Download A. Point Charge Model for Ion Wakefields

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1
Mode couplings and conversions for horizontal
dust particle pairs in complex plasmas
Ke Qiao, Member, IEEE, Jie Kong, Zhuanhao Zhang, Lorin S. Matthews, Member, IEEE, and Truell
W. Hyde, Member, IEEE

Abstract—The normal modes for horizontal dust particle pairs
in a complex plasma are investigated using two methods, a
numerical calculation and a molecular dynamics (MD)
simulation. The ion wakefield downstream of each particle and
the variation of charge with particle levitation height are taken
into account independently. It is shown that both mechanisms
can create mode couplings, or hybrid modes. As the modes
hybridize, their frequencies are altered. Mode conversions and
resonance instabilities are also observed and examined near the
resonance points. The resulting power spectrum clearly
resembles the experimentally observed high energy density
signature for large crystals. The criteria for the occurrence of
both the mode conversion and resonance instability are found to
be accurately calculated by the double derivative of the
interparticle potential, for both the ion wakefield and charge
variation cases.
Index Terms—Complex plasma, dusty plasma, mode coupling,
mode conversion
I. INTRODUCTION
P
LASMA crystals were first observed in the laboratory in
1994 [1]-[3] and for the almost two decades since have
been an integral part of the complex plasma research field. A
typical laboratory plasma crystal is formed of micron-sized
dust particles, each negatively charged to 103-104 elementary
charges and then levitated in the plasma sheath above the
lower electrode due to the balance between gravity and the
sheath electric field. For dust particles located in the plane
parallel to the lower powered electrode, the interparticle
potential is of a repelling Debye–Hückel (or Yukawa) type,
allowing the formation of a plasma crystal under the proper
external horizontal confinement.
Recently, investigations into the dynamics of dust particle
pairs have created great interest in both the experimental [4][10] and theoretical [11]-[16] complex plasma communities.
The simplicity of a two-particle system makes it ideal for
studying the interparticle force between dust grains and often
provides the ability for direct measurement of this force [4][7]. However, the underlying physics discovered employing
this “simple” system has proven surprisingly complex. For
example, particle pairs were initially employed to examine the
Authors are with the Center for Astrophysics, Space Physics and
Engineering Research at Baylor University in Waco, TX, 76798 USA, e-mail:
[email protected].
non-Hamiltonian influences that can be created by the ambient
plasma [6]-[10]. Probably the best known of these is the nonreciprocal ion wakefield force. This force is created by the
vertical ion flow within the sheath [6]-[19] and has been
shown to be the cause of many interesting effects in plasma
crystals such as the vertical alignment of particles [6]-[10],
[17], horizontal and vertical wave coupling [21]-[27] and twodimensional crystal melting [23], [25], [26]. Particle charge
variation within the sheath, which is dependent upon the
height above the powered electrode, can also be induced by
the surrounding plasma [14], [15], [21], [28]-[32]. This effect
can lead to interesting instabilities [14], [15], [21], [28], [29],
[32]. In each of the cases above, the system’s normal modes
and their related instabilities lie at the core of the physics
involved.
To date, normal modes and their resulting instabilities have
been examined theoretically using either a Hamiltonian or
perturbation theory approach. A Hamiltonian approach
typically involves the introduction of an effective
Hamiltonian, allowing the stability of the system to be
analyzed by mapping its effective potential [10]-[12]. On the
other hand, a perturbation theory approach involves solving
Newton’s equations for the mode frequencies, employing a
linear perturbation method [10], [13]-[16]. Regardless of
which method is employed, the majority of studies referenced
here [10]-[12] focus on an examination of the horizontalvertical transition rather than development of detailed mode
spectra. Of the few that do examine the mode spectra, each
disregarded either mode coupling [13], particle screening
and/or charge variation [16] or required a priori assumptions
such as assuming a smaller ion wakefield length as compared
to the interparticle distance [14] or the assumption of a
generalized interparticle potential rather than employing a
specific model [15]. Since recently observed mode coupling
and melting [23], [25]-[27] are based on just such an
investigation of the power spectra, a numerical simulation for
these systems is sorely needed. Such a simulation would allow
the power spectra to be obtained in the same manner as from
experiment and then compared directly to experimental data.
Finally, in all previous research the influence from the ion
wakefield and charge variation have always been treated
together. The ability to examine the influence of the ion
wakefield and charge variation independently provides the
ability to determine whether either produces independent
effects which can be observed experimentally.
