VOLUME 86, NUMBER 12 PHYSICAL REVIEW LETTERS 19 MARCH 2001

Longitudinal and Transverse Waves in Yukawa Crystals

Xiaogang Wang,1 A. Bhattacharjee,2 and S. Hu2
1
Department of Physics, Dalian University of Technology, Dalian, China 116024
2
Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242
(Received 22 June 2000)
A unified theoretical treatment is given of longitudinal (or compressional) and transverse modes in
Yukawa crystals, including the effects of damping. Dispersion relations are obtained for hexagonal
lattices in two dimensions and bcc and fcc lattices in three dimensions. Theoretical predictions are
compared with two recent experiments.

DOI: 10.1103/PhysRevLett.86.2569 PACS numbers: 52.35.Fp, 52.27.Lw, 82.70.Dd

In a Yukawa system, charged microparticles interact dynamics simulation of longitudinal and transverse
with each other through a Yukawa or a screened Coulomb waves in a Yukawa liquid [24,25]. Our method is based
potential. For a point charge Q located at the equilib- on the harmonic approximation, which is standard in the
rium position R 苷 x̂X 1 ŷY 1 ẑZ, this potential is theory of crystal lattices and has been used in earlier
given by f共r兲 苷 共Q兾r兲e2r兾lD , where r 苷 jr 2 Rj 苷 studies of two- and three-dimensional Coulomb crystals
关共x 2 X兲2 1 共 y 2 Y 兲2 1 共z 2 Z兲2 兴1兾2 . Various physical [17,26]. We derive dispersion relations for the different
systems, including colloidal crystals and strongly coupled waves from a master dispersion relation by considering
dusty plasmas, can be modeled by the Yukawa potential. different regimes of the wave number k and the so-called
In a dusty plasma, highly charged microparticles are sus- screening parameter k ⬅ a兾lD , which is the ratio of the
pended in a gaseous electrical discharge. Such a system is interparticle spacing and the Debye length lD . [Here,
said to be strongly coupled when the coupling constant lD 苷 共l22 22 21兾2
De 1 lDi 兲 , where lDe and lDi are the elec-
G关⬅ Q 2 兾共aT 兲兴 for microparticles is equal to or greater tron and ion Debye lengths, respectively.] We compare
than unity. (Here, T is the temperature of each micropar- quantitatively the predictions of the theory with available
ticle, and a is the interparticle spacing.) When G ¿ 1, experimental data on two-dimensional monolayers pro-
the particles have been shown to crystallize [1–4], that is, duced in the laboratory. It has been demonstrated experi-
organize themselves into an ordered spatial structure that mentally that, in a two-dimensional system, the Yukawa
we call a Yukawa lattice. potential is a good approximation for the interaction
Plasma crystals allow direct optical imaging of particle between microparticles [27]. We assume that this poten-
motion. Compared with colloidal suspensions, the par- tial also holds in a three-dimensional crystal under
ticles are weakly damped. Particle motion can be excited microgravity conditions. Based on the analogy with
by laser manipulation. This makes it possible to excite OCPs [28], it is generally believed that in three dimen-
and test the dispersion relations of certain types of lon- sions the preferred crystal structure is a bcc or fcc lattice,
gitudinal [5–10] and transverse waves [11–15] predicted depending on the value of k. Indeed, there is experi-
by theory. (These waves have antecedents in the theory mental evidence in three-dimensional laboratory dusty
of one-component plasmas (OCPs) [16–20].) The ex- plasmas of the appearance of bcc and fcc structures,
perimentally measured waves in dusty plasma crystals in- mixed with hexagonal structures [1,29]. In anticipation
clude the acoustic [21] and lattice waves [22], both of of future microgravity experiments [30] where pure bcc
which are longitudinal waves, and, most recently, a trans- and fcc lattices may be realized, we make predictions
verse wave [23]. for two-dimensional damped waves in three-dimensional
In laboratory experiments involving waves in two- bcc and fcc lattices.
dimensional Yukawa lattices (or monolayers), the micro- For horizontal linear modes, we write r̃ 苷 x̂j 1 ŷh
particles form a hexagonal lattice. The waves experience as the two-dimensional displacement. The linear displace-
a frictional drag due to the background neutral gas as ment obeys the equation of motion,
well as ions. This drag has a significant effect on the
dispersive properties of the waves. In order to compare d 2 r̃共X, Y , Z, t兲
1
experimental data with theory, it is necessary to develop dt 2
theoretical models in which the structure of the crystal as d r̃共X, Y , Z, t兲 Q
well as damping are included as essential elements. n 苷 E关r共X, Y , Z兲, t兴 , (1)
dt Md
In this Letter, we present a unified analysis of lon-
gitudinal and transverse waves in a Yukawa crystal, where n represents the frictional drag coefficient, Md
including the effects of damping. Our work complements is the mass of a point particle, and the electric field is given
and extends a recent theoretical study and molecular by E 苷 2=f. We assume that the zx plane is a plane

