ARTICLE IN PRESS

Journal of Crystal Growth 264 (2004) 223–231

Formation and growth of NbSe3 topological crystals
T. Tsuneta*, S. Tanda
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Received 14 October 2003; accepted 2 December 2003
Communicated by T. Hibiya

Abstract
We investigated the self-assembled growth of mm-scale topological crystals of NbSe3 based on the chemical vapor.
transportation method. The shapes of these materials, including cylinders, Mobius
strips and ‘‘ﬁgure-eight’’ strips, arethe physical realization of topological conﬁgurations deﬁned by wedge and twist disclinations as well as dispirations.
The crystal structure is investigated by X-ray powder diffraction technique and it is revealed that they are NbSe3
crystals with a considerable strain. The variations in topology and growth form (e.g. rings, disks and tubes) are
explained based on a growth model that a NbSe3 whisker encircles a selenium droplet by the surface tension to form a.
loop. The twisted topologies of Mobius
strips and ﬁgure-eight strips originates from either the symmetry of the elastic
property of NbSe3 or bend-twist transformation.
r 2003 Elsevier B.V. All rights reserved.
PACS: 81.10.h; 61.72.Lk
Keywords: A1. Crystal morphology; A1. Defects; A1. Growth models

1. Introduction
Controlling the global geometry of a material is
now regarded as an important method for obtaining
new materials. In particular, a topological deformation, such as the rolling of a planar crystal into acylinder, can signiﬁcantly affect physical properties
via changes in spatial symmetry (rotational, translational and/or chiral symmetry). Such new materials
include fullerenes [1] and related nested onion-likestructures [2], carbon nanotubes [3] and various*Corresponding author. Tel.: +81-11-706-7126; fax: +8111-706-7293.
E-mail address: tsuneta@eng.hokudai.ac.jp (T. Tsuneta).
URL: http://exp-ap.eng.hokudai.ac.jp.

inorganic tubular systems [2–6]. These are called
topological materials because, in mathematicalterms, the spherical surface (fullerene) and thecylindrical surface (nanotube) correspond to phasespaces S2 and S1 I; respectively. These multiconnected spaces are topologically different fromthe two-dimensional space R2 of the original ﬂatmaterials. They are useful for fundamental studies ofboth the physics of multi-connected geometries [7] aswell as nano-physics, due to the reduction in size that
accompanies the new topology. However, no
topological materials other than originally twodimensional layered crystals, like graphite, havebeen made into curved surfaces.
Recently, we developed the self-assembled
growth of micrometer-scale NbSe3 crystals that0022-0248/$ - see front matter r 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.jcrysgro.2003.12.020

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T. Tsuneta, S. Tanda / Journal of Crystal Growth 264 (2004) 223–231

have various multi-connected geometries [8].
NbSe3 is originally a quasi-one-dimensional material and its ordinary crystal shape is a whisker or athin ribbon. These new geometries include rings,
disks, tubes and two twisted versions of a cylinder:.
Mobiusstrips and ‘‘ﬁgure-eight’’ (8-) strips, which
are characterized by twists of p and 2p, respectively. These are not simply hollow crystals carved
out of a bulk material: their crystal axes are
mapped onto the curved geometries. Thus thesecrystals form a new group of topological crystal ofa one-dimensional material. They are remarkable
examples of a mm-scale monocrystal with unlocalized, complete wedge and twist disclinations aswell as dispirations. Aside from crystallography, it
is interesting how the multi-connected topology
affects the electronic properties of NbSe3, especially its characteristic charge-density-wave(CDW) states [9–11].
A variety of crystal topologies under virtually
the same conditions is a signiﬁcant feature of thismaterial. A variety of topological materials:.
untwisted rings, p-twisted Mobius
strips and 2ptwisted 8strips can all be produced together in asingle growth batch. In addition, there are further
varieties within the same topological class, e.g. ﬂat
disks, ﬁne tubes, paraboloids or more complex
objects have the topology of a cylinder. In thispaper, we focus on the mechanism of the selfassembled growth with this unusual morphology.
We report the effects of the growth conditions on
the morphology as well as structural properties ofthese materials as studied by X-ray diffraction,
which are apparently related to their formationand growth.

