Journal of Mathematical Chemistry, Vol. 43, No.

4, May 2008 ( 2007)

DOI: 10.1007/s10910-007-9276-0

Equilibrium locations for nested carbon nanocones

Duangkamon Baowan and James M. Hill
Nanomechanics Group, School of Mathematics and Applied Statistics, University of Wollongong,
Wollongong, NSW 2522, Australia E-mail: db898@uow.edu.au
E-mail: jhill@uow.edu.au

Received 1 February 2007; Revised 23 February 2007

Potentially, carbon nanostructures are very important as ideal components to

create many novel nano-devices. Such devices including nano-oscillators, ultra-fast
optical filters and nano-bearings, are based on the unique mechanical and electronic
properties of carbon nano-structures. Common carbon nanostructures used are usually
C60 -fullerenes, carbon nanotubes, carbon nano-bundles and carbon nanotori. In the
synthesis and production of carbon nanostructures, carbon nanocones tend to occur
less frequently, and it is known that five different size cones may occur, depending on
the number of pentagons in the atomic network. However, the simple geometric structure of carbon nanocones certainly facilitates calculations for their potential energy.
Here, the LennardJones potential energy function and the usual continuum approximation are employed to determine the energy for two such nested carbon nanocones
which are located co-axially. We show graphically the energy profiles for any two carbon nanocones arising from the five possible structures. For both two distinct cones and
two identical cones, we find that the equilibrium location moves further away from the
vertex as the number of pentagons is increased. However, we observe that the equilibrium position occurs such that one cone is always inside the other, and therefore, we
might expect that nested double-cones are formed according to these results.
KEY WORDS: carbon nanocones, LennardJones potential, hypergeometric functions
AMS subject classification: 33C05 Classical hypergeometric functions, 2 F1 , 74G65 Energy
minimization

1.

Introduction

Carbon nanostructures, such as C60 -fullerenes, carbon nanotubes and

carbon nanocones, are attracting much attention as a means to create novel
nano-devices. This is due to their unique properties including mechanical, electronic and their small size, high flexibility and low friction [13]. The most
important issue in producing such devices is to understand the interatomic van
der Waals potential energy behaviour between two carbon nanostructures. Much
research utilizes molecular dynamics simulation to calculate the energy of carbon
Corresponding author.

1489
0259-9791/08/0005-1489/0 2007 Springer Science+Business Media, LLC

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D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

nano systems. Girifalco and his collaborators [46] employ the LennardJones
potential energy function to determine the universal graphitic systems. Zheng
et al. [2, 3] use the LennardJones potential energy function to determine the
energy for multi-walled carbon nanotubes and propose the gigahertz frequency
oscillator. This phenomena is confirmed by Legoas et al. [7]. The C60 -fullerenes
systems are studied by Qian et al. [8, 9]. Moreover, the present authors and their
colleagues employ elementary mathematical modelling to determine analytically
the energy and the resulting force for such systems, (see [1012]).
All of these investigations emphasize systems involving C60 -fullerenes, carbon nanotubes and carbon nanotori, but very little of the existing literature deals
with carbon nanocones. Carbon nanocones have received less attention primarily
because only a small amount tend to occur in the production process [13].
There are five possible ways to construct carbon nanocones depending on
the number of pentagons which are needed to close the vertex, and most research on nanocones deals with their electronic structure [14, 15]. It is believed that
the different number of pentagons in carbon nanocones is the key to the puzzle
of nucleation in atomic construction [14, 16]. Kim et al. [17] utilize the catalytic
chemical vapor deposition method to synthesize carbon nanocones inside carbon
nanotubes, and they find that the resulting structures have different physical and
electronic properties from the original carbon structure. Charlier and Rignanese
[14] use molecular dynamics simulation to examine the local density of states
for the five possible carbon nanocones and propose that carbon nanocones are
ideal candidates for nanoprobes in scanning tunneling microscopy. The electronic structure of carbon nanocones is examined by Pincak and Osipov [15]. They
employ the effective-mass theory for a graphite monolayer and gauge theory of
disclinations on fluctuating elastic surfaces to obtain a mathematical equation,
and finally they find that the electron states are dependent on the position of
the pentagons. The mechanical properties of carbon nanocones are investigated
by Jordan and Crespi [18]. They obtain the nonlinear mechanical behaviour for
both the original shape and the inverse carbon nanocone which is obtained from
the original cone by inversion.
However, from our knowledge, there is little work examining the potential energy behaviour of carbon nanocones. In this paper, we determine the
potential energy for two carbon nanocones which are assumed to be co-axial.
The LennardJones potential energy together with the continuum approximation, which assumes that carbon atoms are uniformly distributed over the surface of each molecule, is utilized to calculate the potential energy of the system.
Details for carbon nanocones are presented in the following section, and in the
section thereafter we examine the geometry of cones. The interaction energy and
numerical solutions for two generally distinct nested carbon nanocones are presented in sections 4 and 5, respectively, and conclusions are given in section 6. In
Appendices A, and B we present mathematical details for the derivation of the

