Page 38 - Carbon Nanotubes

Physics of carbon nanotubes 29
Table 1. Parameters of carbon nanotubes

Symbol Name Formula Value
___
carbon-carbon distance 1.421 *4 (graphite)
length of unit vector 2.46 A

unit vectors in (x,y) coordinates

reciprocal lattice vectors in (x,y) coordinates

chiral vector Ch = na, + ma2 = (n,m) n, m: integers
circumference of nanotube L= /C,I =uJn~+m2+nm Oslmlsn
L Jn’+m’+nm
diameter of nanotube d,= - z li
7r 7r
chiral angle

2n + m
cos 0 =
2Jn2 + m2 + nm
am
tan 0 = -
2n + m
the highest common divisor of (n, m)
the highest common divisor of d if n - m not a multiple of 3d
(2n + m,2m + n) dR=(
3d if n - m a multiple of 3d.
translational vector of 1D unit cell T = t,a, + f2a2 = (11,12) t, , t,: integers
2m + n
t, = ~
dR
2n + m
1, = -~
dR
aL
length of T T= -
dR
2(n2 + in2 + nm)
number of hexagons per 1D unit cell N= 2N = n,/unit cell
dR
symmetry vector$ R =pa, + qa2 = (nq) p, q: integers?
d = mp - nq, 0 5 p s n/d, 0 5 q 5 m/d
number of 2n revolutions M= [(2n + m)p + (2m + n)q]/d, M: integer
NR = MCh + dT
basic symmetry operation$ R = ($17)

rotation operation 6: radians
dT
translation operation 71- T,X: length
N
t (p, q) are uniquely determined by d = mp - nq, subject to conditions stated in table, except for zigzag tubes for which
C, = (n,O), and we definep = 1, q = -1, which gives M= 1.
$R and R refer to the same symmetry operation.

(n,O) tubules], where the translational and rotational this volume for further details regarding the non-
symmetry operations can each be executed indepen- symmorphic space groups for chiral nanotubes.
dently, or the symmetry group can be non-symmorphic The symmetry operations of the infinitely long
(for a general nanotube), where the basic symmetry armchair tubule (n = m), or zigzag tubule (rn = 0), are
operations require both a rotation $ and translation described by the symmetry groups Dnh or Dnd for
r and is written as R = ($1 r)[7]. We consider the even or odd n, respectively, since inversion is an ele-
symmorphic case in some detail in this article, and ment of Dnd only for odd n, and is an element of Dnh
refer the reader to the paper by Eklund et al.[8] in only for even n [9]. Character tables for the D, groups