World Journal of Nano Science and Engineering, 2011, 1, 45-50
doi:10.4236/wjnse.2011.12007 Published Online June 2011 (http://www.SciRP.org/journal/wjnse)
Geometrical Model Based Refinements in
Nanotube Chiral Indices
Levan Chkhartishvili, Tamar Berberashvili
Department of Physi cs, Georgian Techni cal University, Tbi lisi, Georgia
Received March 28, 2011; revised April 6, 2011; accepted April 13, 2011
Abstract
There is demonstrated how it is possible to refine graphitic nanotubes’ chiral indices based on the appropri-
ate geometrical model for their structure.
Keywords: Graphitic Nanotube, Diameter, Chiral Indices, Analytical Polyhedral Model
1. Introduction
Nowadays there known many types of nanostructures.
But certainly graphitic carbon C nanotubes discovered by
Iijima [1] are the most prominent member of a nanoma-
terials family with technologically interesting properties.
Many researchers have been successful in producing
single- and multi-walled carbon nanotubes with various
diameters and lengths showing that they exhibit number
of new properties and have the potential to be very useful
in the design of nanomechanical and nanoelectronic de-
vices. Nowadays carbon fibers are used as conductive
fillers and mechanical structural reinforcements in com-
posites, e.g., in the automobile and aerospace industry,
materials for sporting-goods production, scanning probe
tips, electrode additive materials exploiting their resi-
liency, e.g., in lithium ion battery applications, electros-
tatic shielding materials, specialized medical appliances,
like the catheters, etc. Nanotubular carbon has revealed a
potential to be used also as supercapacitor electrodes,
single-tip electron guns, multi-tip array X-ray sources,
3D-electrostatic dampers, probe array test systems,
fuel-cell electrodes and catalyst supports, brush contacts,
transparent conducting films, sensor devices and biosen-
sors, field emission displays/lightings, electromechanical
memory devices, inks for printing, thermal-management
systems, power transmission cables, nanoelectronic and
flexible electronic devices, photovoltaic devices, filtra-
tion/separation membranes, drug-delivery systems, etc.
The most recent survey on carbon nanotubes’ applications
one can find in [2].
Based upon the similarity of structures of the boron
nitride BN layered phases with graphite, it was assumed
that along with carbon nanotubes, stable BN nanotubes
—fragments of hexagonal BN layers wrapped into cy-
linders—could also exist [3]. Soon after synthesizing the
first BN nanotubes, it was proposed a number of their
possible applications in technique, like the fibers, com-
posites, nanotubular heterojunctions, nanoreservoirs,
piezoelectrics, hypersound quantum generators, etc (e.g.,
see [4-13]). Interestingly that there are also some other
binary compounds not crystallizing in layered graphite-
like bulk structures, but nevertheless forming hexagonal
nanotubes: AlN [14], GaN [15], ZnO [16-19], and ZnS
[20,21].
Carbon nanotubes and other (binary compounds) gra-
phitic ones are a heterogeneous group of materials. Such
nanotubular material can comprise structurally perfect
single-walled nanotubes with a quite wide range chiral
indices, multi-walled nanotubes, nanotube bundles, and
single-walled nanotubes those change chirality along
their length due to structural defects. Furthermore, they
can interact with the environment and other neighboring
nanosystems. These two are main reasons why various
spectroscopic techniques used to characterize individual
nanotubes usually require long data acquisition times and
complicated sample preparation. New insights into the
structure-lines in graphitic single-wall nanotubes can
provide a combination of experimental data with theo-
retical calculations based on the geometrical models for
their atomic structure.
Present work aims to illustrate how an appropriate
geometrical model can be used in structural data refine-
ment recently obtained for single-walled carbon nano-
tubes. Namely, indices will be specified for nanotubes
with given diameters. Paper is organized as follows. Af-

L. CHKHARTISHVILI ET AL.
46
ter Introduction the brief overview is given on geome-
trical models up to this point suggested for graphitic sin-
gle-walled nanotubes. Then, one of them—analytical
polyhedral model—is applied to the recent experimental
data analysis. And finally, some concluding remarks are
given.
2. Geometrical Models for Graphitic
Nanotubular Structures
2.1. Conventional Rolled-up Model
In all but a few of studies dealing with carbon nanotubes
curvature effects are ignored and such structures are as-
sumed to be constructed by rolling up a plane sheet of
graphene, which in its plane state is assumed to comprise
a network of perfect hexagons, in the sense that all bond
lengths and all bond angles are assumed to be identical.
The conventional rolled-up model [22-24] implies that
the nanotube diameter (, )nm D is assumed to be given by
the expression