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In this research, two independent methods are employed to
examine the mode spectra, mode couplings, mode conversions
and resonance instabilities for a horizontal dust particle pair
under typical laboratory plasma conditions. In both methods,
the ion wakefield and charge variation are considered
independently of one another. The first method uses a
numerical calculation starting from Newton’s equations, thus
treating the system as non-Hamiltonian from the outset. This
allows equilibria to be determined by simply mapping the
particle positions over coordinate space and then finding the
force balance points. Normal modes and instabilities (for
oscillation modes producing negative frequencies) can then be
obtained naturally from the dynamical matrix. The numerical
nature of this method requires far fewer a priori assumptions
than have previous analytical methods [10], [13]-[15] and
allows examination of particle motion for each mode directly
from the eigenvectors. Thus, mode conversions can be
examined in detail, especially around critical points.
The second method employs a molecular dynamic (MD)
simulation using a box_tree code [33]-[35]. In addition to the
advantages listed above, this method also provides the normal
mode power spectra, indicating mode couplings and resonance
instabilities and allowing direct comparison with data obtained
experimentally for larger crystals [25]-[27]. This method
additionally has the advantage of allowing for the examination
of nonlinear effects and provides a natural extension to many
particle systems.
Section 2 introduces the physical models used to examine
the ion wakefield and charge variation for both the numerical
calculation and MD simulation method described. Results are
given in Sections 3 and 4, and then discussed and
quantitatively compared with previous research in Section 5.
Conclusions can be found in Section 6.
II. PHYSICAL MODELS
A. Point Charge Model for Ion Wakefields
The most detailed theoretical model for the ion wakefield to
date obtains the potential using linear response theory (Fig. 1)
[11], [18], [19]. Although robust, the mathematical
representation required by this theory is extensive. A much
simpler model is the point-charge model, where one assumes
the ion wakefield around a dust particle with negative charge
Q takes the form of the potential created by the particle itself
and a positive point charge q located at a distance l beneath it
(Fig. 2) [8],[10],[14],[16],[17],[21],[22],[24]. Selecting the
location of the point charge to correspond with the first
minimum in the wake field potential as obtained from linear
response theory (Fig. 1) and selecting appropriate values of q,
the potential calculated using the point charge model agrees
qualitatively with the potential determined employing linear
response theory. In this case, the distance between the dust
particle and the point charge (or the potential minimum) is on
the same order of magnitude as the vertical particle spacing
found in experiments. This model will be used in this research
since it provides a reasonable analytical approximation
2
[8],[10],[14],[16],[21],[22],[24] while also having the
advantage of highlighting qualitative features of the physical
process, such as the positive space charge effect beneath the
dust particle.
In the point charge model the interaction potential between
two particles is given by [14], [21]
U (r , z )  Q
exp(  r 2  z 2 /  )
r2  z2
q
exp(  r 2  ( z  l ) 2 /  ) , (1)
r 2  ( z  l )2
where r and z represent the radial and vertical distance
between the two dust particles respectively, and λ is the
screening length. In the sheath region where the majority of
dust particles levitate in experiments on Earth, the ion-flow
velocity is such that the mean kinetic energy of the ions equals
or exceeds that of the bulk electrons [18]. For this case,
experimental measurements have shown the screening length
to be roughly equal to the bulk-electron Debye length [4], [5];
thus, screening is often attributed to the electrons rather than
the ions [19], [20]. However, due to the complexity of the
plasma sheath, the origin of this screening remains an open
question, and will not be specified here [16]. The interaction
between the particles as defined above is assumed to include
the dust-dust interaction, the interaction between dust particle
1 and virtual particle 2, and the interaction between dust
particle 2 and virtual particle 1 [8], [16]. A primary advantage
of this method is that it properly addresses the nonHamiltonian nature of this interaction from the outset [24][26].
B. Model for Charge Variation with Height
The variation of charge with height is modeled
independently of the ion wakefield; thus, the interparticle
potential is assumed to be Yukawa in nature but with the
charge Q given as a function of height z,
U (r , z )  Q( z )
exp(  r 2  z 2 /  )
r2  z2
,
(2)
where r, z and λ are defined as above. It will be assumed the
particle charge can be linearly approximated for small
oscillations around the equilibrium height as Q(z) = Q0 + Q′zz
with Q′z = (dQ/dz)0, where the subscript ‘0’ denotes the
derivative taken at the equilibrium height, z = 0 [14]. In this
research, as in [14], the charge is considered to be an
instantaneous function of height, i.e. charge delay is ignored.
This is because the charging time scale (10-7 s) [36] is much
smaller than that for dust oscillation (> 10-2 s). Under this
condition, as shown in [29], the charge delay only affects the
damping rate, not the mode frequencies.
