[lambda]]D*

‚[[gamma]] [[carriagereturn]]0.15 +/- 0.03

Çæ+0.18 +/- 0.02

Ç+r0.24 +/- 0.02

Çq+"0.61 +/- 0.12

Figure Captions

Fig. 1. Schematic diagram of the apparatus. (a) Perspective view of the electrode showing locations of the excitation wire and particles, the coordinate system, and block diagram of the wave-driving electronics. (b) Side view showing particle location with respect to the electrode's sheath.

Fig. 2. Dispersion relation. Experimentally measured *kr* (J) and
*ki* (E) of compressional waves in a Kr plasma are shown as a function of
driving frequency for gas pressures of (a) 55, (b) 100, (c) 220, and (d) 300
mtorr. The curves represent fits by the theoretical DAW dispersion relation of
Eq. (2). Free parameters (plasma frequency *[[omega]]pd*, Debye length
*[[lambda]]D*, and damping rate *[[nu]]*) are listed in Table I.

Fig. 3. Experimental data from Fig. 2 scaled by *[[omega]]pd *and
*kD* = 1/*[[lambda]]D*. The curves would be expected to coincide if
the ratio *[[nu]] / [[omega]]pd,* which varies from 0.7 to 1.2, were
constant.