22
(, )
(, )
3
π
nm
nm
nnmm
D
d

,
where (, )nm
d denotes the bond length, and 1, 2, 3,n
and 0mn are nanotube indices.
Mentioned model with same success can be applied to
nanotubes of other materials with graphitic structure, like
the boron nitride BN nanotubes [25].
2.2. Idealized Polyhedral Model
But, in the cylindrical rolled-up state, the bond lengths
and bond angles are no longer equal and if such discre-
pancies are not small they should not be ignored in re-
gard to the calculation for the diameter of the carbon
nanotube. Taking into account these circumstances, Cox
and Hill have constructed a new geometrical model for
carbon nanotubes [26-28] called by authors as idealized
polyhedral model. It adopts as its basic hypothesis that
the hexagons in the cylindrical rolled-up state should be
ideally perfect in the sense that all bond lengths and all
bond angles are identical and furthermore that all atoms
are equidistant from a common axis of symmetry. These
three assumptions give rise to a geometric structure for
which all bonds play a truly equal role, unlike conven-
tional theory. It has been demonstrated that the idealized
polyhedral model gives rise to the conventional expres-
sion of the diameter as the leading term, but in addition
one may include correction terms. In general, the smaller
the tube diameter, the larger these correction terms be-
come.
In order to find the nanotube diameter expression in
the idealized polyhedral model, one need to introduce a
further two angles which are termed as the subtend
semi-angle

and the incline angle

, and a further
three parameters

,

and

, which are defined in
terms of the various angles. The first step in determining
the exact geometric structure is to determine the subtend
angle 2

, subtended at the axis of the cylinder by the
base of a single equilateral triangle. This angle is deter-
mined as a root of the following transcendental equation






 
22 2
22
sin π
2sin π2sin0.
nm nmm
nnmnm mn m





This equation may have many roots but specifically
the one required here must also satisfy the inequalities

πnnm

  .
Once the correct root is found all other expressions
can be determined. The chiral angle

is given by the
expression


  
2
2
2222
2sin
cos
sin sinπ
nn m
nmm nmm





 
and the angle of incline

of the pyramidal compo-
nents of the surfaces is given by
22
cot4cos3 cot
sin 3cos




.
With all the necessary angles now determined the
bond angle 2

is determined from the formula

2
sin3 4k  ,
where k is related to the perpendicular height of the con-
stituent pyramids and is given by the positive root of
20kk


.
Here the coefficients

,

and

are given by
equations:
22
1sin cos

  ,


2coscossincos3 cot

  ,


22
1cos4sin23sin coscot

 

 ,
which in turn lead to
24
2
k



 
.
Substituting for k yields the equation, from which the
angle

can be determined. The task of determining all
the angles for the problem is now complete and therefore

L. CHKHARTISHVILI ET AL.
47
the diameter can be calculated:
(, )
(, )
2sin cos
sin
nm
nm
D
d



.
Summarizing above description, it should be stated
that calculation of the nanotubes’ diameter in the frames
of idealized polyhedral model seems to be a tedious nu-
merical procedure.
2.3. Analytical Polyhedral Model
In general, to describe a graphitic nanotube as a polyhe-
dron one must begin with the tessellation of regular hex-
agons where the vertices of the tessellation represent the
atoms and lines of the hexagons represent chemical
bonds.
In their model, Cox and Hill overlay on this a second
tessellation of equilateral triangles where the vertices of
the triangles are the atoms and every second triangle also
has an atom located at its center. The net effect of these
two tessellations is a single tessellation of equilateral and
isosceles triangles and by fixing the lengths of the sides,
which represent bonds, it is possible to construct a truly
facetted polyhedron where all vertices are equidistant
from an axis of symmetry and all the bond lengths and
bond angles are equal for all atoms.
At the same time, one of us—Chkhartishvili—sug-
gested another version of polyhedral model for achiral,
i.e., zigzag and armchair, boron nitride nanotubes with
equal bond lengths and also rolled up from a hexagonal
plane sheet, in which however sites were alternatively
occupied by boron and nitrogen atoms [29-32]. Our
model uses different method of tessellation: equilateral
hexagons in zigzag and armchair sheets are divided into
two isosceles trapeziums or one rectangle and two iso-
sceles triangles, respectively. It means that all of lines of
tessellation are parallel to the tube axis.
Nanotubes’ cylindrical symmetry accounted in this
way allows us obtaining of the explicit expressions for
atomic sites co-ordinates, inter-atomic distances, and dia-
meters of zigzag
(,0)
(,0)
3
2sinπ2
n
n
D
dn

and armchair
(,)
(,)
54cosπ2
2sinπ2
nn
nn
Dn
dn


nanotubes. From these expressions it is easy to construct
an interpolation formula