In a typical experiment on earth, the dust particle pair is
confined in the vertical direction within a potential well
formed by the electrostatic and gravitational fields, and in the
horizontal direction by the electric field which is assumed to
be parabolic as produced by the experimental setup [37]. For
this case, the external forces acting on the particles are
Fz ,ext  Ez x, z  Qz   Mg
Fx,ext  Ex x, z  Qz 
(3)
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where x and z are representative particle coordinates. For
simplicity, and taking into account the symmetry of the
system, the y-coordinate is ignored here. The electric field is
assumed to be linear for small oscillations; thus Ez = E0 - Ez′z
and Ex = -Ex′x, where E0 is the vertical electric field at the
equilibrium position. Equation (3) allows parabolic
confinement from electric fields in both the vertical and
horizontal [38] directions.
Dimensionless variables will be applied throughout the
remainder of the paper unless specifically stated otherwise,
with Q0 defined as the unit charge and λ as the unit length. For
example, x and z will be normalized by λ, Q and q by Q0. For
simplicity and without loss of generality, a particle mass of M
= 1 will also be employed.
III. NUMERICAL CALCULATIONS
A. Screened Coulomb Interactions
To determine the equilibria of the system, the force
components Fαi were numerically calculated over two-particle
coordinate space. In our notation, Greek letters (α, β …) are
used to denote coordinate indices x (α, β… = 1) or z (α, β… =
2) while i, j… denote the particle number index. In this case,
equilibria correspond to Fαi = 0. To determine the normal
modes around a specified equilibrium point, the 4 × 4 dynamic
matrix Mαi,βj = dFαi/drβj must be established, where rβj
represent particle positions. The eigenvalues ω2n (where n is
the mode number and can be 1, 2, 3 or 4) correspond to the
squared mode frequencies, and the eigenvectors Vαi,n
correspond to the oscillation amplitudes and directions for a
given mode; i.e., Vαi,n is the oscillation amplitude for the ith
particle in direction α, for the nth mode.
In order to establish a baseline, numerical calculations were
initially conducted assuming a simple screened Coulomb
interaction (in other words, disregarding both the ion
wakefield and any possible charge variation) between two
particles, each having charge Q0. Values of Ex′ varying over a
range 1 ≤ Ex′ ≤ 10 while holding Ez′ constant at Ez′ = 10 were
used. In agreement with previous research [16], four normal
modes were obtained for each combination of Ex′ and Ez′ (Fig.
3), corresponding to the horizontal center of mass (COM)
motion (Hc mode, ω1 = ωHc), horizontal relative motion (Hr
mode, ω2 = ωHr), vertical relative motion (Vr mode, ω3 =
ωVr), and vertical COM motion (Vc mode, ω4 = ωVc). In this
case, a pure mode is defined as that consisting of only one
uncoupled mode, as opposed to a hybrid mode, which is
composed of two or more modes coupled to each other.
Obviously all modes resulting from a screened Coulomb case
are pure modes.
The mode frequencies as a function of Ex′ are shown in Fig.
3. When Ex′ = 1, the frequencies of the relative and COM
modes in the vertical direction, ωVr and ωVc, are approximately
equal, hinting at a degeneracy when the horizontal
confinement is much weaker than the vertical confinement. On
the other hand, when Ex′ = Ez′ = 10, ωVr = 0; at this point the
system becomes unstable and a horizontal-vertical transition
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occurs. For the simple Yukawa (screened Coulomb) case, the
threshold condition for this transition is Ez′/Ex′ = 1.
Experimentally ωHc and ωVc are much easier to measure than
are Ex′ and Ez′. Thus, this condition is more commonly
referred to as ωVc/ωHc = 1. As Ex′ increases from 1 to 10, ωVr
decreases, ωHr and ωHc increase, and ωVc remains constant.
These trends agree with earlier derived results for a dust
particle pair assuming only an unscreened Coulomb
interaction [16].
B. Ion Wakefield
With a baseline now in place, the point charge model was
employed to include the effect of the ion wakefield. Three
values of q were chosen, q = 0.2, 0.5, and 0.8, corresponding
to a positive space charge much smaller, half as much, and
approximately equal to the dust particle charge. A distance l =
0.5 was used in order to match typical experimental values [6],
[8]-[10]. Inspection of the resulting eigenvectors produced by
the dynamic matrix revealed that the COM modes (Hc and Vc)
remain as pure modes having frequencies ωHc and ωVc,
unchanged from the screened Coulomb case. However, the
relative modes have now become hybrid modes, with the Hr
mode coupled to the Vc mode, producing a Hr-Vc hybrid
mode and the Vr mode coupled to the Hc mode, producing the
Vr-Hc hybrid mode. These mode couplings are in agreement
with previous results derived for a generalized interparticle
potential [15].