2. Experiments
2.1. Sample preparation
Since the ﬁrst report in 1975 [12], single crystalsof NbSe3 have been synthesized by the chemical
vapor transportation (CVT) method [13,14]. We
used this method to synthesize topological crystals
and studied the effects of the experimental conditions on crystal morphology. Since we observedthat strong convection of selenium (Se) vapor/mist

inside the reaction apparatus is crucial for theformation of topological crystals, we used areaction furnace with inadequate insulation to
induce a large thermal inequilibrium. We usedniobium (Nb) in the form of either powder or rods
and granular Se. Both elements were obtained in apurity of greater than 4N from the manufacturer.
The starting material was a mixture of Nb and Se
with about 3% excess Se as a transport agent. Nb
and quartz ampoules were baked at around 900 C
in a vacuum for degassiﬁcation. Se was also
degassed by boiling in vacuum. Nb rods weretreated with a solution of hydroﬂuoric acid andnitric acid prior to baking. Shortly after baking,the mixture was sealed in a quartz ampoule,1.75 mm in internal diameter and 20 cm in length,that was evacuated to the order of 106 Torr. The
ampoule was placed horizontally in an open-end
furnace with the starting material placed on thehigh-temperature side. The temperature of thestarting-material end of the ampoule was raised to
the reaction temperature, typically 740 C, at a rateof 10 C/min. A temperature gradient 1.5 C/cm
was applied to the ampoule. The temperature wasthen maintained for 0.5–100 h before the ampouleis quenched.

2.2. Effects of growth conditions
Using the above procedure, we can reproduciblysynthesize dozens of topological crystals, equivalent to a yield of around 103 wt%, along with
NbSe3 whisker. The growth region for topologicalcrystals is always on the colder side of theampoule. A certain growth stage of surrounding
NbSe3 whiskers seems to be crucial for theformation of topological crystals. Whiskers grownin the region are so ﬁne and soft that most of them
are rumpled. They also tend to radiate outward
from several apparent growth points. Growth islikely to originate from Se droplets, since Se
condenses into a liquid as it circulates around theampoule as a transport agent. The growth region
in the ampoule is typically 1–5 cm long in the caseof powders. The temperature in this region isusually 710–720 C regardless of the maximum/
minimum temperature at both ends.

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Topological crystals are most efﬁciently produced within the ﬁrst few hours of the reaction.
Fig. 1 shows the yield (in number) of topologicalcrystals for different reaction times, where the open
and ﬁlled circles represent Nb powders and rods
used as a starting material, respectively. In the caseof powders, the formation of topological crystals
starts within 60 min and gradually slows. Afterreaching a peak at 300–400 min, the yield begins todecrease, possibly due to decomposition. With Nb
rods instead of powders, the yield of topologicalcrystals is improved and they are produced for
longer time. In addition, the growth region of
topological crystals increases to 10–20 cm. A rod
reacts with a Se atmosphere at a slower rate. While
all of Nb powder is crystallized into NbSe3 whiskers
within 30 min, with a rod, more than 90% of thestarting material remains unreacted after thousandsof minutes. This difference is likely due to thegreater surface area of Nb powders, where thereaction with Se gas takes place, and powder
scattered on the wall of ampoule provides efﬁcient
nucleation centers for whiskers.
2.3. Classification of crystal topologies
The morphology of our topological crystals canbasically be represented by a framed strip rolled

Fig. 1. Production rate of topological crystal samples pickedfrom growth batches at different reaction times. Open and solid
circles represent niobium as a starting material in the form of apowder and rod, respectively.

