PHYSICS OF PLASMAS
VOLUME 6, NUMBER 7
JULY 1999
Dust–dust and dust-plasma interactions of monolayer plasma crystals
H. Schollmeyer, A. Melzer, A. Homann, and A. Piel
Institut für Experimentelle und Angewandte Physik, Christian-Albrechts-Universität Kiel,
24098 Kiel, Germany
~Received 30 October 1998; accepted 23 March 1999!
The horizontal and vertical oscillation modes of monolayer plasma crystal are investigated and are
related to dust–dust and dust-plasma interactions. It is found that the dust particles in the plasma
sheath excited by a biased wire show parametric resonances. This parametric resonance is identified
as the reason for the observed ‘‘sublimation’’ transition of the plasma crystal from a solid to a
gaslike state. © 1999 American Institute of Physics. @S1070-664X~99!00807-1#
oscillation, ~2! a global horizontal oscillation of all particles,
and ~3! intrinsic longitudinal waves in the horizontal plane.
I. INTRODUCTION
Plasma crystals are an ideal model system for the investigation of strongly coupled dusty plasmas. Plasma crystals
usually are generated in parallel plate rf discharges with
monodisperse plastic particles of micrometer size trapped in
the sheath above the lower electrode.1–3 The particles charge
up due to the inflow of plasma electrons and ions. In the
sheath the weight of the particles is balanced by the electric
field force acting on them. Due to their mutual Coulomb
repulsion the dust particles can form ordered structures, i.e.,
the plasma crystal.
The plasma crystal has been investigated in view of
phase transitions,4,5 structure,6,4,7 defects8,9 and waves10–13
and a quite reasonable understanding of the intrinsic modes
of the crystal has been achieved.
The situation is more complicated for the coupling between intrinsic modes and oscillations of the dust particles in
the confining plasma trap. This coupling becomes evident
when the plasma crystal is excited by a probe wire inserted
in the plasma sheath which results in a number of horizontal
and vertical type of oscillations and waves.10,11,14–16 It is the
aim of this paper to analyze the different oscillations and
wave type motions and to relate them quantitatively to the
various dust–dust and dust-plasma interactions.
Here, the excitation of monolayer, two-dimensional ~2D!
plasma crystals and linear, one-dimensional ~1D! particle arrangements shall be investigated in detail. These systems are
not subject to the ion streaming instability17 and can therefore be studied at relatively low gas pressure where damping
of waves and oscillations is weak. Finally, single-layer crystals represent easier systems which are easier to handle in the
theoretical analysis.
A. Vertical oscillations
In the vertical direction z, the dust particles are trapped
in the sheath by the balance of gravitational force and electric field force. Since the electric field in the sheath of an rf
discharge, to a good approximation, increases linearly with z,
there is almost always a position where the gravitational
force can be balanced,
(m d is the dust mass, g the gravitational acceleration, Z the
number of elementary charges on the particle, and z 0 the
position of force balance.! The ~time-averaged! electric field
can be written as
E ~ z ! 5E ~ z 0 ! 1E 8 ~ z2z 0 ! ,
~2!
where it is assumed that E 8 5 ] E/ ] z is constant and that the
change in the horizontal plane (x,y) is much smaller than in
the vertical direction ( ] E/ ] x, ] E/ ] y! ] E/ ] z). This is justified in an ion matrix sheath model,18 where the horizontal
extension of the electrodes is much larger than the sheath
width.
The electric field force and gravitational force then form
a parabolic potential well for the particles
V5 12 ZeE 8 ~ z2z 0 ! 2 ,
~3!
with a resonance frequency
v 20 5
ZeE 8
.
md
~4!
This vertical resonance has been used by Melzer et al.3,19 in
their resonance method for the determination of the dust
charge. There the lower electrode was modulated at a lowfrequency voltage which exerts a periodical force F el sin(vt)
on the particles in the well. The equation of motion then
reads
II. HORIZONTAL AND VERTICAL OSCILLATIONS OF
THE PLASMA CRYSTAL
The frequencies of different horizontal and vertical resonances in the dust particle motion allow for the quantitative
determination of dust–dust and dust-plasma interactions.
