Journal of Crystal Growth 371 (2013) 17–22

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Journal of Crystal Growthjournal homepage: www.elsevier.com/locate/jcrysgro

Cycloid crystals by topology change
T. Matsuura a,b,n, T. Matsuyama c, S. Tanda a,babc

Department of Applied Physics, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan
Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan
Graduate School of Education, Nara University of Education, Nara 630-8528, Japana r t i c l e i n f oabstract

Article history:
Received 1 August 2012
Received in revised form
19 December 2012
Accepted 25 January 2013
Communicated by Y. Furukawa
Available online 7 February 2013

A cycloid, formerly known as a roulette, is a curve arising from a point on a circumference that rolls on
its plane over a ﬁxed straight line, and the cycloid shape is observed in many natural objects formed
under a constraint. Here we report the discovery of cycloid-shaped crystals of TaSe3 obtained via the‘‘topology-change surgery’’ of thin ring-shaped crystals. We cut the ring-shaped crystals with a focusedion beam. After being cut, they formed a cycloidal shape similar to Cyclotron trajectories of electrons
under magnetic and electric ﬁelds. We conclude that the inhomogeneous curvature distribution
minimizes the bending energy and shear modulation, which corresponds to our shear-less model.
& 2013 Elsevier B.V. All rights reserved.

Keywords:
A1. Crystal morphology
A1. Defects
A2. Single crystal growth
B1. Alloys

1. Introduction

2. Experimental

The mathematical study of the cycloid (named by Galileo) wasvery fashionable in the 17th century. Pascal calculated its centreof gravity and its area, Huygens calculated the movement of amaterial point along a cycloid, and Bernoulli studied a cycloid as abrachistochrone (curve of fastest descent). In nature, the cycloid
shapes have been observed, such as the cycloidal patterns on
Europa (Jupiter’s moon) [1], concave proﬁles of mountains [2],and topological defect structures in two-dimensional liquid crystals [3], since the cycloid shape is a universal structure derived
from Lagrangian calculus of variations under some constraint,
namely boundary conditions.
Constraints relate to topology of systems closely, and mustchange when the topology changes. Thus topology must be a
powerful controllable parameter to investigate topological matters in nature, such as a Möbius strip of soap ﬁlm [4], nanotubes[5] and fullerenes [6], and knots and links of crystals [7–12]. In
this paper, we show the cycloid-shaped crystals of TaSe3, whichwere obtained by ‘‘topology-change surgery’’ for ring-shaped
crystals with diameters of 1–100 mm. We demonstrated to cut
the rings with a focused ion beam [13–15], then they opened forrelaxation of elastic energy and formed a curve of the cycloid
family, namely trochoid curves.

Ring-shaped crystals, a group of crystals with topologically
nontrivial disclinations, such as Möbius strip (p twisted ring),
ﬁgure-of-8 (2 p twisted ring), polyhedral, and Hopf-link crystals,
have also been synthesized in MX3 (M: Ta, Nb, and X: S, Se)
systems [7–12]. These crystals have pled to new scientiﬁc interestin the macroscopic quantum states of charge density waves[13,16–20] and superconductivity [14,21,22] on multi-connected
space, and in the problems of interplay between local crystal
order and global topology [15,23,24]. While the topological formsof the crystals are similar to nanotubes, their sizes are relatively
large (carbon nanotubes are typically of the order of a few
nanometers in diameter).
When the ring-shaped crystals are cut, what shape would the
crystals become in the open-ring constraint? In the case of thickring-shaped crystals, the open-ring crystals became arc shapewith constant curvatures [15], since the topological defects
induced in the ring-shaped crystals homogeneously along thecircumferences are dominant. On the other hand, what shapewould be realized in the limit of thin thickness? Suppose that asingle atomic layer is rolled and becomes a seamless ring. Since
the layer is bent and has a ﬁnite curvature, the bending energy
must be increased. Note that the topology of the ring must be
maintained while the bonding energy is lower than the bending
energy. If we can cut a single wall carbon nanotube, it becomes asingle graphene sheet [25,26], and the curvature becomes zeroafter cutting to relax the bending energy. Fig. 1(a) shows aschematic diagram of the topology change of single atomic layernCorresponding author. Tel.: þ81 706 7818.
E-mail address: toru@eng.hokudai.ac.jp (T. Matsuura).