(, )
(, )
32 1cosπ2
2sinπ2
nm
nm
mn n D
dn


useful for chiral nanotubes as well.
Thus, in this approach nanotubes’ diameter is calcu-
lated analytically. By reason of this, let call our model as
analytical polyhedral model.
In general, only geometrical consideration will be in-
sufficient: the equilibrium bond length value in nanotube
should be found maximizing its molar binding energy.
Description made for the nanotubular geometry may
serve as basis for further ground state and electron struc-
ture calculations. In particular, quasi-classical approach
can be used for this purpose (physical theory and related
mathematical problems see in [33] and [34,35], respec-
tively), which successfully have been applied to other
structural modifications of boron nitride. Recently, the
relative stability of the small- and intermediate-diameter
BN nanotubes has been studied by this method, and a
complex dependence of the molar energy on the diameter
was obtained [30,36]. As for the equilibrium bonds
length, for all calculated single-walled boron nitride na-
notubes it is found to be in satisfactory agreement with
experimental value obtained for the intra-layer bonds
length in BN layered crystals. This fact enables us to use
experimental bond length values in all graphitic-type
crystals in estimates made for corresponding nanotubular
structures.
Let draw an analogy with other theoretical (namely,
Monte Carlo) simulations performed in [37-40] to obtain
the dependence of morphology and configuration of the
assembled polymer cylindrical nanostructures confined
in nanopores on their diameters and surface conditions.
3. Recent Experimental Data and Their
Analysis
An ideal experimental characterization method of nano-
tube structures would require little sample preparation,
minimize potential damage to single-wall nanostructures,
and allow the characterized tubes to be used for subse-
quent experiments or device fabrication. Optical imaging
of carbon nanotubes placed directly on a solid substrate
meets these requirements and allows rapid visualization
and spectral resolution of individual nanotubes with rela-
tive ease. Recently, it has been reported [41] a novel on-
chip Rayleigh imaging technique using wide-field laser
illumination to measure optical scattering from individu-
al single-walled carbon nanotubes with high spatial and
spectral resolution. This method in conjunction with ca-
librated atomic force microscopy accurately measures
the resonance energies and diameters for a large number
of tubes in parallel.
The technique was applied for fast mapping of key
parameters, including the chiral indices for individual
single-walled carbon nanotubes. The values of diameters

