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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.