Page 71 - Carbon Nanotubes

60 R. SETTON

b
Fig. 3. The two possible orientations of the hexagons in non-
helical sheets, relative to the cylinder axis (broken line);
(a) case I; (b) case I1 (see text).

Fig. 1. Approximate cross section of a tubule with scroll
helicity. The distance d between successive rolls of the spiral dius ri of the cylinder can conveniently be described
is assumed to be constant, and no is the initial radius of the
innermost fold. in terms of two orthogonal vectors[4], Pi and Qi
(Fig. 4), respectively integral multiples of the unit vec-
tors x and 7 with moduli I XI = (3/2)G and I y I =
(&/2)G. For the ith layer, we can write:
drical sheets which, for simplicity, will be assumed to
have a common axis and a circular cross-section. The
hexagons of the unrolled cylinders all have one of their
three sets of parallel opposite sides respectively per- or, for the scalars Ri, Pi, Qi,
pendicular (case I) or parallel (case 11) to the common
axis of the cylinders (Fig. 3).
The orientation of the hexagons with respect to the
tube axis of the unrolled ith cylinder as well as the ra-
where Pi, Qi are integers of identical parity, and

Ri = 27rri. (4)
The pitch angle ai characteristic of this orientation
of the hexagons is:
ai = arctan(Qi/fiPi), (5)

but one can also distinguish an apparent angle of pitch
pi between the helical rows of unbroken hexagons
and the plane perpendicular to the cylinder axis, with

Although the limits of LY are:

this angle can be limited to 30", since every (Pi, Qi)
doublet is associated with a corresponding doublet
(Pi,-Qi), symmetric of (Pi,Qi) with respect to a =
O", yielding the same value of the radius (Fig. 5) and
characterizing the stereoisomer of the chiral ith cylin-
Fig. 2. A portion of an unrolled cylinder with screw helicity. drical sheet, The doublet (Pi,Qi) is also equivalent to
The broken line is parallel to the cylinder axis, and the cylin- the doublet (Pi, Qj), with Pi = (P, + Q,)/2 and Qi =
drical sheet has been cut along a generatrix (full line paral-
lel to the cylinder axis). a and b are the unit vectors of the (3P, - 2Qi)/2, with the pitch angle ai = 60" - ai.
two-dimensional carbon layer in sp2 hybridization. which also yields the same value of ri.