Visualization and analysis of quasicrystal densities

Several techniques for visualizing and analyzing scatterer densities of quasiperiodic quasicrystals, both in the physical space and in the associated hyperspace, are discussed. In particular, a method for analyzing quasicrystal densities in terms of tilings is introduced and illustrated for the icosahedral Ammann tiling. A specific application to the x-ray and neutron scattering data of i(Al_0.570Cu_0.108Li_0.322) is made. The six-dimensional hyperspace density of i(Al_0.570Cu_0.108Li_0.322) is found to be consistent with the presence of hyperatoms on vertices, edge centers, and body centers of the hypercubic lattice. Some gross features of the hyperatom shapes are suggested. In the physical space, it is shown that densities around some high symmetry points are similar to those in R(Al_0.564Cu_0.116Li_0.320) crystal, while around others, they suggest new atomic clusters. The Ammann tiling is found to be a useful template for the structure of i(Al_0.570Cu_0.108Li_0.322), with the rhombic dodecahedra as important building units. While several structural models of i(Al_0.570Cu_0.108Li_0.322) are generally consistent with the results of the density analysis, some differences are detected. A symmetric decoration of the rhombic dodecahedron, similar to the one found in R(Al_0.564Cu_0.116Li_0.320), that is a basis for several structural models, is not consistent with the density analysis. No sign of the pure Al inside the rhombic dodecahedra, nor of the related Al hyperatom at the body center of the hypercrystal, could be detected.