PHONON MODES IN CARBON NANOTUBULES

The relationship between the phonon modes in carbon nanotubules and th ose in a graphene sheet is studied by use of the zone-folding method. The modes identified with pure translations and rotations are identifi ed. General results relating to the symmetry properties of the Raman a nd infrared activity of the phonon modes are deduced. It is shown that as the diameter of a tubule increases, the number of Raman and infrar ed active modes is essentially unchanged despite the increase in the n umber of atoms per unit cell. Explicit phonon dispersion curves are pr esented for a small diameter armchair carbon tubule.