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

1491

Figure 1. Five possible nanocones with vertex angles 19.20 , 38.90 , 60.00 , 83.60 and 112.90 .

potential energy for two cones which involve both hypergeometric and Legendre
functions.
2.

Carbon nanocones

Carbon nanocones as one of the graphitic structures were formerly discovered by Ge and Sattler [19] and subsequently synthesized by Krishnan et al.
[16]. Typically, carbon nanocones are observed together with carbon nanotubes
and nanotube bundles during the synthesis process [13], and carbon nanocones
tend to be found at the cap of carbon nanotubes. There are five possible structures for nanocones, as shown in figure 1, because the cone angle depends on the
number of pentagons needed to close the structure. Cones are formed from hexagons on a honeycombed lattice by adding fewer pentagons than the six which are
needed by Eulers theorem [20] for a closed structure. In C60 -fullerenes, a hexagonal lattice of any size or shape can only form a closed structure by precisely
twelve pentagons. The carbon nanotube cap which is a half C60 -fullerene contains six pentagons and therefore, carbon nanocones must have a number of pentagons which is less than six.
The disclination number of pentagons on the graphene gives the change
with in the form

= Np ,
3
where Np is the number of the pentagons which ranges from 0 to 6. From the
diagram of the cone shown in figure 2, it is clear that sin(/2) = r/R and c =
2 r = 2(1 Np /6)R. Therefore, we obtain the relation of the cone angle and
the number of pentagons as
sin(/2) = 1

Np
.
6

(1)

1492

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

(a)

(b)

r
c

Figure 2. Graphene sheet (a) forming the carbon nanocone (b).

Table 1
Relation of number of pentagons Np and open angle for carbon nanocones.
Number of pentagons (Np )

Angle of cone ()
180
112.90
83.60
60.00
38.90
19.2
0

0
1
2
3
4
5
6

There are seven possible values of the angle depending on the number
of pentagons which are shown in table 1. We note that for Np = 0, we have a
graphene sheet and for Np = 6, we obtain a capped carbon nanotube. Hence
there are only five possible values giving rise to carbon nanocones. We assume
that the mean atomic surface density of carbon nanocones is the mean atomic
surface density of graphene sheet which is 0.3812 2 , due to the fact that the
carbon nanocones are formed from the graphene sheet.
3.

Geometry of cones

The surface shown in figure 3(a) is called a double right cone. A right cone
is one for which the vertex is directly above the centre of its base. However, when
used without qualification, the term cone often means right cone. A right cone is
the surface in three-dimensional space generated by a line that revolves about a
fixed axis in such a way that the line passes through a fixed point on the axis and
always makes the same angle with the axis, and the fixed point is called the vertex of the cone. A cone consists of two parts; called nappes, that intersect at the
vertex [21]. The quadratic equation in Cartesian coordinates (x, y, z) for double
cones is given by

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

1493

ds

Vertex

a
dz

Axis
h

(a)

(b)

Figure 3. (a) Geometry of an elliptical cone and (b) diagram for surface integrations.

z2
x2 y2
+
=
,
a2
b2
c2

(2)

where a, b and c are constants. Alternatively, in cylindrical coordinates (r, , z)

the equation for the right cone can be defined as r = z tan(/2) where is
the cone angle. We need to examine the surface integral of cones to calculate
the Lennard-Jones potential energy in the continuum approach. The surface area
integral of a single cone is given by

h
2
rd ds.
Area =
0

From figure 3(b), we can see that ds = dz/ cos(/2) and r = z tan(/2). Therefore, the surface integral is of the form






tan(/2) h 2
a h 2
Area =
zd dz = 2
zd dz = a,
(3)
cos(/2) 0 0
h 0 0

where  = a 2 + h2 .
4.