0022-0248/$ - see front matter & 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jcrysgro.2013.01.043

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T. Matsuura et al. / Journal of Crystal Growth 371 (2013) 17–22

Fig. 1. Schematics of transformation of the ring-rod by topology-change surgery for single-layered ring (a) and double-layered ring (b). (c) Bending energy density (redcurve) and shear energy density (yellow curve) for ring-rod [(b) left panel] and the cut-ring [(b) right panel]. S0 ¼ 2pR is the total length of the rod, where R is the initial
radius of the ring-rod. (For interpretation of the references to color in this ﬁgure caption, the reader is referred to the web version of this article.)rings. With a rolled double layer, as shown in Fig. 1(b), theinteraction between the layers gives the curvature a ﬁnite valueeven after cutting, since the difference of the numbers of atoms in
the layers causes shear modulation when the double layer ring is
opened. The energy densities of bending and shear for the double
layer ring shown in the left and right panels of Fig. 1(b),respectively, are depicted schematically in Fig. 1(c). The closed
double layer ring has a homogeneous bending energy density, andthere must be no shear modulation. When the ring is forced to bestraight, the difference in the number of atoms induces shear
modulation. In term of symmetry, the shear magnitude must be
zero at the center of the layers, and have its maximum value atthe ends. The ﬁnal form of the multi-layer ring, for example
multi-wall carbon nanotubes and thin ring-shaped crystals
should be determined by the balance of the shear energy andbending energy.
We propose that the shapes with minimized shear areslipping-cycloid curves, showing an experiment of topologysurgery for thin ring-shaped crystals of TaSe3. A TaSe3 crystal
has a monoclinic crystal structure with lattice constants
a ¼ 10:402 Å, b ¼ 3:495 Å, c ¼ 9:829 Å, and b ¼ 106:261 [27] andthe circumference of the ring-shaped crystal is parallel to b axis
[7,8,15,24]. Using a focused ion beam microfabrication technique,
we cut the rings, then found that they opened for relaxation ofelastic energy and formed a curve of the cycloid family. We tookcare to minimize the ion-beam irradiation to whole of the ringshaped crystals to avoid damage except the cutting parts. Thus
the curvatures of the open-rings are dominated by the internalstrains induced by the global constraint while crystal growth, andare not a result of modulation induced by the ion-beam irradiation, which have been reported for composite thin layer structures [28].

3. Fitting analysis and discussions
A cycloid curve is the trace of a point on a circle rotating on
plain, and is also known as a Brachistochrone curve of mass
motion under homogeneous gravity. The cycloid curve can be
expanded by one parameter ks to express a change of curvaturefrom a circle to a straight line, as follows. If a rotating circle withradius a is slipped, the trace is expressed as
xout ¼ aðks y þsin yÞ,

ð1Þ

yout ¼ að1cos yÞ:

ð2ÞThis curve is known as a trochoid curve, which expresses
trajectories of moving electrons in static electric and magnetic
ﬁelds, and the structure of wave proﬁle for the water surface. The
slipping-cycloid (trochoid) curves with various slipping parameters are plotted in Fig. 2(a). When ks ¼ 1, the curve corresponds
to a cycloid. For ks ¼ 0, the curve becomes a circle with radius a,and in the limit of ks ¼ 1, it becomes a straight line. A family of
the slipping-cycloid curves, the second layer separated by distance l from the ﬁrst layer, is expressed as


l
0
0
ðk0s y þsin y Þ,ð3Þ
xin ¼ a
2

 

