Exact phonon Green function for Kirkwood's model of polyethylene

The matrix elements of the lattice Green function for Kirkwood's model (describing the in-plane skeletal vibrations of an extended polyethylene chain) have been calculated exactly as a smooth function of frequency, using a semianalytic approach based on complex integration. The infinite set of matrix elements has been reduced to 12 basic real quantities from which any matrix element of the Green function in space-energy representation (including symmetry states) is obtained by linear combinations. Analytic expressions for these basic quantities are derived containing as parameter the solutions of a cubic equation. In principle, they could be evaluated on a desk calculator. Experimentally accessible frequencies are plotted as a function of the model parameters from which a graphical fit of the model can be easily found. The asymptotic behaviour of the Green function near these critical frequencies is studied in detail. As examples of straightforward applications, the density of states, the incoherent neutron scattering cross section and the localized modes due to an isotopic mass defect are plotted.