0031-9007兾01兾86(12)兾2569(4)$15.00 © 2001 The American Physical Society 2569

VOLUME 86, NUMBER 12 PHYSICAL REVIEW LETTERS 19 MARCH 2001

of reflection symmetry of the crystal. Assuming that the It is then straightforward to show that there are two
components of the linear displacement are independent of vibrational normal modes given by the dispersion
Z, we write 共j, h兲 苷 共j0 , h0 兲 exp关i共kx X 1 ky Y 兲 2 ivt兴. relation,

1
v6 共v6 1 in兲 ⬅ V6
2
共k, k兲 苷 2 关F共k, k兲 1 F共k, k兲兴 6 兵关F共k, k兲 2 F共k, k兲兴2 2 4G 2 共k, k兲其1兾2 . (2)

In Eq. (2), we have redefined the various physical quantities to make them dimensionless. We have rescaled
共v, n兲兾v0 ! 共v, n兲, where the frequency v0 ⬅ Q共Md a3 兲21兾2 is proportional to the dust plasma frequency,
共X, Y , Z, R兲兾a ! 共X, Y , Z, R兲, and ka ! k. The functions F, F, and G represent infinite sums over lattice sites and
are given by
∑ X µ ∂ X µ ∂ X ∏
kx X ky Y
F共k, k兲 ⬅ 4 F共X, 0, Z兲 sin2 1 F共0, Y , Z兲 sin2 1 F共X, Y , Z兲 共1 2 coskx X cosky Y 兲 ,
X.0,Z 2 Y .0,Z 2 X,Y .0,Z
(3a)
X
G共k, k兲 苷 4 G共X, Y , Z兲 sinkx X sinky Y , (3b)
X,Y .0,Z
∑ X µ ∂ X µ ∂ X ∏
kx X ky Y
F共k, k兲 ⬅ 4 F共0, X, Z兲 sin 2
1 2
F共Y , 0, Z兲 sin 1 F共Y , X, Z兲 共1 2 coskx X cosky Y 兲 ,
X.0,Z 2 Y .0,Z 2 X,Y .0,Z
(3c)