225

into a loop with or without a twist along its
circumference. The morphology can be categorized into three classes according to the degree oftwist: 0, p or 2p. Our classiﬁcation is based on
topology, i.e. each crystal belongs to a different
class in the ribbon-knot group in terms ofmathematics. The linking number lk is introduced
to simply characterize all three classes. lk ¼ Wr þ
Tw according to White’s theorem [15], where thewrithing number Wr is derived from the angle ofcurvature of a ribbon integrated over the whole
length, and the twisting number Tw is theintegrated screw angle between the two edges ofa ribbon. lk; Wr and Tw are invariants, so thatoriginally they have integral values in units of 2p:.
However, since we must address p-twisted Mobius
strips, half-integral numbers are also allowed. For
the topology of a ring (cylinder), lk ¼ 1 since it has.
a unit of writhing and no twisting. For a Mobius
strip, Wr ¼ 1 as a loop and Tw ¼ 12 due to its twist,
and therefore lk ¼ 112: Similarly, lk ¼ 2 for a 2ptwisted 8strip.
Fig. 2(a)–(d) shows scanning electron microscope (SEM) photographs of typical lk ¼ 1 samples. (a) is a basic ring or a thin, short cylinder.
The disk (b) is a cylinder that grew radially, while
the tube (c) grew longitudinally. Some samples
with more complex shapes are also categorized ashaving the lk ¼ 1 topology. For example, there areconcentric composite cylinders, like (c), andcylinders for which the diameter changes continuously or abruptly (d). Some lk ¼ 1 samples have abowl-like concave surface, as in (d). The typicaldimensions of lk ¼ 1 samples are 10-300 (around70 on average) mm in diameter, 1 mm in minimum
internal diameter, and 5–600 mm in length. Fig.
2(e) shows an lk ¼ 112 material, the so-called.
Mobiusstrip, which is famous for its characteristic
one-sided, non-orientable surface. Compared totheir two-sided counterparts (lk ¼ 1 and 2), lk ¼
112 crystals are very rare under any growth
conditions. The yield is less than 0.1% of all
topological crystals. Fig. 2(f) and (g) shows thin
and well-grown lk ¼ 2 materials. lk ¼ 2 samples
have a similar (outer) circumference as lk ¼ 1
crystals. On the other hand, they are deﬁnitely
limited in growth in other directions due to their
twisted nature, as demonstrated by sample (g),ARTICLE IN PRESS
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T. Tsuneta, S. Tanda / Journal of Crystal Growth 264 (2004) 223–231

Fig. 2. Scanning electron microscope images of typical topological crystals. (a)–(d) lk ¼ 1; (e) lk ¼ 112; (f), and (g) lk ¼ 2:

which shows the maximum thickness. Right- andleft-handed twists are produced at virtually anequal rate: 82:79 for samples taken from ninebatches. lk ¼ 2 materials represent an average of
4.2% of all topological crystals when Nb powder isused. However, with Nb rods, this value occasionally is as high as 50–80%, although in other cases
it remains in the single digits. The higher-yieldbatches tend to produce lk ¼ 2 samples in a
growth region that is separated from that of lk ¼
1 samples. In the ampoule, while a highertemperature zone, which is about 14 cm long,produced lk ¼ 1 and 2 crystals in a ratio of 7:3, theremaining colder zone only produced lk ¼ 1
crystals. The direction of the growth of topologicalcrystals varies depending on each sample, as seenin Fig. 2(a)–(c). Disk-, ring- and tube-like cylinders
all grow simultaneously in any given batch. Let l, dand t represent the longitudinal length, the outer
diameter and the wall thickness of a cylinder. The
l=d ratio, a measure of variety in shape, varies
more than 400-fold in a single batch. Changes inmean dimensions with time are shown in Fig. 3.
While the average t grows constantly, the average ldoes not increase simply. This indicates that thegrowth rate of l depends more strongly on eachFig. 3. Average dimensions of lk ¼ 1 (cylindrical) topologicalcrystals from a batch with respect to reaction time.

sample. In addition, the outer diameter d gradually
decreases with the reaction time.
2.4. Structural properties
NbSe3 has a quasi-one-dimensional crystal structure based on highly asymmetric atomic bonding. It

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is a monoclinic system: space group P21/m withlattice parameters of a ¼ 10:008; b ¼ 3:478;( and g ¼ 109:4 according to PDF
c ¼ 15:625 A
29-4929. Nb and Se atoms combine by strong
covalent/metallic bonding to form inﬁnitely long
trigonal chains. Chains are aligned along the b-axis
and are weakly bonded to each other; more strongly
in direction c than in a. This asymmetric atomic
bonding is reﬂected in asymmetry in the electric
conductivity, elastic moduli and the growth rate,
which is responsible for the shape of the ribbon.
The crystal structure of powdered topologicalcrystals was analyzed by X-ray diffraction (XRD).
Fifty rather massive rings were ground in a mortar
and then held using adhesive tape. Whiskers fromthe same growth batch were prepared in the sameway. Fig. 4 shows the diffraction angle 2p ( ) vs.
intensity I (count) of rings and whiskers. As

