Comparing different approaches to model the atomic structure of a ternary decagonal quasicrystal

Structure determinations for random-tiling quasicrystals

How, in principle, could one solve the atomic structure of a quasicrystal, modeled as a random tiling decorated by atoms, and what techniques are available to do it? One path is to solve the phase problem first, obtaining the density in a higher dimensional space which yields the averaged scattering density in 3-dimensional space by the usual construction of an incommensurate cut. A novel direct method for this is summarized and applied to an i(AlPdMn) data set. This averaged density falls short of a true structure determination (which would reveal the typical unaveraged atomic patterns.) We discuss the problematic validity of inferring an ideal structure by simply factoring out a "perp-space" Debye-Waller factor, and we test this using simulations of rhombohedral tilings. A second, "unified" path is to relate the measured and modeled intensities directliy, by adjusting parameters in a simulation to optimize the fit. This approach is well suited for unifying structural information from diffraction and from minimizing total energies derived ultimately from ab-initio calculations. Finally, we discuss the special pitfalls of fitting random-tiling decagonal phases.

Decagon covering model and equivalent HBS-tiling model

For decagonal Al—Co—Ni quasicrystals, coverings based on a single decagonally shaped columnar 20 Å-cluster as well as tilings of so-called hexagon-, boat- and star-tiles are used to model the atomic structure. We introduce geometrically defined local transformation rules which enable us to replace a covering of overlapping congruent decagons by an equivalent HBS-tiling at the same scale and vice versa. Starting from an obvious one-to-one correspondence in case of idealized, perfect Penrose order, we show that there is a very similar relation also for random decagon coverings with slightly relaxed structure. The proofs given here complete our statements announced in previous work.

Quasi-unit-cell model for an Al-Ni-Co ideal quasicrystal based on clusters with broken tenfold symmetry

We present new evidence supporting the quasi-unit-cell description of the Al72Ni20Co8 decagonal quasicrystal which shows that the solid is composed of repeating, overlapping decagonal cluster columns with broken tenfold symmetry. We propose an atomic model which gives a significantly improved fit to electron microscopy experiments compared to a previous proposal by us and to alternative proposals with tenfold symmetric clusters.