Interaction energy for carbon nanocones

4.1. Potential energy function

The LennardJones potential, which is widely used for non-bonded atomic structures (see for example, [2, 3, 5, 6]), is adopted here to determine the

1494

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

interatomic potential energy for two carbon nanocones which are located coaxially. The classical LennardJones potential between a pair of atoms at a
distance apart is given by
E() =

B
A
+ 12 ,
6

(4)

where A and B are the attractive and repulsive constants, respectively. Using
the continuum approximation, which assumes that the discrete atoms may be
replaced by a uniform surface density of atoms over the surface, together with
the LennardJones potential, the total potential energy E tot for two non-bonded
molecules can be obtained by performing the double surface integrals



tot
E()d1 d2 ,
(5)
E = n1 n2
where n1 and n2 denote mean surface densities of the first and the second molecule and is the distance between surface elements d1 and d2 .
4.2. Model formulation for two carbon nanocones
With reference to a rectangular Cartesian coordinate system (x1 , y1 , z1 )
with origin located at the vertex of the first cone, shown in figure 4, a typical point on its surface has coordinates (r1 cos 1 , r1 sin 1 , z1 ). Similarly, with
reference to a rectangular Cartesian coordinate system (x2 , y2 , z2 ) with origin
located at the vertex of the second cone, a typical point on its surface has
coordinates (r2 cos 2 , r2 sin 2 , Z + z2 ) with respect to the coordinate system with
origin located at the vertex of first cone, where Z is the distance between their
vertices. The first cone has the open angle with base radius a1 and height h1 ,
and r1 = z1 where = tan(/2). The second cone has the open angle with
base radius a2 and height h2 , and r2 = z2 where = tan(/2). The distance
between two typical points on the first and the second cone is then given by

h1

a1
a2

h2

Figure 4. Geometry for two distinct carbon nanocones.

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

1495

2 = (r1 cos 1 r2 cos 2 )2 + (r1 sin 1 r2 sin 2 )2 + [z1 (Z + z2 )]2 ,

= (z1 cos 1 z2 cos 2 )2 + (z1 sin 1 z2 sin 2 )2 + [z1 (Z + z2 )]2 ,
= ( 2 + 1)z12 + ( 2 + 1)z22 2z1 z2 ( + 1) 2Z(z1 z2 ) + Z 2


2 1 2
.
+4 z1 z2 sin
2
By using the LennardJones potential together with the continuum approximation, the total potential energy is given by




a1 a2 (a12 + h21 )(a22 + h22 )
h2
h1
2
2
A
B
tot

E = n1 n2
z1 z2 6 + 12

h21 h22
0
0
0
0
d1 d2 dz1 dz2 ,
where n1 and n2 are mean surface densities of the first and the second carbon
nanocones, respectively. Further, if we define the integrals In as

2
2

2
2
d1 d2
d1 d2

=
,
(6)
In =
n
2

{ + sin [(1 2 )/2]}n/2

0
0
0
0
where n = 6 and 12, = ( 2 +1)z12 +( 2 +1)z22 2z1 z2 ( +1)2Z(z1 z2 )+Z 2
and = 4z1 z2 . In Appendix A, we show that the integrals In can be evaluated
either in terms of hypergeometric functions or Legendre functions. In terms of
the hypergeometric function, we may deduce



4 2
1

,
F 3, ; 1;
I6 =
2
+
( + )3



4 2
1

.
I12 =
F 6, ; 1;
2
+
( + )6
These two hypergeometric functions are also degenerate hypergeometric
functions, for which the details are presented in Appendix B, in the form of