l
l
0
 a
cos y :
yin ¼ a þ
2
2

ð4Þ
0

Here, we must ﬁnd the relations between ks and k0s , and y and y in
the condition of minimum shear using Fig. 2(b) and (c). In eachﬁgure, the red/blue dotted circle is a circle with a center B/B’rotating with parameter ks /k0s . The radius of the blue circle isgiven by a0 ¼ al=2. The red/blue solid curve corresponds to eachslipping-cycloid curve given by the above equations. The line EF
in Fig. 2(b) [E0 F0 in (c)] is the tangent line at D (D0 ). Line CD (C0 D0 ) isperpendicular to EF (E0 F0 ). In other words, C (C0 ) is the crossing
point of line AB (AB0 ) and line CD (C0 D0 ). If points C and C0 can beon the same point, the shear modulation between two layers iszero when l is sufﬁciently small. Using this condition (C ¼ C0 ), therelations are given by
k0s ¼

l
ks 
2a ,
l
1
2a

y

y0 ¼
1

l
2aks

ð5Þ

:

ð6Þ

T. Matsuura et al. / Journal of Crystal Growth 371 (2013) 17–22

19

Fig. 2. (a) Slipping-cycloid curves for slipping parameters ks ¼ 0, 0.5, 0.75, 1, and 2, respectively. Radius of each rotating circle (a) is set at 1. (b) and (c) drawings of outerand inner layers using the slipping-cycloid curves, respectively. The parameters are described in the text. (For interpretation of the references to color in this ﬁgure legend,
the reader is referred to the web version of this article.)

Surprisingly only slipping-cycloid curves satisfy the above
condition.
To conﬁrm that slipping-cycloid curves exist in nature, we
compare the form of thin open-ring crystals in the topologychange surgery experiment [15] with a slipping-cycloid curve(trochoid) and the following natural curves; circle, ellipse, parabola, catenary, hyperbola, and buckling [see Appendix for detailsof the ﬁtting curves]. If the ﬁnal form was ﬁtted to a circle (arc),
the relaxation of the bending energy was homogeneous. In thatcase, only the bending energy was associated with the relaxation.
If the curvature changed inhomogeneously at the position of thecrystal, it meant that the topology-change surgery caused shear
modulation.
Fig. 3(a)–(c) is scanning ion beam microscopy (SIM) images ofthin ring crystals of TaSe3 before and after topology-change
surgery. Their thicknesses were less than the characteristic
thickness t [15], which means that no cylindrical domain walls
were introduced in these ring-shaped crystals. The surgery was
performed using a FIB microfabrication technique that was capable of cutting micrometer-size ring-shaped crystals while minimizing any damage. The surgery opened the thin rings andrelaxed their bending energy, and their radius of curvature morethan double. The coordinate data (x,y) for open-ring crystals wereread from the SIM images shown in Fig. 3(a)–(c), and are plottedwith open dots in Fig. 3(d)–(f), respectively. Note that eachcoordinate (x,y) is normalized by the initial radius of each ringshaped crystal. Fitting analysis was performed while minimizingPthe average values of ðDxÞ2 [1=N N ðDxN Þ2 , where DxN is thedifference between the x coordinate of the Nth experimental data
and that of each theoretical ﬁtting curve; DxN ¼ xN xfit ðyN Þ]. The
ﬁtting parameters and the average values of ðDxÞ2 are summarized in Table 1.
We found that a trochoid, circle, and ellipse ﬁt the form of the
open-rings better than a parabola, catenary, hyperbola, or buckling. The average values of ðDxÞ2 for a trochoid, circle, and ellipse
are one order smaller than other curves. We note that this result