Here, the oscillatory motions and resonances of single-layer
plasma crystals and linear particle arrangements shall be discussed in more detail. Under the conditions here, three different types of oscillations can be found: ~1! a global vertical
1070-664X/99/6(7)/2693/6/$15.00
~1!
m d g5ZeE ~ z 0 !
z̈1 b ż1 v 20 z5F el sin~ v t ! ,
~5!
which describes the resonance of a damped harmonic oscillator with the resonance near v 0 . The Epstein coefficient b
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2694
Schollmeyer et al.
Phys. Plasmas, Vol. 6, No. 7, July 1999
describes the friction of the dust particles with the neutral
gas. Relating E 8 with the ion density n i via Poisson’s equation E 8 5(e/ e 0 )n i , the resonance frequency allows us to determine the dust charge.3,19 Anharmonic effects due to distortion of the potential well by variation of the dust charge Z
with height z in the sheath20 can be neglected because of
small electrode modulation voltages used in the experiment.
In conclusion, this vertical resonance is related to the
interaction of dust particles with the vertical structure of the
plasma trap.
q is the ~complex! wave vector and k 5a/l D the screening
strength in the plasma crystal. k n is the linearized stiffness of
the electrostatic ‘‘spring’’ between two particles of distance
na. Since these plasma crystal waves are driven, v has to be
taken as real and q5q r 1iq i as complex. Boundary effects
due to finite particle chains are neglected, here. The calculation of the dispersion relations can be found in more detail
for linear and two-dimensional systems in Refs. 21,13.
The investigation of waves allows a quantitative analysis
of dust–dust interactions. Characteristic frequencies are of
the order of the dust-plasma frequency
B. Horizontal oscillations
In the horizontal plane, the crystal usually is confined by
a ring on the electrode in order to provide a plasma crystal of
definite size and location. The ring on the electrode raises the
equipotentials of the sheath environment in a region next to
the ring. In that way the dust particles cannot overcome the
potential barrier formed by the ring. Hence, the confinement
potential in the horizontal plane is shaped like a ‘‘bathtub’’
with a flat potential in the middle of the ring and a rather
steep potential rise at the borders. In this potential well global horizontal oscillations of all particles can be excited,11,14
where the particles bounce between the confining potential
walls. Due to the flat potential in the horizontal plane, the
resonance frequency of these oscillations is much smaller
than in the vertical direction.
v Pd5
A
3Z 2 e 2
4 p e 0m da 3
.
~9!
In frictionless systems, the maximum excitation frequency,
obtained for the shortest possible wavelength q r a5 p is the
Debye frequency
v D 5 A 43 exp~ 2 k !~ 212 k 1 k 2 ! v pd
~10!
~for next neighbor interaction N n 51). At frequencies higher
than v D , the wave vector becomes imaginary ~and the wave
is damped! in a frictionless system, similarly to the ‘‘cutoff’’ in usual plasma wave theory ~see Fig. 5 below!. A
system with friction does not show such an abrupt change in
behavior and the wave vector q r a5 p may not be reached.
C. Waves in the horizontal plane
III. EXPERIMENTAL SETUP
In the strongly coupled plasma crystal the particles interact by means of their mutual ~shielded! Coulomb repulsion. Under these conditions wave type motions of the particles relative to each other are possible and have been
observed in different experiments.10,12,13 In linear particle arrangements as well as in monolayer plasma crystals these
waves have been identified as dust lattice waves ~DLW!12,13
under strong coupling conditions. For simplicity of the calculation of the dispersion relation and the experimental realization, the dispersion of waves in a linear particle chain
shall be discussed, here. The equation of motion for a dust
particle then reads
The experiments are performed in a parallel plate rf discharge operated in helium. The lower electrode is powered
with 11 W at a frequency of 13.56 MHz. The upper electrode
and the discharge vessel is grounded.
Monodisperse plastic spheres of 9.47mm diameter are
trapped in the sheath above the lower electrode. Laser diodes
with their light expanded into horizontal and vertical laser
fans allow for the illumination of perpendicular cross sections of the plasma crystal. The scattered light is recorded
with two CCD ~charge coupled device! cameras from top
and from the side. In the experiments presented here, the
amount of trapped particles and the confining barrier on the
lower electrode were chosen in such a way that either singlelayer plasma crystals or linear particle arrangements are produced.
