Flexure modes in carbon nanotubes

Phonons are calculated for single wall carbon nanotubes. Eigenvalues and eigenvectors are presented for armchair and zig-zag tubes. The model contains just three adjustable spring constants: two for first and second nearest neighbor directed bonds, and a third for radial bond-bending interactions. There are four low frequency modes at long wavelength: a longitudinal acoustical, a torsional mode, and two flexure modes.

Figure 1

The three neighbors of atom $A$, at angles ${\theta }_1$, ${\theta }_2$ from the center of the tube.

Figure 2

Coordinate labeling in the armchair tube. The tube axis is up.

Figure 3

(a) Phonons in a (10,10) armchair nanotube. (b) Larger scale presentation of the region at low frequency and small wave vector.

Figure 4

$(m,l)$ labeling of atoms in a zigzag tube.

Figure 5

(a) Phonons in a (10,0) zigzag nanotube. (b) Larger scale presentation of the region at low frequency and small wave vector.