227summarized in Table 1, although each diffraction
peak for rings shows one-on-one correspondence
to those for whiskers, the lattice spacing shows
signiﬁcant deviation from that of whiskers. The
lattice parameters of rings were determined to be( and g ¼ 107:1 bya ¼ 9:55; b ¼ 3:34; c ¼ 15:59 A
extrapolation [16]. Those of whiskers were a ¼
10:37; b ¼ 3:47; c ¼ 15:67 and g ¼ 109:6: a, b and cof rings were reduced by 5.1%, 3.6% and 0.2%,
respectively. The very large values of strain
obtained by the powder method should be treatedcarefully. Our recent XRD measurements on few
individual, untreated samples showed that theirstrain is not as much as these values. This suggeststhat ring crystals are plastically deformed bymechanical compression, probably during theirformation. The c direction seems to be less affectedby rolling of a crystal because it is perpendicular to

Fig. 4. X-ray powder diffraction (CuKa ) pattern of (a) NbSe3 whiskers and (b) lk ¼ 1 topological materials held on adhesive tape.
Sample (b) consists of about 50 rather thick topological materials.

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228

the axis of the bending moment so that it remains
straight after being rolled. The value of the strain
is fairly large even if it is considered that theaccuracy of the results is rather low. However, weTable 1Main XRD peaks of NbSe3 whiskers and lk=1 topologicalcrystals
Index (hkl)

100
002
102%
102
003
200
004
300
205%
301
210
213%
302
006
400
303
015
107%
216%
016
108%
408%
020
602%
600
601
800

2p ( )

9.20
11.95
12.45
17.45
17.85
18.75
23.40
28.30
29.70
30.85
31.75
33.45
34.50
36.45
38.10
38.65
40.00
40.80
43.55
45.35
47.10
51.75
52.55
55.45
58.65
61.30
81.70

Whisker

Ring

(
d (A)

I (%)

2p ( )

(
d (A)

I (%)

9.58
7.41
7.11
5.08
4.97
4.76
3.80
3.15
3.01
2.89
2.82
2.68
2.60
2.46
2.36
2.33
2.25
2.21
2.08
2.00
1.93
1.77
1.74
1.66
1.57
1.51
1.18

53
4
4
12
17
100
40
39
94
21
15
15
25
39
21
69
19
35
26
34
8
14
27
20
9
19
7

9.73
12.60
—
18.13
18.67
19.4
—
29.44
—
32.07
33.06
34.71
35.74
37.88
39.63
40.25
41.73
—
45.77
47.04
—
—
54.55
57.63
—
63.62
—

9.08
7.02
—
4.89
4.75
4.57
—
3.03
—
2.79
2.71
2.58
2.51
2.37
2.27
2.24
2.16
—
1.98
1.93
—
—
1.68
1.60
—
1.46
—

30
20
—
10
20
100
—
40
—
20
20
20
20
90
20
100
10
—
10
50
—
—
60
10
—
30
—



didn’t ﬁnd any evidence of structural breaches or
the polycrystalline nature in most topologicalmaterials through SEM or transmission electron
microscopy measurements. By observing the two
characteristic CDW states for lk ¼ 1 and 2
materials by low-temperature electron diffraction
technique and transport measurements, we conﬁrmed that the crystalinity of the topologicalmaterials is rather good. In addition, the residual
resistance ratio (RRR) of an lk ¼ 1 material canbe as high as 29, while that of a NbSe3 single
crystal synthesized by us has a comparable value,i.e. around 100.

3. Formation and growth mechanism
3.1. Formation of a looped whisker
We found that pre-reacted selenium acts as atemplate that can bend NbSe3 whiskers by the forceof surface tension. Fig. 5(a) shows a solidiﬁed Se
droplet and NbSe3 whiskers taken from a postquench batch. Some whiskers in the ﬁgure cling to
the spherical surface of the droplet. Consequently,
they form arcs. With further growth, these whiskers
could eventually wrap around the droplet. Notethat the droplet has a diameter similar to a typical
topological crystal. The two ends of a whisker could
easily coalesce by loose van der Waals’ bonding of
NbSe3 to form a seamless ring. This model isconsistent with the fact that topological crystals areproduced most efﬁciently before the reaction andtransportation proceed, since it is crucial in thisFig. 5. (a) An SEM micrograph of NbSe3 whiskers spontaneously rolling around a solidiﬁed selenium droplet. (b) An SEM image of a
NbSe3 whisker that has twisted by itself on the surface of a Se drop. It is ﬂipped over at the two points indicated by arrows.