Due to mode coupling, the mode frequencies are altered.
The Hr-Vc hybrid frequency, ωHr-Vc, increases while the Vr-Hc
hybrid frequency, ωVr-Hc, decreases below the pure mode
frequencies ωHr and ωVr (as obtained from the simple screened
Coulomb case) (Fig. 4). This difference is greater when the
point charge q increases. Lowering ωVr-Hc causes the threshold
value for Ex′th (the horizontal-vertical transition) to decrease
dramatically (Fig. 4 (b) ).
Mode conversions can most easily be examined via the
mode polarization, defined as P ≡ Vh/Vv, where Vh (Vv) is the
horizontal (vertical) oscillation amplitude of a specified hybrid
mode. A peak in the polarization P (or reverse polarization
1/P) as a function of Ex′, occurs for both the Hr-Vc and Vr-Hc
hybrid modes as shown in Fig. 5. Comparing Fig. 5 with Fig.
4, it can be seen that in this case the peak sits at the
intersection between the hybrid mode and the pure mode to
which it is coupled. For example, when frequencies of the HrVc and Vc modes are far apart (Δω2 > C, where Δω2 is the
difference between the squared frequency of the two modes
and C is the coupling term, the significance of which will be
discussed in Section 5), the vertical amplitude of the Hr-Vc
mode is much smaller than the horizontal amplitude, Vv << Vh.
However, when ωHr-Vc is close to ωVc (Δω2 < C), resonance
effects can create a sudden increase in the vertical amplitude
of the hybrid mode, causing it to become much larger than the
horizontal amplitude, that is, Vv >> Vh (Figs 4(a), 5(a)). This
indicates a mode conversion at the point of resonance (when
the two frequencies are equal). Similar arguments can be made
for the Vr-Hc mode (Figs 4(b), 5(b)). For both modes, the
position of this peak shifts as q increases.
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C. Charge Variation
Next, the effect of the charge variation was examined as
defined in 2 and 3, independent of the ion wakefield. To
accomplish this, the electric field E0 was first estimated at the
equilibrium point employing representative experimental
parameters. Assuming a particle mass and charge, M ≈ 1 × 1013
kg, Q0 ≈ 1 × 10-15 C [5], E0 was calculated using the force
balance equation, E0Q0 = Mg to be E0 ≈ 1 × 103 N/C. Since Q0
and λ are the unit charge and length respectively, the unit
electric field is defined as Eunit = (1/4πε0)Q0/λ2, which gives a
value for Eunit ≈ 40 N/C for a typical screening length λ ≈ 500
μm [5]. Thus, E0 ≈ 20 in normalized units represents a
reasonable value for the calculations at hand. In addition,
values of Q′z = 0.1 and 0.2 are assumed.
With the charge variation introduced, the horizontal modes
(Hc and Hr) remain pure with frequencies ωHr and ωHc, the
same as those found for the simple screened Coulomb case
(Figs 6, 7). The vertical modes however, become hybrid
modes with the Vr mode now a Vr-Hc hybrid mode and the Vc
mode becoming an Vc-Hr hybrid mode. The frequencies ωVr-Hc
and ωVc-Hr both exhibit a difference from ωVr and ωVc by a
value of –E0Q′z, thus increasing for negative Q′z and
decreasing for positive Q′z (Figs. 6 and 7).
Examining the mode polarization (Fig. 8), it is again found
that when the hybrid mode frequency ωVr-Hc(ωVc-Hr) is far from
that of the pure mode, ωHc(Hr) (Δω2 > C), the horizontal
amplitude of the hybrid mode is much smaller than its vertical
amplitude, i.e. Vh << Vv. When ωVr-Hc(ωVc-Hr) is approximately
equal to ωHc(Hr) (Δω2 < C), resonance effects create a sudden
increase in the horizontal amplitude, making it much larger
than the vertical amplitude, i.e. Vh >> Vv, which again denotes
mode conversion occurring at the resonance point. The
maximum exhibited by the mode polarization as a function of
Ex′ again shifts as Q′z changes (Fig. 8).
IV. MD SIMULATIONS
Finally, MD simulations were conducted with the individual
algorithms for the ion wakefield (as modeled by the point
charge method) and the charge variation due to height built
into the box_tree algorithm [33]-[35]. This allowed them to
be turned on and off independently. All simulations begin
from an initial random distribution of two particles in a box of
10×10×10 mm3. The initial center of mass of the two-particle
system is at the center of the box, which is also the origin of
the coordinate system. The system is unbounded other than the
trapping provided by the parabolic confinement created by the
electric fields in both the vertical and horizontal directions.