At a height of 7 mm above the electrode a tungsten wire
of 150 mm diameter and 10 cm length was placed close to
the plasma crystal @see Fig. 1~a!#. A low-frequency sinusoidal voltage is applied to this wire in order to excite the
particle oscillatory motions. Hence, the experimental setup is
similar to that presented in Refs. 11, 10, 14–16. Additionally, the rf voltage applied to the lower electrode can be
modulated by a low-frequency sinusoidal voltage in order to
excite pure vertical oscillations of the dust particles in the
sheath3,19 in order to investigate the dust-sheath interaction
and to determine the dust charge. This setup is used for the
excitation of oscillations in monolayer crystals as well as in
linear particle arrangements.
For measuring the dispersion relation of the waves the
laser beam is focused onto the first particle in a linear chain
Nn
ẍ1 b ẋ52
(
n52N n ,nÞ0
¹ f ~ na ! ,
~6!
where f (x) is the particle–particle interaction potential and
2N n neighbors with interparticle distance a are taken into
account. Assuming a Debye–Hückel interaction potential
f (x)5Zeexp(2x/lD)/(4pe0x), where l D is the shielding
length, the dispersion relation is obtained using the standard
approach for longitudinal waves on an infinite linear chain.
The dispersion relation for these waves then reads21,12
Nn
v 2 1i b v 54
k
( n sin2
n51 m d
S D
nqa
2
~7!
with
k n5
Z 2e 2
4 p e 0n 3a 3
exp~ 2n k !~ 212n k 1n 2 k 2 ! .
~8!
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Phys. Plasmas, Vol. 6, No. 7, July 1999
FIG. 1. Experimental setup ~a! for the electrostatic excitation of dust particles by a wire or by a low-frequency modulation of the rf voltage, ~b! for
the excitation of horizontal waves in a linear particle arrangement by the
radiation pressure of the focused laser light. The laser for the illumination of
the dust particles and the CCD camera for a side-on observation of the dust
particles have been omitted here for clarity.
of dust particles @Fig. 1~b!#. This particle is pushed by the
radiation pressure of the laser beam. Switching the laser periodically ‘‘on’’ and ‘‘off’’ leads to oscillations of the first
particle and to the excitation of waves in the linear particle
arrangement.12
In conclusion, this experiment combines several experimental setups in order to provide a detailed analysis of the
horizontal and vertical oscillations found in the plasma crystal which allows to quantify the dust–dust and dust-plasma
interactions.
IV. RESULTS AND DISCUSSION
A. Monolayer crystals
First, we present measurements in a monolayer plasma
crystal at a gas pressure of 10 Pa. The crystal consists of
about 200 particles that are arranged in a crystal lattice with
hexagonal order. Multi-layer plasma crystal would be in a
gaslike state under these conditions.4,5 A sinusoidal voltage
of U 0 528 V pp is applied to a tungsten wire that is positioned
close to the crystal. The frequency f is varied between 0.1 Hz
and 50 Hz. Figure 2 shows the oscillation amplitudes of the
dust particles in the horizontal and in the vertical direction as
a function of f. At low frequencies ~up to 5 Hz! almost pure
horizontal oscillations are observed. These are global oscillations of all particles in the crystal, which have a resonance
at about 0.8 Hz. This oscillation mode is identified as the
‘‘bathtub mode’’ where the particles bounce back and forth
in the horizontal potential well provided by the barrier on the
electrode. This kind of oscillation has been observed also in
Refs. 11, 14 with very similar resonance frequencies.
When the frequency is increased, vertical oscillations become more and more prominent and horizontal oscillations
are nearly undetectable. At a frequency of about 14 Hz a first
resonance in the vertical oscillations can be seen and a second resonance appears at f 528 Hz.
Dust-dust and dust-plasma interactions of monolayer . . .
2695
FIG. 2. Horizontal and vertical oscillation amplitudes of the dust particles in
a monolayer plasma crystal and in a linear chain arrangement. The vertical
resonances of plasma crystals and linear chains coincide very well.
It is anticipated here that the monolayer crystal stays in
an ordered, crystalline state when the particles show the global horizontal and vertical oscillations, but at the first resonance at f 514 Hz and, more pronounced, at the second resonance at f 528 Hz the ordered plasma crystal performs a
sudden transition to a gaslike state, where the particles are
completely disordered and swirl around at very high speed.
