Phason strain in an energetic growth model of a quasicrystal.

The mathematical descriptions of quasicrystal' show that for a given quasicrystalline symmetry, there is an uncountably infinite number of distinct packing of unit cells. At present, these packings [different local isomorphism (LI}classes] and the unit-cell decoration cannot be determined simultaneously from diffraction data. Steinhardt and co-workers introduce three criteria for the identification of LI classes realizable in solids ' using growth algorithm: structures should be restorable, gromable, and rapidly gromable. Since these criteria are chosen to make the packing stable and physically accessible, and that experimental data ' suggest real quasicrystals always have some defects, a local realization of these criteria is to grow the quasicrystal using energetic models. Atomic decoration can then be explored to compare with experiments. This approach is also supported by the icosahedral glass school, ' which asserts that a sensible description of quasicrystal is to start from a random packing of clusters with appropriate constraints of local order. The energetic model here attempts to interpolate these two schools: quasicrystalline models and icosahedral glass models, by employing energy parameters to control local order. Here we address only one of the many features revealed by the energetic models: topological disorder which produces power-law decay of the phason correlation function. In a previous paper, we have studied the energetic growth of two-dimensional quasicrystals using two kinds of disks, with local configuration dictated by tenfold symmetry. The structures depend critically on the interaction energy parameters between the disks. For certain range of energy parameters, we found a perfect (compact) structure which looks like a multiple-twinned sample describable as the image of projection' '" of a threedimensional staircase on two dimensions. We now generalize this energetic growth algorithm to the fat and skinny Penrose tiles with angles (72,108') and (36,144'). (This choice of units can be generalized to other units such as squares and triangles. ' ) These units are grown with a fixed concentration Cf =0.618=2cos72' and with pairwise interaction energy: (Ett, Et, =E,t,E„). (Here the subscripts f and s stand for fat and skinny tile. ) Although our preliminary exploration in energy space has not found a perfect Penrose tile, nor any quasicrystalline ground state, we do find interesting topological structures in the phason coordinate space. A standard analysis of phason strain' ' reveals a power-law decay of the phason correlation function with the size of the sample, rather than a logarithmic decay expected from entropic quasicrystal.