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**Carbon Nanotubes**
Sources:

- Thomas W. Ebbesen, "Carbon Nanotubes,"
*Physics Today*
June 1996, pp. 26-32.
- Gene Dresselhaus, Peter Eklund, and Riichiro Saito, "Carbon
Nanotubes,"
*Physics World* January 1998, pp.1-11.
- Cees Dekker, "Caron Nanotubes as Molecular Quantum Wires,"
*Physics Today* May 1999, pp. 22-28.

Carbon nanotubes combine the confinement of electrons in two
dimensions to regions comparable to their de Broglie wavelength,
but in addition carbon nanotubes have a special periodic structure
so the quantum effects may be somewhat different from a simple
quantum wire. The tubular structure may lead to wavefunctions
which involve Bessel functions rather than trigonometric functions.

Moreover carbon nanotubes come in a variety of
internal structures characterized by two integers, say (n,m).
The pair of integers define a *chiral vector* given by:

#### C_{h} = na_{1} + ma_{2}

a_{1} and a_{2} are the normalized primitive vectors of
the two dimensional hexagonal lattice constituting the surface of the
tube.

The angle between C_{h} and a_{1} is called the
chiral angle and its value is determined by (n,m); i.e.,
#### cos(θ) = C_{h}.a_{1}/|C_{h}|

where

C_{h}.a_{1} = n + ma_{1}.a_{2}

|C_{h}|^{2} = n^{2} + 2nm(a_{1}.a_{2}) + m^{2}

Since the primitive (unnormalized) vectors for a hexagonal lattice are:
####
A_{1} = (3^{1/2}a/2)i + (a/2)j

A_{1} = (3^{1/2}a/2)i + (a/2)j

where a is the lattice constant,

their magnitudes are both
equal to

(3/4)a + (1/4)a = a

and thus the corresponding unit vectors
are

a_{1} = (3^{1/2}/2)i + (1/2)j

a_{1} = (3^{1/2}/2)i - (1/2)j

and hence

a_{1}.a_{2} = 1/2

The previous expressions for
cos(θ) and C_{h} are easily evaluated; i.e.,
Likewise the diameter of the nanotube is determined by the
chiral vector.
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