This ‘‘sublimation’’ transition was previously reported in
Refs. 14, 15. We will return to this point below.
The vertical resonance at 14 Hz is easily identified as the
resonance at v 0 due to the potential well formed by gravity
and electric field force. This is shown by applying the lowfrequency modulation voltage to the lower electrode. Figure
3 shows a comparison of the resonance curves by wire excitation and electrode voltage modulation. One can clearly see
that the first resonance at 14 Hz coincides for both types of
excitation, but that the second resonance at 28 Hz can only
be excited by the wire. We therefore propose that the second
resonance is a parametric excitation of vertical particle oscillations due a distortion of the potential trap by the periodic
voltage on the probe wire.
From the excitation by modulating the electrode voltage,
the resonance frequency is found at v 0 52 p 313.5 s21 . The
Epstein friction coefficient is determined as b 57.4 s21 ,
which is slightly larger than the expected theoretical value of
b 54.0 s21 . From the resonance frequency v 0 and the measured ion density, the dust charge is determined as Z
511 400.
B. Parametric resonance
Although a complete analysis of the parametric excitation by the wire is difficult in view of the complicated geometry and the sheath environment, we give a simplified
analysis here, that is based on the reasonable assumption that
the probe bias U5U 0 cos(vt) affects the sheath width, at
least in the vicinity of the wire. In addition, the probe bias
can lead to a temporal change of the dust charge by influ-
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2696
Schollmeyer et al.
Phys. Plasmas, Vol. 6, No. 7, July 1999
FIG. 3. Vertical resonance curves of the monolayer plasma crystal due to
excitation by a voltage on the lower electrode. For comparison the resonance by wire excitation is also shown. The resonances at about 14 Hz
coincide very well, the second resonance at 28 Hz can be excited by the
wire, only.
encing the plasma parameters and, hence, the particle charging currents. A change in the sheath width or in the dust
charge results in a periodic modulation of the potential well
and its resonance frequency
v 20 → v 20 @ 11h cos~ v t !# ,
~11!
where h,1 is the modulation depth. The equation of motion
for a dust particle in the sheath then becomes
z̈1 b ż1 v 20 @ 11h cos~ v t !# z50,
~12!
which for z̃5exp(2bt/2)z results in the standard form of the
well-known Mathieu equation.22 This equation is the paradigm for the study of parametric resonances. The parametric
resonances then occur at frequencies of
v n5
2v0
,
n
n51,2, . . . ,
~13!
where the frequency width for the excitation of resonances
decreases as h n .23 So the broadest resonance will be observable at the highest frequency v 52 v 0 , which is two times
the frequency of the vertical resonance used for the charge
measurement. Another resonance is also expected at v 5 v 0 ,
whereas the resonance at v 52 v 0 /3 usually is too small for
detection.23
In the presence of damping ~as in the experiment! there
exists a threshold h c for the modulation depth in order to
excite parametric resonances
h c5
2b
v0
~14!
for the resonance at v 52 v 0 ~see Ref. 22 and references
therein for details!.
FIG. 4. ~a! Resonance curves for vertical oscillations of the dust particles in
a monolayer plasma crystal for different excitation voltages U 0 on the probe
wire. The curves have been shifted vertically for clarity. The corresponding
zero is marked with an arrow on the right hand side. ~b! Resonance amplitude at 2 v 0 as a function of excitation voltage. At voltages higher than
approximately U c 514 V a sudden onset of this resonance occurs. ~c! Variation of the excitation voltage necessary for the appearance of the parametric
resonance as a function of gas pressure.
In the experiment by wire excitation, resonances were
found at v 0 52 p 314 Hz and at 2 v 0 which is a first indication of parametric resonance. In order to prove that the resonance at 2 v 0 indeed is due to parametric excitation the
threshold behavior also has to be shown.
