Properties of random tilings in three dimensions

Abstract

Three-dimensional icosahedral random tilings with rhombohedral cells are studied in the semientropic model. We introduce a global energy measure defined by the variance of the quasilattice points in orthogonal space and justify its physical basis. The internal energy, the specific heat, the configuration entropy, and the sheet magnetization $($as defined by Dotera and Steinhardt [Phys. Rev. Lett. 72, 1670 (1994)]$)$ are calculated. Since the model has mean-field character, no phase transition occurs in contrast to matching-rule models. The self-diffusion coefficients closely follow an Arrhenius law, but show plateaus at intermediate temperature ranges, because there is a correlation between the temperature behavior of the self-diffusion coefficient and the frequency of vertices which are able to flip (simpletons). We demonstrate that the radial distribution function and the radial structure factor depend only slightly on the random tiling configuration. Isotropic interactions lead to an energetical equidistribution of all configurations of a canonical random tiling ensemble and do not enforce matching rules.