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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:
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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