The existence of a threshold is investigated by the variation of the amplitude of the voltage applied to the wire. Figure 4~a! shows the vertical resonance curves for different
voltages. One can clearly see that below a voltage of U 0
512 V pp no resonance at 2 v 0 is found, whereas above U 0
516 V pp the parametric resonance can be excited @Fig. 4~b!#.
Furthermore, the critical voltage and corresponding
modulation depth for the excitation of the resonance at 2 v 0
as a function of gas pressure is shown in Fig. 4~c!. The
excitation voltage increases with increasing neutral gas friction. At gas pressures higher than 15 Pa the damping was too
strong to excite parametric resonances. The increase of the
critical voltage with gas damping found in the experiment is
stronger than linear calculated from Eq. ~14!. A full quantitative agreement, however, cannot be expected because
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Phys. Plasmas, Vol. 6, No. 7, July 1999
Dust-dust and dust-plasma interactions of monolayer . . .
2697
Mathieu’s equation is used as a standard model for parametric excitation, here, to reveal the basic ideas of parametric
resonance. However, the appearance of a resonance at 2 v 0 ,
where v 0 is the eigenfrequency of the potential well of the
dust particles in the sheath, and the threshold behavior of this
resonance with respect to gas damping and excitation voltage
clearly shows that this resonance is of parametric nature.
The excitation by electrode voltage modulation obviously does not lead to a parametric excitation of the vertical
resonances, in contrast to the wire excitation. The detailed
analysis of the difference in the excitation mechanisms and
their coupling to the sheath environment cannot be done in a
simple model and is beyond the scope of this work.
C. Linear particle chain
The resonances of the dust particle motion are, so far,
related to dust-plasma sheath interactions. In order to compare these resonances dust–dust interactions the behavior of
a linear particle arrangement is studied in the following, because of simpler geometry and easier wave excitation.
First, the vertical resonances due to excitation by a voltage on a wire are compared with the situation of the monolayer plasma crystal. Figure 3 shows the resonance curves for
vertical oscillations. The resonances are found at the same
frequencies as in the monolayer crystal: One finds the resonance at the eigenfrequency v 0 of the potential well in the
plasma sheath and the parametric resonance at 2 v 0 . Therefore, the results from the wave experiments can also be used
for monolayer plasma crystal.
The dust–dust interactions are measured by the excitation of longitudinal waves in the linear chain, where the
beam of a laser diode focused onto the first particle. The
laser power is square wave modulated at a frequency f. The
motion of each particle in the chain is recorded and split into
the time-averaged equilibrium position x (n)
0 in the chain and
the time-dependent displacement about this equilibrium j (n) .
A Fourier analysis of the particle motion gives the phase and
amplitude for the oscillation of each particle. The complex
wave vector q is determined according to j (n) 5exp(i(qr
1iqi)x(n)
0 ) from the linear increase of the phases and the exponential decay of the amplitudes with position in the linear
chain. Finally, the complex wave vectors are determined for
different excitation frequencies, thus yielding the dispersion
relation q( v ) of the wave. A detailed discussion of this procedure is found in Refs. 12, 13.
Figure 5 shows the dispersion relation of the wave in a
linear chain arrangement of 10 dust particles at a gas pressure of 10 Pa. The real part of the dispersion relation shows
an almost linear increase with frequency, whereas the imaginary part is almost constant for excitation frequencies below
2 Hz and exhibits a quite pronounced increase at higher frequencies. The theoretical dispersions of a dust lattice wave
are fitted to the experimental results. The best fit is obtained
for a damping coefficient b 56.3 s21 , a charge number Z
59000 and a screening strength k50.9. These values agree
well with those determined from the resonance curve ~Fig.
3!. The value of k coincides with those of previous
experiments.12,13 The theoretical dispersion for these values
FIG. 5. Dispersion relation of the waves in the linear chain arrangement.
The experimental values of the real and imaginary part of the wave vector
are represented by the symbols. The best fit theoretical dispersion for
k50.9, Z59000 and b56.3 s21 is shown by the lines. The shortest wavelength is obtained at f max53.35 Hz. Theoretical dispersions for the same
values of k and Z, but without friction are also shown for comparison. The
cutoff wavelength is then obtained at a Debye frequency f D 53.21 Hz.
is shown for comparison in Fig. 5 along with the experimental data.