L. CHKHARTISHVILI ET AL.
48
and chiral indices experimentally determined in [41] are
shown in Table 1.
In the present work, we have calculated same nanotu-
bular diameters based on the analytical polyhedral mod-
el’s interpolation formula for given chiral indices and for
following bond lengths values 0.140, 0.141, 0.142, 0.143
and 0.144 nm from the vicinity of 0.142 nm, the bond
length value in the graphene (see for example [2]). The
best agreement with measured and calculated values was
found just for the bond length of 0.142 nm characteristic
for carbon honeycomb plane sheet. These theoretical
results also are listed in Table 1.
One can see that for three, two, two and three species
relative difference between experiment and theory are
almost 0, <1, <3, and <12%, respectively. Agreement
can be improved—made all deviations less than 1%—
slightly (not more than in ±2) changing chiral indices.
Refined chiral indices, diameters and corresponding lo-
wered deviations are shown in brackets.
The possibility of refinement in chiral indices reveals
the fact experimentally obtained in [41] that frequent
chirality-changing structural defects accompanied with
only slight diameter-changes are characteristic for grown
single-walled carbon nanotubes. Consequently, it is not
improbable that measured values of diameter, on the one
hand, and chiral indices, on the other hand, are attributed
with different parts of the same nanotube.
In general, electronic and other significant properties
of the single-walled nanotubes depend on their structure,
which may be characterized by two parameters: the di-
ameter and the chirality, which can be encoded by two
integers—nanotube indices (, )nm . Usually, for the syn-
thesis of carbon nanotubes one may achieve some con-
trol over their diameters but little control over their chi-
ralities. As such tubes may be either metallic or semi-
conducting this poor structural control implies a rather
poor control over their electronic properties. And for
nanotubular systems other than carbon, one faces almost
the same situation. The basic reason for this defect has
been clearly explained in [42].
According to this explanation, the noted fact that the
chiral indices vary for an almost fixed tube diameter is a
direct consequence of the carbon nanotube strain energy
dependence upon its diameter. By virtue of this depen-
dence, the synthesis conditions fix a certain range of di-
ameters but leave the chirality practically unspecified.
Strain energy quantifies 1) the difference in cohesive
energy among different nanotubes, 2) the deformation
(curvature) energy per atom which is necessary to roll up
a single sheet into a nanotube of certain diameter and
chirality, and 3) a measure of the mechanical tension of a
nanotube—this tension stabilizes the tubular shape and
makes the tube round.
In particular, for carbon nanotubes the strain energy
refers to a graphene sheet and the strain energy effec-
tively depends on the diameter but not on the chirality.
The radial dependence is easy to understand: the diame-
ter is just a measure for the curvature of a tube, and the
smaller its diameter the more energy is needed to bend a
graphene sheet. But the independence of strain energy of
chirality may be attributed to the nearly isotropic in-
plane mechanical properties of the graphene sheet, as
quantified by its elastic moduli for stretching and bend-
ing. For example, the diagonal elements of the elastic
tensor of graphene are the same, due to a hexagonal
symmetry of the honeycomb lattice. Therefore, when
stretching a graphene sheet along different in-plane lat-
tice directions, one will observe the same stiffness, and
the physical systems will behave like a homogeneous
two-dimensional continuum. Thus, when rolling up a
Table 1. Experimental and theoretical value s of single-walled carbon nanotubes’ diameters.
#


,nm (, )
Theo
nm
D, nm (, )
E
xp
nm
D, nm

, %
1 (12,11) [(14,11)] 1.40 [1.57] 1.58 11.4 [0.6]
2 (20,4) 1.76 1.77 0.0
3 (22,2) 1.83 1.83 0.0
4 (15,14) [(17,16)] 1.76 [2.00] 1.99 11.6 [0.5]
5 (20,9) [(22,7)] 1.98 [2.06] 2.04 2.9 [1.0]
6 (16,15) [(18,17)] 1.88 [2.12] 2.13 11.7 [0.5]
7 (13,1) 1.07 1.07 0.0
8 (15,0) 1.18 1.19 0.8
9 (23,5) 2.05 2.05 0.0
10 (25,10) [(25,11)] 2.42 [2.47] 2.48 2.4 [0.4]

L. CHKHARTISHVILI ET AL.
49
graphene sheet along different in-plane directions to
form various nanotubes with similar diameters, this
process will require almost same deformation energy.
A similar behavior is also known for boron nitride BN
and other binary compounds nanotubes graphitic struc-
ture. It was shown [43] that main point defects formed in
the BN single-walled nanotubes are vacancies, which
tend to associate in divacancies. Due to the partially io-
nic character of the chemical bonding in BN, divacancies
behave like a Schottky pair, with very high dissociation
energy. Clustering of multiple vacancies in boron nitride
nanotubes being energetically favorable leads to ex-
tended defects formation, which locally change the na-
notube diameter and chirality. Nevertheless, these de-
fects do not alter significantly the band gap energy and
electronic structure in whole, from which are originated
optical properties of nanotubes used in structural studies.
Finally, it should be mentioned that asymptotical
analysis (1n) of the nanotube diameter formula ac-
cording to our model, as well as numerical estimates
made based on same formula for boron nitride reveal
number of groups of nanotubes with almost same diame-
ters, but different chiralities [32].
4. Conclusions
Summarizing obtained results, we can conclude that due
to the isotropic in-plane mechanical properties of the
honeycomb plane sheet, the energy needed to roll up a
nanotube is almost independent of the roll up direction,
i.e. chiral indices. It leads to the chiralty-changing struc-
tural defects and as a result to experimental ambiguities
in determination of chiral indices. These ambiguities can
be avoided characterizing nanotube structure by the ana-
lytical polyhedral model.
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