2 
3

4 2

+
,
1

+
8 +
2 ( + )




2

15

4 2
5
=
+
I12
1

2 +
4 +
5 ( + )

3

4

5 

25
175
63

.
8 +
128 +
256 +
I6 =

Then the total potential energy becomes

1496

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

h
Figure 5. Geometry for two identical carbon nanocones.

a1 a2 (a12 + h21 )(a22 + h22 )
h2
h1

= n1 n2
z1 z2 (AI6 + BI12
)dz1 dz2 .
h21 h22
0
0

E tot

(7)
However, for the final solution for E tot , we need to integrate (7) with
respect to z1 and z2 which we determine numerically. Although clearly complicated numerical values for these integrals may be readily evaluated using the algebraic computer package MAPLE.
4.3. Model formulation for identical carbon nanocones
For the particular case, we consider the two identical carbon nanocones, as
shown in figure 5, which the distance between their vertices is denoted by Z.
They both have the cone angle which corresponds to base radius a and height
h. The relations between r and z for both cones are given by r1 = z1 and
r2 = z2 , respectively, where = tan(/2) = a/ h. In the cylindrical polar coordinates (r, , z), the parametric equations for the first and the second cone can
be written as (r1 cos 1 , r1 sin 1 , z1 ) and (r2 cos 2 , r2 sin 2 , Z + z2 ), respectively.
Then the distance between two typical points from the first cone to the second
cone is given by
2 = (r1 cos 1 r2 cos 2 )2 + (r1 sin 1 r2 sin 2 )2 + [z1 (Z + z2 )]2 ,
= (z1 cos 1 z2 cos 2 )2 + (z1 sin 1 z2 sin 2 )2 + [z1 (Z + z2 )]2 ,
= ( 2 + 1)(z1 z2 )2 2Z(z1 z2 ) + Z 2 + 4 2 z1 z2 sin2 [(1 2 )/2].

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

1497

Therefore, the total potential energy can be written as





4
2
h
h
2
2
A
a
B
a
tot
2
E = n
+
z1 z2 6 + 12 d1 d2 dz1 dz2 , (8)
h4 h2

0
0
0
0
where n is the mean surface density of the carbon nanocone. We also need to
evaluate the integral

2
2

2
2
d1 d2
d1 d2
In =
=
, n = 6, 12,
n

{ + sin2 [(1 2 )/2]}n/2

0
0
0
0
where in this case = ( 2 + 1)(z1 z2 )2 2Z(z1 z2 ) + Z 2 and = 4 2 z1 z2 . By
precisely the same method, we may deduce




h
h
a4
a2
tot
2

+
z1 z2 (AI6 + BI12
)dz1 dz2 ,
(9)
E = n
h4 h2
0
0
which we have also evaluated numerically to obtain the final solution for E tot .
5.

Numerical solutions

In this section, we show graphically the potential energy for two carbon
nanocones of the five possible structures. Using the algebraic package MAPLE,
we plot the relation between the potential energy and the distance between their
vertices Z. Due to the lack of the LennardJones constants, namely the attractive constant A and repulsive constant B, specifically for carbon nanocones, we
then employ the LennardJones constants for carbon-carbon atoms for the carbon nanocones. Following the work of Girifalco [4], we obtain A = 19.97 eV6
and B = 34.81 103 eV12 . The values of a and h are given by a = sin(/2) =
3ma sin(/2)/2 and h = cos(/2) = 3ma cos(/2)/2, where m is a positive integer which is indicated the size of carbon nanocones. Here, we choose m = 50.
We examine the potential energy for the system of the two carbon nanocones which are explicitly shown in figures 68. For the combination of the
cone containing one pentagon, the equilibrium location for the other four cones
occurs at approximately at 3 for the distance between their vertices, illustrated in figure 6, and the most stable equilibrium location happens for the cone
Np = 2. The equilibrium location increases and the system has less stability as
the number of pentagons increases, and this is shown in figure 7. Of particular interest for two identical carbon nanocones, the increased number of carbon
atoms at the vertex causes the distance between their vertices at the equilibrium
position to increase as demonstrated in figure 8. Values of the distances between
their vertices Z at the equilibrium location for any two carbon nanocones are
given in table 2.