with the spring constant matrices, strong-coupling models in the strongly coupled regime
共G ¿ 1兲 [7–9,24,25]. This dispersion relation also
F共X, Y , Z兲 苷 R 25 e2kR 关X 2 共3 1 3kR 1 k 2 R 2 兲 follows in the limit k ! 0, k ø 1 of the exact unified
dispersion relation derived in [10] for an infinite sequence
2 R 2 共1 1 kR兲兴 , (4a) of parallel sheets in three-dimensional space interacting
via an electrostatic force that depends only on the coordi-
G共X, Y , Z兲 苷 共XY 兾R 5 兲e2kR 共3 1 3kR 1 k 2 R 2 兲 . (4b) nate perpendicular to the sheets. This dispersion relation
has been confirmed in a number of experiments in the
In Eqs. (3a)–(3c), the summations over X, Y , and Z are moderately strong [31] as well as strong-coupling [21]
carried out over their entire range except when specified regimes.
otherwise. In particular, the specification X . 0 共Y . 0兲 For the two-dimensional hexagonal monolayer, we
implies that the summation is carried only over positive 2
p
obtain c22 苷 0, and c21 苷 kCl jk!0 p 苷 4p兾 3. In this
values of X 共Y 兲.
case, it follows that Cl 艐 共4p兾 3 k兲1兾2 苷 2.69k 21兾2 ,
Equations (2)–(4) give the general dispersion rela-
which is in agreement with the longitudinal acoustic speed
tions in three-dimensional Yukawa crystals. If we set
calculated numerically in [11].
Z 苷 0 and omit the sums over Z, we obtain the two-
A one-dimensional longitudinal acoustic speed, valid
dimensional dispersion relation for a monolayer. Special
for a chain of point particles, can be derived from the
cases of the two-dimensional dispersion relation have
dispersion relation (2) by setting Y 苷 Z 苷 0 and omitting
been discussed earlier in [11] and [22].
the sums over Y and Z. The infinite sum over X then can
Acoustic limit.—We choose k to be parallel to the x
be carried out exactly to obtain
axis. The longitudinal (or compressional) and transverse µ ∂
“acoustic” speeds can then be calculated in the long- k关ek 共k 1 2兲 2 2兴
2 Cl 苷 2 k 2 ln共e 2 1兲 1
2 k
, (5)
wavelength limit from the relation Cl,t 共k兲 苷 关V1,2
2
共k, k兲兾 2共ek 2 1兲2
k 兴k!0 . In the k ø 1 regime, the longitudinal acoustic
2
which reduces to Cl2 苷 22 lnk in the limit k ø 1. We
speed can be expanded as a Frobenius series, Cl2 苷 thus note that the k dependence of the acoustic speed
c22 k 22 1 c21 k 21 1 cln lnk 1 c0 1 c1 k 1 c2 k 2 1 . . . . for compressional modes depends sensitively on the
In a three-dimensional lattice, it is straightforward to geometry of the crystal: It is proportional to k 21 for a
show analytically from the dispersion relation (2) that three-dimensional lattice, to k 21兾2 for a two-dimensional
Cl2 艐 c22 k 22 , where c22 苷 k 2 Cl2 jk!0 苷 8p 共16p兲 for lattice, and to 共2 lnk兲1兾2 for a one-dimensional chain.
a bcc (fcc) lattice. However, if we scale v and n by The acoustic velocity of transverse modes in a two-
the standard plasma frequency vpd 苷 共4pnd Q 2 兾Md 兲1兾2 dimensional hexagonal lattice is also isotropic, as shown
instead of the frequency v0 苷 Q共Md a3 兲21兾2 and note numerically in [11]. We can demonstrate this as a spe-
that nd 苷 2兾a3 共4兾a3 兲 for a bcc (fcc) lattice, we ob- cial case of the general dispersion relation (2) for the
tain the so-called dust-acoustic-wave dispersion relation transverse modes in the limit k ! 0, k ø 1. We obtain
vl 共vl 1 in兲 苷 Cl2 k 2 兾k 2 , in agreement with the dis- vt 共vt 1 in兲 苷 Ct2 k 2 , with Ct2 苷 c0 1 c1 k 1 c2 k 2 1
persion relation obtained from other three-dimensional . . . 艐 0.26 2 0.02k 2 . In three-dimensional lattices, we
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VOLUME 86, NUMBER 12 PHYSICAL REVIEW LETTERS 19 MARCH 2001