(See Eq. 3). Simulation parameters are chosen to represent
normative experimental values, with the particle diameter d =
8.89 µm, the particle density ρ = 1.51 g/cm3, the particle
charge Q = 2.4 × 10−15 C and the screening length λ = 300 µm.
In order to compare with numerical calculation results, q, l, E0,
Q′z, Ex′ and Ez′ are all chosen to be the values used in the
numerical calculations. Since this work is focused on the
oscillation modes and mode coupling, frictional damping is
not included.
4
Particle pair systems were formed, stabilizing
approximately 3 s after the start of simulations. System
oscillations were tracked for 16 s producing data files showing
each particle’s position and velocity at a time interval of 1/60 s
for analysis.
Mode spectra were obtained employing two methods. The
first employed a direct Fourier transformation of the average
particle velocity (Figs. 9, 10), while the second method is
similar to one used by Melzer [39], where the time series of
the particle velocities is projected onto the direction of the
eigenvectors for each pure mode (i.e. with no coupling due to
either the ion wakefield or charge variation). The mode power
spectrum is then obtained in the form of an energy density
through Fourier transformation [39] (Fig. 11 (a)).
For both the ion wakefield (Fig. 9) and the charge variation
(Fig 10) cases, the normal modes can be clearly seen in the
spectra obtained from the first method (Fig. 9(a)(b), Fig.
10(a)(b)). The peaks in the spectra can be identified through
comparison with the numerically calculated frequencies (Fig.
9(c), Fig. 10(c)). Excellent agreement between mode
frequencies obtained from numerical calculation and MD
simulation can be seen. The spectra also clearly identify the
hybrid character or coupling between modes. For example, for
the ion wakefield case, three peaks appear in the spectra
obtained from the horizontal motion of the particles (Fig. 9
(a)): the two peaks for the horizontal Hc and Hr modes and a
third suppressed peak. Comparing the suppressed peak with
the spectra obtained from the vertical motion of the particles
(Fig. 9 (b)), one finds that it falls at the exact frequency of the
Vr mode. Thus, the Vr mode actually has both vertical and
horizontal components, i.e. it is a Vr-Hc hybrid mode. Similar
arguments can be made for the Hr-Vc mode for the ion
wakefield case (Fig. 9 (b)), and the Vc-Hr and Vr-Hc modes
(Fig. 10 (a)(b)) for the charge variation case.
It is interesting to note that for the charge variation case, the
vertical modes are always coupled to horizontal modes
(through the Vr-Hc and Vc-Hr hybrid modes). This results in
any spectrum obtained from the horizontal motion showing
both horizontal and vertical modes (Fig. 10(a)), while the
spectrum produced from vertical motion shows only vertical
modes (Fig. 10(b)). This may be related to recent experimental
results in 2D Coulomb crystals, where a high frequency
optical branch was observed in the dispersion relations
obtained from horizontal motion [27]. Experimental research
on the relationship between the two is currently underway at
CASPER.
Mode spectra peaks can also be identified by the
eigenvectors obtained via Fourier transformation (Fig. 9 (d),
Fig. 10 (d)). Again, the horizontal and vertical components of
these modes can be clearly seen. Furthermore, by inspecting
the manner in which these components change at the points of
resonance (Fig. 9(d), 10(d)), any conversion of the hybrid
modes can be easily recognized. Fig. 9(d) (Fig. 10(d)) shows
the change of direction of the eigenvector of the Hr-Vc (VcHr) mode as its frequency moves closer to that of the pure Vc
(Hr) mode. The Vc (Hr) component increases dramatically in
magnitude as the two frequencies approach one another,
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dominating the Hr (Vc) component, clearly demonstrating a
mode conversion, in agreement with the numerical calculation
results.
Power spectra for the pair system discussed were also
obtained employing the second method. As a representative
example, Fig. 11 (a) shows the power spectrum including the
ion wakefield, when the Vr-Hc and Hc mode frequencies are
equal. The coupling between the Hr and Vc modes (Hr-Vc
hybrid) can be clearly seen in the spectrum, as indicated by the
light lines at mode number 2 and 4 with the same frequency
(ω ≈ 3.7). Resonance between the Vr-Hc and Hc modes can
also be clearly seen in that the spectral line for the pure Hc
mode is much brighter than those of all the other modes, as
shown in Figure 11 (a). It is interesting to note that this clearly
resembles the experimentally observed high energy density
signature at the intersection point of the horizontal and vertical
dispersion relations, in the power spectra for large crystals
[25]-[27]. At the same time, the average particle velocity
increases dramatically in a direction corresponding to the pure
Hc mode (the horizontal direction), exhibiting a sharp peak at
3 < Ex′ < 4 (Fig. 11 (b)). In combination, these show that the
resonance instability occurs as the frequency of the Vr-Hc
mode approaches that of the Hc mode. Similar types of power
spectra can be obtained for all of the hybrid modes (the Hr-Vc
mode for the ion wakefield case, and the Vc-Hr and Vr-Hc
modes for the charge variation case) as their frequencies
approach those of the pure mode with which they are coupled,
exhibiting resonance instabilities.