1498

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

Figure 6. Variation of potential energy versus the distance between vertices for the cone Np = 1
and Np = 2, 3, 4 and 5 for the second cone.

Np = 1 and Np = 2
Np = 2 and Np = 3
Np = 3 and Np = 4
Np = 4 and Np = 5

Potential energy (eV)

0
2

10

12

14

-2

-4

-6

Distance between vertices ()

-8

Figure 7. Variation of potential energy versus the distance between vertices for various combinations of two different cones.

The shortest distance between the two cones at the equilibrium location is
denoted by x as illustrated in figure 9 and values of x are shown in table 3.
For the two identical carbon nanocones, we obtain the value of 3.4 , which is
the inter-spacing distance for two graphene sheets. Otherwise, we get values of
approximately 2.6 which is less than the equilibrium location for two graphene
sheets due to the fact that the two surfaces are not parallel. We observe that x
increases with increasing number of pentagons.

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones
300

Np = 1
Np = 2
Np = 3
Np = 4
Np = 5

200

Potential energy (eV)

1499

100
0
5

10

15

20

25

30

35

-100

Distance between vertices ()

-200
-300
-400
-500

Figure 8. Variation of potential energy versus the distance between vertices for five identical carbon
nanocones.
Table 2
Distance between vertices Z at the equilibrium position for any two carbon nanocones.
Np of inner cone
Np of outer cone
1
2
3
4
5

6.

4.1631

3.1161
5.2663

3.1614
3.9256
6.9211

3.2001
3.9973
5.2785
10.3732

3.2110
4.0197
5.3703
8.0004
20.6060

Conclusions

In this paper we determine the potential energy for two carbon nanocones for both cases of identical and non-identical cones. Using the LennardJones
potential energy and the continuum approximation, which assumes that the discrete carbon atoms can be replaced by an average distribution over each surface,
we perform the integration of the LennardJones potential energy. Due to the
lack of specific data for carbon nanocones, we employ for the carbon nanocones the known LennardJones constants determined for plane sheets of carbon
carbon atoms. We obtain a value 3 for the distance between the two vertices at
the equilibrium position for the system of two identical one pentagon cones, and
if this cone is paired with the other four possible cones, then this distance increases with decreasing cone angle or increasing number of pentagons. Moreover,
in the case of two identical carbon nanocones, the equilibrium position moves
away from the vertex as we reduce the cone angle. However, the equilibrium
location is always inside the cone, and therefore as a result, we might construct

1500

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

Figure 9. Shortest distance x between two carbon nanocones at the equilibrium location.
Table 3
Shortest distance x between the two cones at the equilibrium position for any two carbon
nanocones.
Np of inner cone
Np of outer cone
1
2
3
4
5

3.4695

2.5970
3.5101

2.6347
2.6165
3.4606

2.6670
2.6643
2.6393
3.4541

2.6761
2.6793
2.6852
2.6640
3.4364

nested double-cones from any two possible combinations of carbon nanocones

as shown in figure 9.

Acknowledgements
The authors are grateful to the Australian Research Council for support through
the Discovery Project Scheme and JMH is grateful for the award of an Australian Professorial Fellowship. They are also especially grateful to their colleagues
Barry Cox, Ngamta Thamwattana and Tamsyn Hilder for many helpful comments and discussions on this work.