significantly from this limit for nonzero values of k. The

TLW is, in general, strongly anisotropic. In the regime
k # 5, whereas the speed of the parallel TLW increases,
that of the perpendicular TLW decreases — a feature that
can be tested by changing the direction of manipulation by
a laser in an experiment.
Effects of damping and comparison with experi-
ments.—We now consider the effects of frictional damp-
ing and demonstrate its important role in experiments.
Both the experiments discussed in this Letter use laser
pressure on particles to excite waves and measure the
real and imaginary parts of the complex wave number
k 苷 kr 1 iki . Without damping, one obtains ki 苷 0 and
the waves propagate unattenuated in space. In view of the
experimental uncertainties in crucial parameters such as
k and n, it is important to carry out experimental tests
FIG. 1. The phase speed of undamped transverse lattice waves
propagating parallel (u 苷 0, solid line) and perpendicular (u 苷 of the real as well as the imaginary part of k predicted
p兾2, dashed line) to a primitive translation vector. by theory.
In Fig. 2, we compare the predictions of the theoretical
can calculate the acoustic velocity of transverse modes dispersion relation (6a) with experimental data points (rep-
as a power series in k, Ct2 艐 0.037 2 0.0013k 2 1 resented by squares) for the approximate parameters given
0.000 018k 4 which is in approximate agreement with the in [22]. These results confirm the interpretation that the
theoretical calculation for Yukawa liquids [25]. CLW was excited in the Kiel experiment [22].
Lattice waves.—We have seen above that the acoustic We now revisit the data from the Iowa experiment on
(or k ! 0) limit is isotropic in wave number space. Except transverse waves in a two-dimensional hexagonal lattice
in this special limit, the waves are generally anisotropic, [23,32]. The dispersion relation for acoustic waves without
and we call them lattice waves. We distinguish two types: damping (that is, n 苷 0), obtained earlier in [11], was
the compressional (or longitudinal) lattice wave (CLW) used in [23] to fit the experimental data on kr 共v兲. In
and the transverse lattice wave (TLW). To obtain disper- the experiment, ki 共v兲 is also measured but a theoretical
sion relations for lattice waves in hexagonal monolayers, fit was not attempted because the results given in [11]
we choose the x axis to be parallel to a primitive trans- did not include the effect of damping. If the dispersion
lation vector. If u is the angle between the wave number relation is of the form k 苷 k0 共v兲 for n 苷 0, we can obtain
k and a primitive translation vector, then kx 苷 k cosu and perturbative corrections to this dispersion relation for small
ky 苷 k sinu. If u 苷 0, the dispersion relation (2) simpli- damping by writing
fies to produce two branches, where
kr 1 iki 艐 k0 共v 1 in兾2兲
Ç µ ∂
v1 共v1 1 in兲 苷 vl 共vl 1 in兲 ≠k0 in
µ ∂ 艐 k0 共v兲 1 1 O共n 2 兲 . (7)
X ≠v 2
2 kX
n苷0
苷2 F共X, Y , Z兲 sin (6a)
X,Y ,Z 2 Equating the real and imaginary parts of Eq. (7), we obtain
kr 艐 k0 共v兲 and ki 艐 共n兾2兲 共≠k0 兾≠v兲n苷0 苷 n兾共2Vg0 兲,
describes the longitudinal modes, and where Vg0 is the group speed in the absence of damping.
Clearly, while the correction to kr due to small n is of
v2 共v2 1 in兲 苷 vt 共vt 1 in兲 order n 2 , the correction to ki is of order n. This explains
X µ ∂
2 kX
why, if the damping is weak, the dispersion relation for
苷2 F共Y , X, Z兲 sin (6b) zero damping can be used to fit kr 共v兲, but such a fit
X,Y ,Z 2
cannot detect deviations from theoretical predictions of
describes the transverse modes. The particle motion in order n, reflected in ki 共v兲.
these two cases is parallel and perpendicular to k, re- In Fig. 3, for experimentally relevant parameters
spectively. Similar dispersion relations can be written for [23,32], we show the fit for the theoretical dispersion
longitudinal and transverse waves when u 苷 p兾2, with relation including the effect of damping for two different
v2 苷 vl and v1 苷 vt . values of the angle u. On the basis of this analysis,
In Fig. 1, we plot the phase speeds of the parallel 共k兲 taking into account the size of the error bars, it appears
and perpendicular 共⬜兲 TLW as a function of k for k 苷 1 that the waves excited in the Iowa experiment correspond
when the frictional damping is zero. In the limit k ! 0, approximately to u 艐 0.
the phase speeds of the parallel 共u 苷 0兲 and perpendicular One of the important features of longitudinal as well as
共u 苷 p兾2兲 TLW tend to the acoustic limit but deviate transverse modes in the lattices is “negative dispersion”:
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VOLUME 86, NUMBER 12 PHYSICAL REVIEW LETTERS 19 MARCH 2001

This research is supported by NASA Grant No. NAG5-

2375. We thank J. Goree and S. Nunomura for helpful
discussions.

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