V. DISCUSSION
In this study, it was shown that the introduction of the ion
wakefield interaction causes the frequency of the Hr-Vc mode
to increase (Fig. 4 (a)). and the Vr-Hc mode to decrease (Fig.
4 (b)) , while the Hc and Vc modes remain unaffected. The
decrease in the Vr-Hc frequency causes the horizontal-vertical
transition threshold Ex′th to decrease from ten for q = 0, to
approximately seven for q = 0.5 and six for q = 0.8 (Extending
the frequency curves provides Ex′th as the x-axis value at the
intersection point.) These values of Ex′ correspond to the ratio
between the vertical and horizontal sloshing frequencies, i.e.,
ωVc/ωHc =
E z ' / E x ' = 1.2 and 1.35. These results are in
qualitative agreement with previous experimental results [8][10], although the quantities predicted here are smaller than
those observed. The most likely reasons for this are: (1)
experimentally, friction can become a factor impacting the
horizontal-vertical transition threshold [9]. (2) The values of q
and l used in this study may well differ from those present in a
given experiment, or (3) there may be some limitation on the
point charge model itself.
Note that the current results do not contradict those reported
in [14], which shows a decrease in the Hr frequency and
increase in the Vr frequency when the ion wakefield is
introduced. The reason for this is that the frequencies are
treated as functions of Δ (where Δ is defined to be the
interparticle spacing) in [14], allowing the frequencies for
5
different ion wakefields to be compared for equal Δ, thus
different horizontal confinements. In this study, frequencies
are treated as the functions of horizontal confinement Ex′; thus
frequencies are being compared for equal confinements, and
therefore different Δ. For the same reason, in [14] the Hc
mode frequency was found to be modified by the ion
wakefield force, while it was found to be unaffected in this
study. It is also important to note that Ex′ is an experimentally
tunable controlling parameter and does not depend on the
particle system, while Δ is determined by the structure of the
particle system itself. Thus it is more natural to study the
alteration of mode frequencies induced by varying ion
wakefields while keeping the horizontal confinement Ex′
constant.
The charge variation with height (Fig. 6 and 7) also alters
the frequencies of the Vr-Hc and Vc-Hr modes while the Hc
and Hr mode frequencies are unchanged. Both the Vr-Hc and
Vc-Hr mode frequencies decrease for positive Q′z and increase
for negative Q′z by an increment of –E0 Q′z/M. This frequency
change causes the vertical sloshing mode shown in Fig. 6 (b),
to become unstable when its squared frequency ωVc2 < 0,
corresponding to a region of the plasma sheath close to the
electrode, where (EQ)′z > 0. For a single particle system, this
causes the particle to drop, in agreement with theoretical
predictions [29] which have been recently verified
experimentally [32]. This type of instability is not affected by
charge delay, as shown in [29]. For the particle pair system,
the horizontal-vertical transition occurs due to instability of
the Vr-Hc mode (Fig. 6 (a)) when its squared frequency is
zero. However, although positive Q′z causes the decrease of
the Vr-Hc frequency, unlike the ion wakefield case, this does
not cause the horizontal-vertical transition to occur for ωVc/ωHc
> 1, i.e., the condition ωVc/ωHc = 1 for the simple Coulomb
case remains. The reason for this is that the frequency function
shifts for different values of Q′z are parallel, as shown in Fig. 6
(a). Thus the experimentally observed horizontal-vertical
transition threshold change shouldn’t be caused by the charge
variation with height.
The criterion for two coupled modes to have a resonant
instability is Δω2 < C [15], [21], where Δω2 is the difference
between the squared frequency of the two modes and C is the
coupling term. Fig. 12 (a) shows Δω2 between the Vr-Hc and
pure Hc modes for the ion wakefield case with q = 0.8. Fig. 12
(b) shows Δω2 between the Vc-Hr and pure Hr modes for the
charge variation case with Q′ = 0.2. The solid dots represent
the values of Δω2 while dashed lines represent the values of
the second derivative U′′rz, where U is the interparticle
potential as shown in (1) and (2). Comparing Fig. 12 (a), (b)
with Fig. 12 (c), (d), which show the resonant instability
directly by the particle velocities obtained from simulation, it
can be seen that the position and range of Ex′ for the
occurrence of instability agree exactly with the range of Ex′
where the dots lie above the dashed line, i.e. the differences in
frequencies squared are greater than U′′rz. Figs. 12 (a), (b) can
also be compared with Fig. 12 (e) and Fig. 12 (f), which shows
the occurrence of the mode conversions for the two cases
respectively. It can be seen that the occurrences of mode
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conversion also fall in exactly the same range of Ex′ for both
cases. Hence the conclusion that the coupling term C, as the
criteria for the occurrence of both the instabilities and the
mode conversions, can be accurately calculated by the double
derivative U′′rz. This value is on the same order as predicted in
[15] for a generalized interparticle potential.
