Hydrodynamic structure factor of quasicrystals

A theory of the hydrodynamic structure factor for quasicrystals is developed and exploited. Based on the hydrodynamic equations for icosahedral quasicrystals we introduce the terms of dynamic correlation and response. The phononic and phasonic diffuse part of the dynamic structure factor are examined in detail in frequency and time domain. We present a complete set of solutions for the hydrodynamic equations. Out of the diffusive modes we separately study the phasonic diffusion, the anisotropy of the phasonic diffusion constants, and the general solution for phason wall diffusion. All results include phonon-phason coupling.

Figure 1

(Color online) The intensity distribution of the phononic diffuse part for the $20/32\left(\text{x}\right)$ reflection along the twofold direction for three different frequencies. The minima are located at $q_x=0$ while the maxima lie on the curve given by Eq. (2.20). For smaller values of the kinetic viscosity, such as ${\eta }_L=0.25{\text{cm}}^2/\text{s}$, $q_{x,\text{max}}$ approaches the linear-dispersion relation $c_L{q}_x$.

Figure 2

The zooming effect of the frequency $\omega$ illustrated on the phasonic isointensity contours of the $20/32\left(\text{x}\right)$ twofold reflection. Since the form of the contours for small values is the same anisotropic as in the static case, for higher ones they take an almost circular shape.

Figure 3

(Color online) The directionality of the diffusion coefficients without and with coupling, $D_i^0$ and $D_i$, as a function of the polar angle $\phi$. An increase in $K_3$ lowers the coefficients. There are systematic degeneracies along the threefold and fivefold axes, the intersection point of the latter being independent of $K_3$. At the maximum value of $K_3$ a softening occurs.

Figure 4

(Color online) Broadening of the phason walls.