The maximum frequency for this damped system is
found for f max53.35 Hz ( v max521.4 s21 ) but at a wave
vector of q r a52.25. So in the damped system the shortest
principal wavelength corresponding to q r a5 p is not
reached. Taking the values of Z and k for a frictionless system ~b50!, also shown in Fig. 5, the dispersion relation results in the Debye frequency of f D 53.21 Hz ( v D 520.2
s21 ) for q r a5 p . The Debye frequency for a monolayer
plasma crystal for these values of Z and k is calculated to be
f D 53.4 Hz. Taking into account the experimental errors in
determining the wave vector q and the accuracy of the fitting
procedure in neither case a Debye frequency above 4.0 Hz
was found under these conditions. In conclusion, the dust–
dust interactions give rise only to frequencies decisively below the vertical resonances.
D. Parametric resonance and ‘‘sublimation’’
When the monolayer plasma crystal is excited by a lowfrequency voltage on a wire a phase transition of the plasma
crystal at the parametric resonance at 2 v 0 /(2 p )528 Hz
from an ordered solidlike structure to a gaslike structure is
observed, a ‘‘sublimation’’ transition in classical terms. This
‘‘sublimation’’ transition was previously reported in Refs.
14, 15 in a very similar experiment performed in krypton.
There the ‘‘sublimation’’ transition was found around a frequency f 540 Hz. In Ref. 15 this transition was attributed to
the Debye frequency according to Eq. ~10!. It was assumed
that this excitation frequency f is the Debye frequency with
the corresponding wave vector q r a5 p and that a further
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2698
Schollmeyer et al.
Phys. Plasmas, Vol. 6, No. 7, July 1999
increase in frequency leads to the phase transition since
shorter wavelengths cannot exist in the system.
In Refs. 14,15 two observations are reported that support
our interpretation in terms of the parametric resonance model
also for those experiments. First, it is stated that at the border
of the frequency range, where the phase transition takes
place, the transition occurs only after a transient time of
some seconds. Second, the frequency range for the transition
grows with the voltage applied to the wire. Both findings are
distinctive features of a parametric resonance where the
growth rate of the unstable oscillations is very small ~corresponding to large rise times! at the border of the instability
regions and where the width of the unstable regions grow
with increased modulation.
Our experiments, however, clearly show that the ‘‘sublimation’’ transition is related to a parametric excitation at
2 v 0 of the dust particles in the vertical direction. The dispersion relation was measured almost up to the Debye frequency f D which is found to be an order of magnitude less
than the parametric resonance and no phase transition is observable near f D . Even though a complete description of the
horizontal and vertical oscillations would require us to include the effects due to a finite chain and particle charge
variations in the sheath, we can conclude from our findings
that the two frequencies 2 v 0 and f D belong to different processes in the plasma crystal.
The question arises as to why the plasma crystals turn
into a gaslike state at the parametric resonances. When the
resonance is excited, the dust particles have vertical oscillation amplitudes of 0.5 mm and more. At a frequency of 28
Hz this corresponds to a kinetic energy of the dust particle of
23104 eV. From plasma crystal experiments4,5 it is known
that a phase transition from an ordered structure to a liquid
phase occurs at particle energies of a few eV and gaslike
structures are found at kinetic particle energies of about 20
eV. This energy is one thousandth of the kinetic oscillation
energy at the vertical parametric resonance. Therefore, even
a small disturbance of a purely vertically oscillating particle
transfers a large amount of energy into its horizontal motion
which then influences neighboring particles, thus leading to a
transition into the gas phase. Exactly, this behavior was reported in Ref. 15 where the transition starts with a single
particle leading to an avalanche-like particle heating. Since
this phase transition is in no way related to intrinsic dust–
dust interactions, the term ‘‘sublimation’’ is misleading. The
phase transition is externally induced by a parametric heating
of the particles in the plasma sheath.
V. SUMMARY
We have presented experiments on the vertical and horizontal oscillations of monolayer plasma crystals in the sheath
of an rf discharge. From the different resonances properties
of the dust-plasma and dust–dust interactions have been
measured. From the excitation of waves in the plasma crystal, the charge and the screening strength for the dust–dust
interactions can be derived; from the vertical resonances in
the plasma sheath the dust charge can be determined. The
use of a biased wire as an excitation method leads to parametric resonances which show that the wire itself strongly
distorts the plasma trap. We conclude that the phase transition of monolayer plasma crystals from an ordered to a gaslike state is due to the parametric heating of the dust particles
in the sheath rather than to stagnating wave energy at the
Debye frequency.
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