is consistent with the statement in Ref. [15], where the forms of
all open-ring crystals are considered to be circle arcs. An ellipse
ﬁts rather than a circle, an ellipse has two ﬁtting parameters.
Since a trochoid also has two ﬁtting parameters, ks and a, an
ellipse and trochoid become similar ﬁtting curves. The maindifference between trochoid and ellipse can be seen at the endsof the crystals. Shown in the insets of Fig. 3(d)–(f), which focuseson the ends, the trochoid curves ﬁt the experimental data better
than the ellipse. These parts are the main difference between atrochoid and an ellipse. The radius of curvature of a trochoid (forks r 1) becomes small near the ends, and this feature is the sameas that of the experimental data. The average values of ðDxÞ2 onlyat the two ends of the crystals for trochoid curves were obviously
smaller than those of an ellipse or a circle. These results indicatethat the ﬁnal form of the open-ring crystals is determined not
only by the bending energy, but also by the shear modulation.
Since the shear modulation must increase near the ends, thinring-shaped crystals adopt forms from the slipping-cycloid family
to reduce the shear modulation and bending energy.
The shear modulation must be increased when the ringshaped crystals open. The free length of the crystal axis along
the circumference with the inner radius is smaller than that withthe outer radius, because the crystal dislocations have been
initially introduced into the ring-shaped crystals to reduce thebending energy. After the topology-change surgery, the differenceof the free length, however, disturbs relaxation of the bending
energy of the crystals. The ﬁnal curvature must be ﬁnite, becauseof the competition of bending stress and internal bending torquegenerated from edge dislocations [15]. Thus the strain Ex ¼ E0x þ Enx ,where E0x is the strain proportional to stress, and Enx is the eigen
strain associated with plastic deformation. Enx represents thedislocation distribution inside the crystals introduced while thecrystals were growing due to the global constraint of the ringshape. To release the bending strain E0x , the crystals can makeshear modulation gxy via the strain compatibility equation@2 Ex =@y2 þ @2 Ey =@x2 @2 gxy =@x@y ¼ 0. Furthermore, the rigidity
modulus G of the TaSe3 crystals must be small because they arebundles of covalently-connected one-dimensional crystal chainswith a relatively weak van der Waals force, and the shear
modulation could be induced easily.
The distribution of the shear modulation inside the crystals is
expected as follows. The strain and stress of the open-rings are
symmetric against the center, and the shear modulation is zero atthe center, because the shear in the closed-ring before surgery is
assumed to be macroscopically zero. After the surgery, the openring crystals have two ends. Due to the boundary conditions, at theends of the crystals both the stress along the one-dimensional
chain and the shear modulation must be zero. Thus the shear
modulation could be increased only at the middle of the cut-ring
crystals, where the bending energy is more relaxed, and at the endsof crystals, it could not contribute to releasing the bending energy,
and the curvature closes to the initial curvature of the ring-shaped
crystal before cutting. Therefore, the curves of the cut-ring crystals
become the slipping cycloid curves. The curvature distribution

