Environmental Engineering Reference
In-Depth Information
Appendix B: Structure of Single-Walled Carbon
Nanotubes
Here, the structure of SWNTs is briefly summarized. Figure B.1
shows the graphene layer and the primitive lattice vectors
a
1
and
a
2
, which are defined as
√
3
a
2
√
3
a
2
,
a
2
a
2
(B.1)
Here,
a
is the Bravais lattice constant,
a
=
√
3
a
cc
=
2.46A,
where
a
cc
=
1.42A is the bond length between neighboring carbon
atoms. As seen in Fig. B1, an SWNT can be conceptually obtained by
folding the dashed line containing points O and B to the dashed line
containing point A and B'. An SWNT structure is characterized by
three geometrical parameters: the chiral vector
C
h
, the translation
vector
T
, and thechiral angle
a
1
=
,
a
2
=
,
−
, as shown inFig. B1.
The chiral vector
C
h
or
−
OA
is expressed as
C
h
=
n
a
1
+
m
a
2
(B.2)
that connects two crystallographically equivalent carbon atoms on
the graphene. The SWNT structure is uniquely determined by the
pair of integer (
n
,
m
) in Eq. B2. Thus, an SWNT is described as
an (
n
,
m
) SWNT. For example, the circumference and diameter of
an (
n
,
m
)
SWNT are giv
en by the absolute val
ue of chiral v
ector
|
C
h
| =
θ
a
√
n
2
nm
a
√
n
2
+
+
nm
and
d
t
= |
C
h
|
/π
=
+
+
/π
,
respectively. Other two geometrical parameters (
T
and
θ
) can be
alsodeterminedfromthechiralvector.Thechiralangle
θ
istheangle
between the chiral vector and the primitivelattice vector
a1
:
m
2
m
2
C
h
·
a
1
|
C
h
||
a
1
|
=
2
n
+
m
2
√
n
2
cos
θ
=
(B.3)
+
nm
+
m
2
Due to the sixfold hexagonal symmetry of graphene, the chiral
angle is restricted to 0
≤
θ
≤
30
◦
. Next, the translational vector
T
=
(
t
1
,
t
2
) can bedetermined by the orthogonality condition:
C
h
·
T
=
t
1
(2
n
+
m
)
+
t
2
(2
m
+
n
)
=
0
(B.4)
Thus, the integers
t
1
and
t
2
are given as
2
m
+
n
g
d
2
m
+
n
g
d
T
=
(
t
1
,
t
2
)
=
,
−
(B.5)
