Hydrodynamics of icosahedral quasicrystals

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Abstract
The equations governing long-wavelength, low-frequency excitations in icosahedral quasicrystals are derived. It is found that while the speeds of the propagating modes are isotropic, the attenuations are not, implying that purely macroscopic experiments can in principle distinguish quasicrystals from crystals, glasses, or conventional incommensurate systems. The coefficient of the anisotropy is, regrettably, quite small. The complete spectrum consists of three diffusive phasons, two pairs of transverse and one pair of longitudinal sound modes, a vacancy diffusion mode, a heat diffusion mode, and, in a material with n atomic species, n-1 additional particle diffusion modes. The diffusion times of the vacancy and phason modes are expected to be comparable and very long. It is shown that propagating phasons, even at short wavelength, are an unlikely prospect. The static, equilibrium elastic properties are also anisotropic, but are approached very slowly, and in many situations, the elastic response is isotropic on experimentally accessible time scales. Our results also imply that nonlinear fluctuation corrections to the linearized hydrodynamics presented here are finite as q and ω→0, i.e., there is no breakdown of conventional hydrodynamics in icosahedral quasicrystals.