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Carbon Nanotubes

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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:

Ch = na1 + ma2
a1 and a2 are the normalized primitive vectors of the two dimensional hexagonal lattice constituting the surface of the tube.

The angle between Ch and a1 is called the chiral angle and its value is determined by (n,m); i.e.,

cos(θ) = Ch.a1/|Ch|
where
Ch.a1 = n + ma1.a2
|Ch|2 = n2 + 2nm(a1.a2) + m2

Since the primitive (unnormalized) vectors for a hexagonal lattice are:

A1 = (31/2a/2)i + (a/2)j
A1 = (31/2a/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
a1 = (31/2/2)i + (1/2)j
a1 = (31/2/2)i - (1/2)j
and hence
a1.a2 = 1/2

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

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