Charge polarization (dressed electrostatic interaction) effects in dusty (complex) crystals

I. Kourakis a;¤, P. K. Shukla b;¤ and B. Farokhi†
¤Ruhr-Universität Bochum, Institut fürTheoretische Physik IV, D-44780 Bochum, Germany
a www.tp4.rub.de/»ioannis ; b www.tp4.rub.de/»ps
†Department of Physics, Bu-Ali Sina University, Hamadan, Iran
Abstract. The influence of dust charge polarization (dressing) on lattice vibrations is investigated. Both one-dimensional
(1D) and hexagonal (2D) monolayer configurations are considered. It is shown that dressed interactions lead to a reduction
(increase) in the frequency of lattice vibrations, as regards longitudinal (transverse) degrees of freedom. The possibility of a
new crystal instability (melting) entirely due to the dressing effect is pointed out. On the other hand, the occurrence of crystals
consisting of opposite (…+-+-+-+…) charge dust grains may be anticipated via this mechanism.
Keywords: Dusty plasma crystals, complex plasmas, dust-lattice waves, electrostatic interactions, phonons, solitons.
PACS: 52.27.Lw, 52.35.Fp, 52.25.Vy
It is now established that the presence of massive mesoscopic (micron-sized, typically) particulates (“dust grains”)
may modify plasma properties substantially [1]. Of particular importance is the occurrence of strongly-coupled
crystalline-like dust configurations, due to strong inter-grain interactions [2]. These dust quasi-lattices are now known
to support a variety of linear and nonlinear excitations, which may be of potential use in future applications.
At a first approach, ab initio studies show that inter-grain electrostatic interactions may be considered to be of
the screened Coulomb (Debye – Hückel) type [3]. More refined theoretical studies have later shown that taking into
account plasma polarization due to the sheath region (near the grain surface) associated with the grains [4, 5] results
in a strong modification of the (oppositely charged) charge cloud surrounding the particles. This “dressing” effect
leads to a change in the very nature of the inter-particle interactions, which may even become attractive for equal-sign
charged particles (inversely, repulsive interactions may appear in the case of opposite neighboring grain charges).
The influence of dust charge polarization (dressing) on lattice vibrations is investigated in this brief report. Both onedimensional
(1D) and hexagonal (2D) monolayer configurations are considered. It is shown that dressed interactions
lead to a reduction in the frequency of lattice vibrations [6, 7], as regards both longitudinal and transverse degrees of
freedom. The possibility of a new crystal instability (melting) entirely due to the dressing effect is pointed out. On the
other hand, the occurrence of crystals consisting of opposite (…+-+-+-+…) charge dust grains may be anticipated [8].
DUST-LATTICEWAVES IN ONE-DIMENSIONAL (1D) DUST CRYSTALS
The potential (energy) of interaction between two particles (charges Q1 and Q2) located at a distance r reads [1, 4, 5]

where x = r=lD ´kr0, and lD denotes the effective Debye radius [1, 3]; here, we have defined the lattice parameter
k = r0=lD and the reduced space variable r0 = r=r0. The parameter s = sgn(Q1Q2) = §1 is equal to 1 (-1) for
equal- (opposite-)-charge-sign particles, respectively. The parameter d simply takes the values 1 (for “dressed” Debye
interactions) and 0 (recovering the familiar unperturbed Debye form); unless otherwise stated, d = 1 in the following.
The potential form (1), studied in Refs. [4, 5], is depicted in Fig. 1. For s = 1 (equal charge-sign grains), it changes
sign at x = 2, shifting from repulsive to attractive interactions (among equal charge signs, here). Furthermore, it bears
a minimum at x = 1+
p
3 ¼ 2:732, which may play the role of a potential well for neighboring particles located at
an appropriate distance; naturally, this potential form was suggested as a simple model for dust molecule formation in
earlier works [1, 5].

FIGURE 3. The TDLW dispersion curve: the square frequency w2
T , as given by Eq. (2) (scaled by w2
g ) vs. the reduced
wavenumber kr0), for arbitrary values of all parameters except s (here s = +1) and d . d = 0 (1) in the lower (upper) curves.
Stabilization of LDL waves in crystals of alternating charge-sign grains. An interesting consequence of the
electrostatic potential “dressing” effect is the following. Let us consider the 1D alternating charge sign pattern:
:::;+;¡;+;¡;+;¡;+;¡;+:::
Coulomb-like interactions are attractive here, giving rise to unstable longitudinal displacements [8].
Taking into account the dressed Debye potential given by Eq. (1), for opposite grain charge-signs, i.e. for s = ¡1,
one essentially obtains an reversed, qualitatively speaking, picture, as compared with the case s = +1 treated above;
cf. Fig. 1, upon setting U ! ¡U, which yields the mirror-symmetric plot, with respect to the horizontal axis; the
corresponding figure is omitted here, for brevity. Most interestingly, considering this type of interaction among one
(any) grain and its first order neighbors, we see that the total force Fn = Fn¡1;n +Fn+1;n felt by the n¡th grain, viz.
Fn = ¡
·
¶Un¡1;n
¶ zn
+ ¶Un+1;n
¶ zn
¸
= ¡ ¶
¶ zn
[UdrD(r0+zn)+UdrD(r0¡zn)] ´ ¡
·

¶ xUtotal(x)
¯¯¯¯
x=zn
, derives from a total potential,
say Utotal(x), which may here, for d = 1, present a local minimum (hence a stable equilibrium position for the n¡th
grain). It turns out that the extremum at x = 0, viz. U0
total(0) = 0, is a local minimum (maximum), i.e. U00
total(0) is
positive (negative) for k values above (below) a critical value k3 ¼ 3:4798. This qualitative behavior is depicted in
Fig. 4a. Therefore, the electrostatic dressing effect may result in stabilization of longitudinal grain displacements in a
bi-lattice, consisting of oppositely charged neighboring grains. Remarkably, this possibility is inexistent in the absence
of the dressing effect. Indeed, analyzing the form of Utotal(x) in the case d = 0 (i.e. for simple, unperturbed Debye
interactions), one sees that no stable equilibrium point occurs in this case; cf. Fig. 4b.
DUST-LATTICEWAVES IN TWO-DIMENSIONAL (2D) HEXAGONAL CRYSTALS
Let us now consider a 2D hexagonal dust monolayer; see in Fig. 5. Linear vibrations along the x¡ or the y¡axis,
propagating in an arbitrary direction (0 < q < p=2), have been studied for dressed interactions in [7]. Various

FIGURE 5. (a) Elementary cell in a 2D hexagonal crystalline configuration. (b) The normalized LDL frequency w2
L vs. kr0, for
wave propagation in the x direction. Here k =2.5, and d = 0 (1) in curve 1 (2).
In conclusion, charge polarization (electrostatic “dressing”) results in a significant modification of the propagation
characteristics of dust-lattice waves, which may even be destabilized (for high values of the lattice parameter k,
essentially). Furthermore, transverse off-plane 1D vibrations may shift from a backward- to a forward-propagating
wave, due to the polarization effect. There results may be investigated by appropriate experiments.
Acknowledgements. I.K. acknowledges partial support by the Deutsche Forschungsgemeinschaft (Bonn, Germany)
through the Sonderforschungsbereich (SFB) 591 Programme.
REFERENCES
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