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model that a large amount of selenium is left
unreacted. This model is particularly importantbecause it implies the possibility of a vapor–liquid–
solid (VLS) growth process.

3.2. Generation of a twist in a looped whisker
By adding a twist-generating mechanism to theabove theory, we can model the formation of lk ¼
112 and 2 materials. In fact, we have found somewhiskers attached to Se, as in Fig. 5(b) that are
twisted. Such twisting may originate in thesymmetry of the elastic properties of NbSe3. A
twisting deformation requires an internal shear
strain. The surface tension of a Se droplet acts on awhisker as a bending force and causes only
uniaxial stress. As a monoclinic crystal, however,NbSe3 has off-diagonal shear components in its
elastic compliance matrix, which make the crystal
respond to a uniaxial stress with shear strain in
addition to uniaxial strain [17]. Therefore, it ispossible that a bent NbSe3 whisker could twist by
itself. In this model, a twist would have acontinuous pitch that depends on the dimensionsof the whisker. Furthermore, a twist would be
localized as in Fig. 5(b) to minimize the interface
energy with a droplet. In effect, this mechanism
generates twists in units of p.
Another possible twist-generating mechanism ispremised on over-encirclement. Suppose that awhisker wraps around a droplet twice before its
both ends are connected to each other. The
resultant form (on the left in Fig. 6) has a writhingnumber Wr ¼ lk ¼ 2: According to White’s theorem, the shape is topologically the same as that of
the 2p-twisted lk ¼ 2 material on the right, i.e. theFig. 6. Schematic diagram of lk ¼ 2 topological material(ﬁgure-eight strip) in a pristine shape with double-writhing(Wr ¼ 2; Tw ¼ 0) as formed by double encirclement (left) andwith a twist (Wr ¼ 1; Tw ¼ 1) as it is opened afterward.

229

two shapes can be continuously transformed toeach other. Hence, over-encirclement is equivalent
to the generation of a twist of 2p: The distributionof the circumference of topological crystals reﬂects
this mechanism. The distribution for 61 lk ¼ 2
materials had a peak at around 400 mm, which isroughly double that of about 250 lk ¼ 1 materials,
i.e., 250 mm. This support the above theory
because double-encirclement would result in a
double circumference. This also suggests that the
production of lk ¼ 2 materials is mainly due to the
double-encircling mechanism. Therefore, the spontaneous generation discussed above is likely to be
relevant to only for lk ¼ 112 forms. Most lk ¼ 2
materials have a twisted shape (Fig. 6, right). Thissuggests that there is a transition from the initial
form of Wr ¼ 2; Tw ¼ 0 (Fig. 6 left) to the Wr ¼
1; Tw ¼ 1 form (right). A possible explanation isthat transformation from the writhed form to thetwisted form occurs due to interaction between theshrinking droplet and a topological crystal.
3.3. Growth mechanism
Assuming the above model, a topological crystal
has degrees of freedom about its helicity that causea large variation in the l/d ratio. When the ends ofan encircling whisker connect, a spiral should beformed instead of a perfect circle due to random
misalignment. The exposed area at the tips of awhisker (the {0 1 0} face) is the region for the most
efﬁcient growth of the topological crystal. As aconsequence, the growth of a topological crystal isdominated by 010 growth along the circumference
in a manner that resembles the spiral growth
induced by a screw dislocation. As schematicallyshown in Fig. 7, the helicity of a topological crystal
is either ‘‘radial’’ (a), ‘‘longitudinal’’ (b), orintermediate between them. We can simulate thegrowth of these crystals by assuming the growth
rates in the three basic directions a, b and c. For a
whisker, we estimate the ratio of these growth
rates as 10:1000:1. The same ratio would apply toa topological crystal because asymmetric growth isbased on large inhomogeneity in the bonding
strength of NbSe3. In fact, the volume growth rateof a whisker is on the same order (1015 m3/min) as
that of a topological crystal. A simple calculation

