Valence force model for phonons in graphene and carbon nanotubes

Many calculations require a simple classical model for the interactions between s p2 -bonded carbon atoms, as in graphene or carbon nanotubes. Here we present a valence force model to describe these interactions. The calculated phonon spectrum of graphene and the nanotube breathing-mode energy agree well with experimental measurements and with ab initio calculations. The model does not assume an underlying lattice, so it can also be directly applied to distorted structures. The characteristics and limitations of the model are discussed.