VI. CONCLUSION
In summary, two independent methods were used to
examine a horizontal dust particle pair located within a
complex plasma operating under normative laboratory
conditions.
The first of these methods employed a numerical
calculation starting from Newton’s equations. This allowed
the system to be treated in a non-Hamiltonian manner from the
outset and required far fewer a priori assumptions than have
previous analytical methods. The second used a molecular
dynamics (MD) simulation to examine the normal mode
power spectra, allowing direct comparison with data obtained
experimentally for large crystals. In all cases, the ion
wakefield and the charge variation with height were examined
independently.
Both the ion wakefield and charge variation (with height)
were found to induce mode coupling, creating hybrid modes
within the system. The ion wakefield transformed relative
modes Hr and Vr into hybrid modes, Hr-Vc and Vr-Hc.
Varying the charge with height caused vertical modes Vr and
Vc to become hybrid modes, Vr-Hc and Vc-Hr. In both cases,
as the modes hybridized, their frequencies were also altered.
The ion wakefield created a change in the threshold value at
which the dust pair underwent the horizontal-vertical
transition. As such, it provides one mechanism which could
explain the experimentally observed threshold shift seen
experimentally [8]-[10]. The lowering of the vertical mode
frequencies created by the charge variation, on the other hand,
is related to the experimentally observed unstable region in the
lower plasma sheath [29],[32].
Mode conversions and resonance instabilities were also
examined at the dust particle pair’s points of resonance, i.e.,
where the frequencies of the hybrid modes are almost equal to
the frequency of the pure mode with which they are coupled.
At these points, the hybrid modes are converted, i.e. their
oscillation direction changes suddenly to align with the
direction of oscillation of the pure mode. At the same time, the
kinetic energy of the pure mode increases dramatically,
causing a corresponding increase in the velocity of the
particles in the direction of the pure mode. The resulting
power spectrum was shown to be similar to the high energy
density signature experimentally observed in large crystals at
the intersection of the horizontal and vertical dispersion modes
[25]-[27].
Finally, the criteria for the occurrence of both the mode
conversion and the resonance instability were shown to be
accurately calculated by the second derivative of the
interparticle potential, for both the ion wakefield and the
charge variation cases.
6
As mentioned, the MD simulation allows examination of
nonlinear effects and provide a natural extension to many
particle systems. This is currently underway at CASPER and
will be reported in a future publication.
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7
FIG. 3. Normal mode frequencies as a function of the
horizontal field gradient for a horizontal dust particle pair
(screened Coulomb case).
Fig. 1. Contour plot of the potential around a dust particle
(centered at the origin) obtained using linear response theory.
Dashed contours represent the negative potential surfaces,
with the tick marks indicating the direction of decreasing
potential. Taken from [11].
FIG. 4. The frequencies of the (a) Hr-Vc and Vc modes and
(b) the Vr-Hc and Hc modes as functions of the horizontal
electric field gradient (ion wakefield case).
FIG. 2. The physical model for the dust particle pair. It is
confined in the vertical direction within a potential well
formed by the electrostatic and gravitational fields, and in the
horizontal direction by the electric field produced by the
experimental setup. The ion wakefield is modeled through
inclusion of a virtual point particle with positive charge q a
distance l below the dust particle. The charge is varied with
height.
FIG. 5. Mode polarization for the (a) Hr-Vc and (b) Vr-Hc
hybrid modes as a function of the horizontal electric field
gradient (ion wakefield case).
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8
FIG. 6. The frequencies of the (a) Vr-Hc and Hc modes and
(b) the Vc-Hr and Hr modes as a function of the horizontal
electric field gradient when a positive charge gradient is taken
into account.
FIG. 9. Results obtained from MD simulations for the ion
wakefield case (q = 0.5). Power spectral density (PSD)
obtained from (a) horizontal and (b) vertical average particle
velocity (Ex′ = 3.47). (c) Hr-Vc and Vc mode frequencies in
the vicinity of the mode conversion (2.2 < E x′ < 3.6). (d)
Eigenvectors for the Hr-Vc mode, showing an increase in the
Vc component as the resonant frequencies approach each
other.
