Lumped model for rotational modes in phononic crystals

We present a lumped model for the rotational modes induced by the rotational motion of individual scatterers in two-dimensional phononic crystals comprised of square arrays of solid cylindrical scatterers in solid hosts. The model provides a physical interpretation of the origin of the rotational modes, reveals the important role played by the rotational motion in determining the band structure, and reproduces the dispersion relations in a certain range. The model increases the possibilities of manipulating wave propagation in phononic crystals. In particular, expressions derived from the model for eigenfrequencies at high symmetry points unambiguously predict the presence of a new type of Dirac-like cone at the Brillouin center, which is found to be the result of accidental degeneracy of the rotational and dipolar modes.