Singular behavior of phonon-phonon interactions in molecular crystals

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Abstract
Lattice modes and internal modes do not have the same status with respect to anharmonicity in molecular crystals. Usually the description of lattice phonons is based on a lower-order diagrammatic expansion of temperature Green’s functions, but for internal modes, theories borrowed from gas and liquid states are preferred. In this paper, we give a unified view of phonon-phonon interactions in molecular crystals; the lowest-order diagrams prove inadequate to account for the anharmonic behavior of internal modes. Fourth-order diagrams bring to light the concept of pure dephasing which is particularly emphasized in gas and liquid states; this expansion step holds as long as the anharmonic coupling between internal modes is small compared to their dispersion. For very flat bands, singularities occur and a further step in the expansion is required. We show how singularities may be handled by the resummation of a subset of diagrams; their contribution is identified as an energy-exchange mechanism between internal modes, and the connection is made with previous approaches to this problem. Finally, we point out that spatial dispersion of the internal bands, resulting from their harmonic coupling with the lattice, and temporal dispersion, due to their anharmonic interactions with thermal phonons, have similar consequences on the energy-exchange processes.