Environmental Engineering Reference
In-Depth Information
dynamical matrix given by
1
D
j
β
i
M
i
M
j
K
mj
β
(
k
)
=
exp(
ik
·
(
R
m
−
R
n
)).
(2.28)
α
α
ni
m
Here, the summation index
m
runs over all unit cells,
k
is the
wave vector,
R
m
is the spatial coordinate of the
m
th unit cell, and
M
i
is the mass of the
i
th atom.
D
j
β
(
k
) is independent of
n
owing to
the translational symmetry of the system. The force-constant
K
mj
β
i
α
ni
α
is defined as the second derivative of the potential with respect to
atomic coordinates as follows.
2
V
∂
r
ni
α
∂
r
mj
β
∂
K
mj
β
ni
α
=
,
(2.29)
where
V
is the potential energy of the system,
r
ni
α
is the
α
component (
α
=
x
,
y
,
z
) of the position vector of the
i
th atom in
aunitcell
n
.
For carbon nanostructures, Brenner's empirical potential is
adopted for
V
in Eq. 2.29. The Brenner potential has been
successfully used for molecular-dynamic simulations of carbon
nanostructures. Details of this potential can be found in Brenner's
original paper [19]. Substituting the Brenner potential into Eq.
2.29, an analytical form of the force-constant can be obtained in
a straightforward manner. The derivation is omitted here for the
sake of brevity. As the force constants are only determined for the
equilibrium positions of carbon atoms in the system, it is necessary
to optimize the atomic geometry of the SWNTs using the Brenner
potential.
Figure 2.5a shows phonon energy dispersion curves for the
region near the
point (
k
=
0) for a SWNT with chiral vector
C
h
=
(10,10) [19, 20]. Here the chiral vector (
n
,
m
) uniquely determines
the geometrical structure of SWNTs (See Appendix B for details
of SWNT structure). Four acoustic modes are seen in Fig. 2.5,
as mentioned previously, they are the longitudinal acoustic (LA)
mode, the doubly degenerate flexure (FL) acoustic modes, and a
twisting(TW)acousticmode.Thelowestoptical(E
2
g
Ramanactive)
modes are doubly degenerate. The excitation energy of E
2
g
Raman
active modes depends only on the tube diameter
d
t
and decreases
approximately according to
∼
1/
d
t
[21]. These modes always lie in