Appendix
A Analytical solution for In in (6)
The integral (6) may be evaluated either in terms of hypergeometric of
Legendre functions. First we consider the integral

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

I2m

=

2
0

d1 d2
{ + sin2 [(1 2 )/2]}m

1501

(10)

where m = n/2. However, since the integrand is a symmetric function 1 2 , the

defined by
intermediate integral I2m

2
d1

I2m =
,
2
{ + sin [(1 2 )/2]}m
0
can be shown by differentiation with respect to 2 to be independent of 2 ,
namely



2

dI2m
1

d1 = 0.
=

d2
1 { + sin2 [(1 2 )/2]}m
0
Thus, we may set 2 to be zero and trivially perform the 2 integration so that
(10) becomes,

/2
dx

,
I2m = 8
( + sin2 x)m
0
and we may consider the integral I2m defined by

/2
dx
.
I2m =
( + sin2 x)m
0

(11)

Making the substitution t = cot x we obtain

(1 + t 2 )m1
(1 + t 2 )m1
1
dt
=
dt,
I2m =
( + )m 0 (1 + t 2 )m
( + + t 2 )m
0
where = /( + ). Now on writing this integral in the form


1
dt
1
I2m =
,
m
2
2
m
( + ) 0 [1 (1 )t /(1 + t )] (1 + t 2 )
we are led to make the substitution
z=

t
,
(1 + t 2 )1/2

t=

z
,
(1 z2 )1/2

dt =

dz
,
(1 z2 )3/2

and in the following line we make the substitution u = z2

1
dz
1
I2m =
( + )m 0 [1 (1 )z2 ]m (1 z2 )1/2

1 1/2
u
(1 u)1/2
1
=
du.
2( + )m 0 [1 (1 )u]m

1502

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

From Gradshteyn and Ryzhik [24] (p. 995, equation 9.111) we may deduce



1
,
(12)
F m, ; 1;
I2m =
2( + )m
2
+
where F (a, b; c; z) denotes the usual hypergeometric function. We comment
that Colavecchia et al. [22] examine in some detail the numerical evaluation of
various hypergeometric functions.
From Erdlyi [23] and on recognizing two of the numbers (1c), (ab),
(a + b c) are equal to each other, it can be shown that this result admits a
quadratic transformation and becomes a Legendre function. Using the transformation
F (a, b; 2b; 4z/(1 + z)2 ) = (1 + z)2a F (a, a + 1/2 b; b + 1/2; z2 ),
we obtain
I2m =

(1 + y)2m
F (m, m; 1; y 2 ),
2( + )m

where 4y/(1+y)2 = /(+ ). Using the definitions from Gradshteyn and Ryzhik
[24] (p. 960, equation 8.772.3 and p. 998, equation 9.131.1)
P (z)

1
=
(1 )

z1
z+1

/2 

z+1
2


z1
,
F , ; 1 ;
z+1


and
F (a, b; c, z) = (1 z)cab F (c a, c b; c; z),

where P (z) is a Legendre function of the first kind and in our case is zero,
then we obtain the integral in terms of the Legendre function which is given by




1 + y2
1+y m

Pm1
I2m =
.
(13)
2( + )m 1 y
1 y2
B Degenerate hypergeometric functions
The details of the degenerate hypergeometric function of (12) are presented
in this appendix. The degenerate hypergeometric function is the hypergeometric
function which can be written as the finite summation of the polynomial. For
convenience, we define 4
Jm = F (m, 1/2; 1; z),

(14)

D. Baowan and J.M. Hill / Equilibrium locations for nested carbon nanocones

1503

where m is a positive integer. Following Erdlyi [23], equation (14) admits the
degenerate hypergeometric function in the case number 16 (p.72) with a agenerated solution
F (a, b; c; z) = (1 z)cab F (c a, c b; c; z).
Then we obtain
Jm = (1 z)1/2m F (1 m, 1/2; 1; z).

(15)

In terms of a series, the hypergeometric function is given by

F (a, b; c; z) =


(a)n (b)n
n=0

n!(c)n

zn ,

where
(a)n = (a + n)/ (a) = a(a + 1)(a + 2) . . . (a + n 1)

and

(a)0 = 1.

Here, we need to evaluate J3 and J6 , and from (15) we may deduce



2

(2)n (1/2)n n
3 2
1
1
1z+ z ,
z =
J3 =
n!(1)n
8
(1 z)5/2
(1 z)5/2
J6 =
=

n=0
5


1
(1 z)11/2

n=0

(5)n (1/2)n n
z
n!(1)n

(16)



15 2 25 3 175 4
5
63 5
1
z
+
z
z
z
z
1

2
4
8
128
256
(1 z)11/2

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