S6-2

Configurations of Coulomb Clusters in a Complex Plasma
セミマクロなダスト系プラズマの構造形成と制御
Osamu Ishihara
石原 修
Faculty of Engineering, Yokohama National University,
79-5 Tokiwadai Hodogaya-ku, Yokohama 240-8501 Yokohama, Japan
横浜国立大学大学院工学研究院, 〒240-8501 横浜市保土ヶ谷区常盤台79-5

A complex plasma, known also as a dust plasma, is characterized by a plasma which includes fine particles
of nano to micron meters in size. Fine particles in a plasma are either positively or negatively charged
depending on the environment. In many laboratory experiments, dust particles are highly charged
negatively due to the large mobility of electrons. Since the Coulomb potential of dust particles is much
more than the dust thermal energy, dust particles form strongly coupled Coulomb clusters. The clusters
show a variety of configurations depending on the number of dust particles and also on the strength of theconfinement potential. Structures of the Coulomb cluster are reviewed.

1. Structures by like charges
Like charges have been known to form varietyof structures since the beginning of the twentycentury. Thomson explained the atomic structurefrom the stability conditions of the charges inrings[1], while Madelung discussed the structure ofionic compound from the viewpoint of electrostatic
energy of a nearly infinite crystal [2].
But it is only recently that the detailed
structures formed by the interaction of like chargesof finite numbers become the subject of intensive
research. On the surface of liquid helium electrons
are found to form a Wigner crystal in 1979[3]. In
1980s, the research of quantum dots revealed thatthe confined electrons take the discrete energy as
the energy level of electrons around the nucleus.
Such an analogy introduced the name of artificial
atoms for quantum dots. Research on artificial
atoms is directed to find the behavior of a classicalsystem with finite number of electrons in a confined
potential. The computer simulation showed theWigner crystallization and melting of electrons in
the two-dimensional quantum dots [4]. On the other
hand, advanced technology of laser coolingsucceeded to trap ions in the confined region by theelectromagnetic field. The trapped ions areobserved to form certain structures [5]. Ions trappedin a Paul trap by the electric field and ions trappedin a Penning trap by the magnetic field are known
to form various structures [6]. Dubin and O’Neildiscussed the shell structure of a nonneutral plasma
confined by the Penning trap [7].
Computer simulation was used to study thelaser-cooled ion structure confined in a harmonic
potential. And the stable configurations wereidentified as states of minimum energy [8]. The

stable configuration was foundconcentric shell structures [9].

toinclude

2. Structures by dust particles
In 1994 Coulomb crystal was discovered in aplasma where fine particles are negatively charged
and the Coulomb energy overcomes the thermalenergy [10]. The Coulomb cluster in a plasma could
be observed by naked eyes through the CCDcamera. The structure of the cluster becomes
observable by a very fundamental method, i.e., byhuman vision and the dust plasma becomes thetopic of the research among many plasma scientists[11]. Formation of the Coulomb crystals andmelting process of the Coulomb crystals have been
observed and studied in detail. The phase transition
is now observable by kinetic level through detecting
trajectories of dust particles [12].
Computersimulations by the method of molecular dynamics
and Monte Carlo contributed to the understandingof the Coulomb cluster. Two and three dimensional
structures of the clusters in which repulsive
negative charges are confined in an electrostatic
potential have been studied. The ring structure in
the two dimensional confinement and the shellstructure/BCC structure in the three dimensional
confinement were clarified, where the confinementof dusts was established by the harmonic potential[13, 14]. The inharmonic potential was used to
produce the transition from the two-dimensional tothree-dimensional structures [15].
Our system in consideration may be expressedby the Hamiltonian:N p − ZeA 2 mω 2  2 z 2 iH =∑  i+ ri + 
κ22mi =1


+∑
j <i

(1)

( Ze )
− x −x / λe i j D + V1 ( xi )  ,4πε 0 xi − x j2where N is the number of dust particles, m, p, Ze
are mass, momentum and charge of a dust particle,
respectively, A is the vector potential,ω isthe frequency of the harmonic potential, κ is aconfinement parameter, r 2 = x 2 + y 2 , λD is ascreening length, and V1 (xi ) is a potential related
to all other forces like a gravitational force, a dragforce and a sheath electric field. If we introduce( 2Z e / 4πε mω )characteristic scales [ A =2 1/ 22 20for the length, mω A / 2 for the energy, mω A
for the momentum,mω A / Ze for the vector
potential], the Hamiltonian can be written asN
z22H = ∑  pi − A + ri 2 + iκi =1 (2)2 2

/ xi − x j + V1 (xi )  .
j <i
The ring configuration of dust particles in theabsence of magnetic field (A=0) was studied for+∑ e− x i − x j / λD

n=4

n=3

n=5n=6Fig. 1 Stable ring configurations in
the absence of magnetic field. [Ref.16 ].

N=70

N=70134

N=134

Fig. 2 Stable Coulomb cluster. κ=5,N=70 and 134. Side and top
views[ Ref. 18].

possible stable structures in the presence of ion dragforce. The stable formation was found for dust
particles up to 6, while the dust particles more than 6in a ring become unstable (see Fig. 1). The particle
dynamics in the presence of magnetic field wasalso studied. The rotation of the ring
configuration was observed by the experimentand recovered by the simulation [17]. The three
dimensional dust structures for p=A=V1=0,
λD → ∞ depend only on N and κ . Figure 3
shows the spindle-shape structure for N=70 and 134
with κ = 5 [18].

This research was supported by Japan Society
for the Promotion of Science Grant-in-Aid for
Scientific Research (B)(2) 16340179.
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