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T. Matsuura et al. / Journal of Crystal Growth 371 (2013) 17–22

Fig. 3. (a)–(c) Topology-change surgery for two thin ring-shaped crystals. The images were obtained by scanning ion beam microscopy (SIM) with a focused ion beamsystem. (d)–(f) Final forms of open-ring crystals for (a)–(c), respectively. Coordinates ðx,yÞ, obtained from SIM images (a)–(c), are normalized by the initial radius of eachring-shaped crystal. Trochoid (slipping-cycloid with slipping parameter ks ), circle, ellipse, catenary, parabola, hyperbola, and buckling curves are compared withexperimental data. The origin of the coordinates for each ﬁtting curve is (x,yÞ ¼ ð0,0), and each ﬁtting curve is obtained by minimizing the average values of the squares ofthe difference of the x direction for all data (Ave. ðDxÞ2 ).

T. Matsuura et al. / Journal of Crystal Growth 371 (2013) 17–22

21

Table 1Fitting parameters used for ﬁtting curves in Fig. 3(d), (e) and (f). The averagevalues of the squares of the difference of the x direction for all experimental data

The coordinates (x,y) of a circle or ellipse passing through (0,0) are
expressed as(Ave. ðDxÞ2 ) and that for two ends (Ave. ðDxÞ2 at ends) are displayed.

x ¼ a sin y,

ð7Þ
ð8Þ

Fitting curve

Fitting parameter(s)Ave. ðDxÞ2

Ave. ðDxÞ2 at ends

y ¼ bðcos y1Þ,Trochoid
CircleEllipseParabola
Catenary
Hyperbola
Buckling

ks ¼ 0:95, a ¼ 0.803
a ¼ 2.80
a ¼ 2.80, b ¼ 2.82
a ¼ 2.12
a ¼ 1.62
a ¼ 1.25, b ¼ 1
a ¼ 3.35, b ¼ 5

0.00800.00840.0084
0.020
0.011
0.063
0.022

0.00080.00870.0078
0.060
0.029
0.31
0.11

where y is a parameter. When constant a is equal to b, it is a circle.
The trajectory of a particle under gravity is a parabola, given by

Trochoid
CircleEllipseParabola
Catenary
Hyperbola
Buckling

ks ¼ 0:36, a ¼ 1.34
a ¼ 2.28
a ¼ 2.27, b ¼ 2.23
a ¼ 1.80
a ¼ 1.20
a ¼ 0.90, b ¼ 1
a ¼ 2.70, b ¼ 50.00190.00190.0018
0.018
0.028
0.091
0.026

0.00030.00350.0033
0.068
0.052
0.044
0.045

Trochoid
CircleEllipseParabola
Catenary
Hyperbola
Buckling

ks ¼ 0:0001, a ¼ 1.21
a ¼ 1.21
a ¼ 1.2, b ¼ 1.21
a ¼ 0.87
a ¼ 0.55
a ¼ 0.46, b ¼ 1
a ¼ 1.3, b ¼ 4.3

0.000480.000480.00067
0.090
0.065
0.132
0.096

0.00106
0.001130.00085
0.41
0.206
0.582
0.536

y ¼ ax2 :ð9ÞA catenary is a curve of the string in homogeneous gravity when
its ends are ﬁxed. The coordinates (x,y) of a catenary are describedwith one constant a, such as
y ¼ aðcoshðx=aÞ1Þ:

ð10Þ

A hyperbola expresses the trajectory of particles with Rutherford
scattering, given by
x ¼ a sinh y,

ð11Þy ¼ bðcosh y1Þ:ð12ÞThe shape of a long elastic rod applied on a compressing force iscalled a buckling curve. The coordinate of the buckling curve is
described with constants a and b, such as
y ¼ b cosðx=aÞ:ð13Þ

with the shear modulation is enhanced for thin cut-ring crystals
with thinner than about 1 mm, because, for thicker rings than about
1 mm, cylindrical domain walls introduced into the ring and the
crystal curvatures are dominated by them [15,24].

Using these curves as ﬁtting curves, we compared them with theexperimental data for open-ring crystals of TaSe3.

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In summary, we discussed the ﬁnal form of thin ring-shaped
crystals and proposed the slipping-cycloid model. The model isconsistent with a shear-less condition, and describes the form
from a ring to a straight line with the slipping parameter ks . We
compared the model curves with the experimental results of
topology-change surgery for thin ring-shaped crystals of TaSe3,
and found that the ring-shaped crystals are transformed into
slipping-cycloid (trochoid) curves by topology-change surgery.
Cycloid crystals realized in superconductors and charge density
wave conductors will lead to new electrical transport experiments for quantum brachistochrones [29]. These results also
indicate that the conversion of the bending energy and shear
modulation plays an important role to determine the global shapeof the topological crystals and also topology-change for nanotubes [25,26], and might be a crucial factor of functions with
global shape transformation of in micro- and bio-mechanicalsystems [28].

Acknowledgments
We thank T. Tsuneta, T. Shibayama, K. Yamaya, K. Inagaki, T.
Toshima, H. Ohkawa for discussions and supports of experiment.
This research was supported by the 21 COE program on Topological Science and Technology of the Ministry of Education,
Culture, Sport, Science and Technology of Japan, and the Japan
Society for the Promotion of Science.
Appendix A
Circle and ellipse are found in nature. For example, thetrajectories of planets rotating around the Sun are ellipse curves.

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