Average unit cell for the Generalized Penrose Tiling

The Generalized Penrose Tiling (GPT) can be considered a promising alternative for Penrose Tiling (PT) as a model for decagonal quasicrystal refinement procedure, particularly in the statistical approach (also called the Average Unit Cell (AUC) approach) [1]. The statistical method using PT has been successfully applied to the structure optimization of various decagonal phases [2]. The application of the AUC concept to the GPT will be presented. In the higher dimensional (nD) approach, PT is obtained by projecting a 5D hypercubic lattice through a window consisting of four pentagons, called the atomic surfaces (ASs), in the perpendicular space. The vertices of these pentagons together with two additional points form a rhombicosahedron. The GPT is created by projecting the 5D hypercubic lattice through a window consisting of five polygons, generated by shifting the ASs along the body diagonal of the rhombicosahedron. Three of the previously pentagonal ASs will become decagon, one will remain pentagonal and one more pentagon will be created (for PT it is a single point). The structure of GPT will depend on the shift parameter, its building units are still thick and thin rhombuses, but the matching rules and the tiling changes. In the AUC concept the probability distribution for rhombuses of PT can be obtained as an oblique projection of the ASs on the physical space. This holds true also for the GPT. The derivation of the AUC distribution for a given type of rhomb in a given orientation of an arbitrarily chosen GPT will be presented. In the PT, these distributions are triangular, whereas in the case of the GPT they are triangular (originating from the pentagonal AS) or hexagonal (originating from the decagonal AS). The AUC of GPT for shift parameters 0.2 and 0.5 has been calculated. The derivation of the analytical formula for structure factor using AUC formalism, for the decorated GPT is made similarly to the calculation for the PT [3].

High temperature structural study of decagonal Al-Cu-Rh quasicrystal.

The structure of d(ecagonal)-Al-Cu-Rh has been studied as a function of temperature by in-situ single-crystal X-ray diffraction in order to contribute to the discussion on energy or entropy stabilization of quasicrystals (QC) [1]. The experiments were performed at 293 K, 1223 K, 1153 K, 1083 K, and 1013 K. A common subset of 1460 unique reflections was used for the comparative structure refinements at each temperature. The results obtained for the HT structure refinements of d-Al-Cu-Rh QC seem to contradict a pure phasonic-entropy-based stabilization mechanism [2] for this QC. The trends observed for the ln func(I(T1 )/I(T2 )) vs.|k⊥ |^2 plots indicate that the best on-average quasiperiodic order exists between 1083 K and 1153 K, however, what that actually means is unclear. It could indicate towards a small phasonic contribution to entropy, but such contribution is not seen in the structure refinements. A rough estimation of the hypothetic phason instability temperature shows that it would be kinetically inaccessible and thus the phase transition to a 12 Å low T structure (at ~800 K) is most likely not phason-driven. Except for the obvious increase in the amplitude of the thermal motion, no other significant structural changes, in particular no sources of additional phason-related configurational entropy, were found. All structures are refined to very similar R-values, which proves that the quality of the refinement at each temperature is the same. This suggests, that concerning the stability factors, some QCs could be similar to other HT complex intermetallic phases. The experimental results clearly show that at least the ~4 Å structure of d-Al-Cu-Rh is a HT phase therefore entropy plays an important role in its stabilisation mechanism lowering the free energy. However, the main source of this entropy is probably not related to phason flips, but rather to lattice vibrations, occupational disorder unrelated to phason flips like split positions along the periodic axis.

What periodicities can be found in diffraction patterns of quasicrystals?

The structure of quasicrystals is aperiodic; however, their diffraction patterns comprise periodic series of peaks, which can be used to retrieve essential features of the quasicrystalline structure.

The phase problem for quasicrystals in reciprocal space

The phase problem is very well-known in crystallography and is particularly important for structure solution of quasicrystals. Structure solution ( initial phasing of the diffraction pattern) is the first step of atomic structure determination against the diffraction data. Many tools for solving the phase problem in crystallography were developed over the years. Besides the pioneer Patherson function method or direct methods, also the low density elimination method and, more recently, maximum entropy method or charge flipping algorithm are widely used. We propose another way of phase recovery, directly from the diffraction pattern. It has been shown, that diffraction patterns of aperiodic structures (quasicrystals and modulated structures) consists of periodic series of peaks [1,2]. The peaks are, of course, aperiodically distributed in reciprocal space. However, the envelopes of such peaks are strictly periodic. For Fibonacci sequence the shape of envelopes is given by a sinx/x – type function. In the centrosymmetric case all peaks belonging to a given series have the same phase (0 or π). Moreover, once we rescale peak positions to obtain the reduced envelope of structure factor (intensity), we can easily find the shape of the average unit cell, by applying the inverse Fourier transform. The functionality of such approach has been shown in [3] for d-AlNiCo quasicrystal. In this paper we show a few examples of model structures with recovered phases. Decorated Fibonacci sequence, Penrose tiling and Ammann tiling are used as model structures for 1D, 2D, and 3D quasicrystals respectively. We compare the results with conventionally used charge flipping method.