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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 = aand 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.