FIG. 7. The frequencies of the (a) Vr-Hc and Hc modes and
(b) Vc-Hr and Hr modes as functions of the horizontal electric
field gradient when a negative charge variation is taken into
account.
FIG. 8. Mode polarization for the Vc-Hr (a, c) and the Vr-Hc
(b, d) hybrid modes as functions of the horizontal electric field
gradient when positive (a, b) and negative (c, d) charge
variation is taken into account.
FIG. 10. Results obtained from MD simulations for the charge
variation case (Q′ = 0.2): Power spectral density (PSD)
obtained from horizontal (a) and vertical (b) average particle
velocity (Ex′ = 2.31). (c) Vc-Hr and Hr mode frequencies in
the vicinity of the mode conversion (1.5 < Ex′ < 2.5). (d)
Eigenvectors of the Vc-Hr mode, showing an increase in the
Hr component as the resonant frequencies approach each
other.
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Fig. 11. (a) The spectrum shown by energy density at E x′ =
3.63 for the pair system with ion wakefield q = 0.8, l = 0.5.
The mode numbers 1, 2, 3, 4 correspond to the pure modes
Hc, Hr, Vr, Vc. The bright streak for mode 1 at ω ≈ 2 indicates
the resonance-induced high energy density for the Hc mode;
the light lines at mode numbers 2 and 4 with the same
frequency (ω ≈ 3.7) indicate the hybrid Hr-Vc mode. (b) The
average particle velocity as a function of Ex′, showing a
resonance in the horizontal direction.
Fig. 12. (a) Δω2 (represented by the solid dots) between the
Vr-Hc and pure Hc modes for the ion wakefield case with q =
0.8 (b) Δω2 (solid dots) between the Vc-Hr and pure Hr modes
for the charge variation case with Q′ = 0.2. The dashed lines
represent the values of the second derivative U′′rz. (c-d)
Average horizontal particle velocities obtained from
simulation with identical parameters as in (a) and (b),
respectively. (e-f) The Vr-Hc(Vc-Hr) mode polarization
obtained from numerical calculations with identical
parameters as in (a) and (b), respectively. All values in this
figure are shown as functions of the horizontal confinement
Ex′.
9
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10
1994 and 1998, respectively. She is an Associate Professor in
the Physics Department at Baylor University and Associate
Director of the Center for Astrophysics, Space Physics, and
Engineering Research (CASPER). Previously, she worked at
Raytheon Aircraft Integration Systems as the Lead
Vibroacoustics Engineer on NASAs SOFIA (Stratospheric
Observatory for Infrared Astronomy) project.
Ke Qiao (M04) received the B.S. degree in physics from
Shandong University, Qingdao, China, and the Ph.D. degree in
theoretical physics from Baylor University, Waco, TX. He is
currently with Baylor University, where he is an Assistant
Research Scientist with the Center for Astrophysics, Space
Physics, and Engineering Research. His research interests
include structure analysis, waves and instabilities, and phase
transitions in complex (dusty) plasmas.
Jie Kong received the B.S. degree in physics from Sichuan
University, Chengdu, China, and the Ph.D. degree in surface
analysis from Baylor University, Waco, TX. He is currently
with Baylor University, where he is an Assistant Research
Scientist in the Center for Astrophysics, Space Physics and
Engineering Research. His research interests include complex
(dusty) plasma diagnostics, plasma sheath, waves and
instabilities, and phase transitions in complex (dusty) plasmas.
Zhuanhao Zhang received his B.S. degree in Applied Physics
from University of Science and Technology of China, Hefei,
China in 2007 and his PhD in Plasma Physics within the
Center for Astrophysics, Space Physics and Engineering
Research at Baylor University in 2012. His research interests
include particle-particle interaction, dust acoustic waves and
instabilities, plasma diagnostics, and structural phase
transition in dusty plasmas.
Lorin S. Matthews (M’10) received the B.S. and the Ph.D.
degrees in physics from Baylor University in Waco, TX, in
Truell Hyde (M’01) received the B.S. degree in physics from
Southern Nazarene University, Bethany, OK, and the Ph.D.
degree in theoretical physics from Baylor University, Waco,
TX. He is currently with Baylor University, where he is the
Director of the Center for Astrophysics, Space Physics and
Engineering Research, a Professor of physics, and the Vice
Provost for Research in the university. His research interests
include space physics, dust detectors, shock physics and
waves, and linear and nonlinear phenomena in complex
(dusty) plasmas.