Implicit phonon shifts and thermodynamical properties of rigid carbon nanotube bunches

We calculate phonon shifts of external modes of a bunch of carbon nanotubes. The bunches form a 2-dimensional hexagonal arrangement of lattice with long molecules of carbon nanotubes. A simple model based on atom-atom potential has been used to calculate the implicit anharmonicity in the phonons of carbon nanotube bundles (also called ropes or bunches) having rigid tubes, with the assumption that under hydrostatic pressure only the inter-tube distance in the bunch varies. Various bulk and thermodynamic properties like thermal expansion, bulk modulus and the Gruneisen constants and external phonon shifts which naturally enter into the calculation are also described and compared with the available data. The specific heat capacity has also been calculated.