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demonstrates that the l/d ratio for two extreme
cases (Fig. 7(a) and (b)) differs by 1000-fold. Thisis consistent with the observed wide distribution ofthis value over 400-fold.
Moreover, we found an indication of the VLS
growth of NbSe3, which result in the morphology
of topological crystals. Fig. 8(a), where NbSe3
crystallites appear out of a fractured Se drop,
demonstrates that NbSe3 in the atmosphere canaccumulate into a Se droplet and crystallize inside.
It is quite feasible that a Se droplet can promote
the VLS growth of a topological crystal encircling
it by supplying NbSe3 to the area of mutualcontact. The higher growth rate of a topologicalcrystal at the internal periphery seems to be aconsequence of this, since it is in touch with adroplet at the internal periphery. In addition,
XRD measurements revealed that topologicalcrystals showed only compression.
If topological crystals only grow inward, any
imposed stress would only be compressive. If a Se
drop wrapped by a topological crystal couldchange in size, the resulting growth form would
have countless varieties. In most cases, a Se
droplet seems to rapidly disappear during growth
because very few topological crystals have droplets
within the hole. One of few examples is shown in
Fig. 8(b). The topological crystal has an hourglass
shape possibly because the volume of the enveloped drop has decreased and doesnot match thecircumference of the topological crystal. Thisshrinkage of a drop could cause the formation ofminor varieties like ellipses. Further complex
shapes can be formed by the interaction betweena change in a drop and the growth of a topologicalcrystal, depending on their respective time scales.
The topological crystals in Fig. 2(c) and (d) are
possible examples; topological crystals with varying diameter (d), and composite concentric cylinders (c). Moreover, a few topological crystals havea concave, supposedly spherical, surface. Thissurface is thought to be formed at the interface
between a drop and the topological crystal.

4. Discussion
Fig. 7. Schematic diagram of two extreme conﬁgurations of a
growing NbSe3 whisker rolled into a helix. Growth would be (a)radial and (b) longitudinal.

In terms of crystal defects, our topologicalcrystals can be considered to be a physicalFig. 8. (a) SEM image of a fractured selenium droplet containing fragmentary NbSe3 whiskers. (b) SEM image of a lk ¼ 1 topologicalcrystal (ring, i.e. a short cylinder) transformed into an hourglass shape due to the surface tension of selenium within the ring.

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representation of perfect disclinations [18] anddispirations in crystal solids. In the ideal caseof a whisker encircling a drop to form a perfectcircle (or twisted circle), all three kinds of
topological crystal have a wedge disclination witha power o ¼ þ2p: Furthermore, forms lk ¼ 112 and2 also have a twist disclination with Frank vectoro ¼ 7p and 72p: If a topological crystal is ahelix, it will have a translational defect (dislocation) in addition to a rotational defect (wedge andtwist disclinations). These combinations that havethe symmetry of a screw are called wedge and twist
dispiration.
Both of the two possible types of dislocation(dispiration), global and local [19,20], seem tooccur in our materials. A global disclinationline virtually pierces the hole of a topologicalcrystal while a localized disclination line is a
substantial line defect inside a crystal. A localized disclination in a solid would accompany aplanar defect like a dislocation wall or grainboundary to compensate elastic energy. As aconsequence, a plastic deformation localizes atthe local disclination. The former is seen withthinner materials (less than a few mm), which open
elastically if cut and eventually make an almost
straight, uniformly curved arc. Initially, they havea virtual global disclination that overlaps theircenter axis, and afterward the line slips out
through the cut point. On the other hand, rather
thick materials are formed plastically: such materials retain their initial shapes even when cut.
Therefore, their disclinations are localized almost
at the center.
The topological equivalence between a wedge
disclination (which corresponds to Wr) and a twist
disclination (Tw) can only be applied in ourmaterials if they are delocalized. This ﬁeld of
crystallography has not been previously explored
due to the rarity of delocalized perfect disclinations, especially twist disclinations, in crystal
solids. An open-end tubular system of twodimensional material could have a global wedge
disclination, but would not likely have a twist
disclination. NbSe3 offered an advantage over231

tubular systems due to its quasi-one-dimensional
nature.
Acknowledgements
The authors are grateful to Prof. Yamaya, lateProf. Sambongi, Dr. Inagaki, T. Toshima and T.
Matsuura for stimulating discussions. This study
was supported by Grant-in-Aid for ScientiﬁcResearch (A) and (B), as well as Grant-in-Aid
for Encouragement of Young Scientists of the
Ministry of Education Culture, Sports, Scienceand Technology in Japan.
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