The physical space model of the Tsai-type quasicrystal.

The binary Cd5.7Yb phase representing the Tsai-type category of the icosahedral quasicrystals is solved by the assignment of a unique atomic decoration to rhombohedral units in the Ammann-Kramer-Neri tiling. The unique decoration is found for units with an edge length of 24.1 Šand 3m internal point symmetry. The structural refinement was carried out for two underlying tilings generated by the projection method for 6D space. The difference lies in the location of the origin point which for one tiling is in the vertex and for the second one in the center of the 6D unit cell. The two tilings exhibit mutual duality. The choice of the tiling has a minor effect on the final structural model as both converge to an R factor of ∼11.5%. The main difference is related to the treatment of the Cd4 tetrahedral motif which is either orientationally ordered and aligned with the threefold axis or disordered and modeled as a partially occupied icosahedron. Both models can be presented as a covering by rhombic triacontahedral clusters with identical positions of clusters within rhombohedral units. The shell structure is Tsai-type in the case of the first tiling and Bergman-type for the other.

The decagonal AlCuRh quasicrystal modelled with five atomic surfaces - high-temperature X-ray diffraction data analysis.

Five datasets of high-temperature X-ray diffraction performed upon the decagonal phase of AlCuRh are used to derive the temperature-related structural changes. Two sets of atomic structure refinements are conducted, with four and five atomic surfaces, respectively. The fifth atomic surface emerges as a consequence either of the transition to a tiling with different local isomorphism than the Penrose tiling or of the structure being phason disordered. The correlation of the occupancy of the fifth atomic surface with the thermal dependence of the lattice constants is 80%. In addition, the structural refinement with five atomic surfaces allowed a better agreement with the experimental data to be obtained, compared with the model with four atomic surfaces.

Approximant-based orientation determination of quasicrystals using electron backscatter diffraction.

Orientation mapping of quasicrystalline materials is demonstrated using crystalline approximant structures in the technique of electron backscatter diffraction (EBSD). The approximant-based orientations are symmetrised according to the rotational point group of the quasicrystal, including the visualization of orientation maps using proper colour keys for quasicrystal symmetries. Alternatively, approximant-based orientation data can also be treated using pseudosymmetry post-processing options in the EBSD system software, which enables basic grain size estimations. Approximant-based orientation analyses are demonstrated for icosahedral and decagonal quasicrystals.

A new modulated crystal structure of the ANS complex of the St John's wort Hyp-1 protein with 36 protein molecules in the asymmetric unit of the supercell.

Superstructure modulation, with violation of the strict short-range periodic order of consecutive crystal unit cells, is well known in small-molecule crystallography but is rarely reported for macromolecular crystals. To date, one modulated macromolecular crystal structure has been successfully determined and refined for a pathogenesis-related class 10 protein from Hypericum perforatum (Hyp-1) crystallized as a complex with 8-anilinonaphthalene-1-sulfonate (ANS) [Sliwiak et al. (2015), Acta Cryst. D71, 829-843]. The commensurate modulation in that case was interpreted in a supercell with sevenfold expansion along c. When crystallized in the additional presence of melatonin, the Hyp-1-ANS complex formed crystals with a different pattern of structure modulation, in which the supercell shows a ninefold expansion of c, manifested in the diffraction pattern by a wave of reflection-intensity modulation with crests at l = 9n and l = 9n ± 4. Despite complicated tetartohedral twinning, the structure has been successfully determined and refined to 2.3 Šresolution using a description in a ninefold-expanded supercell, with 36 independent Hyp-1 chains and 156 ANS ligands populating the three internal (95 ligands) and five interstitial (61 ligands) binding sites. The commensurate superstructures and ligand-binding sites of the two crystal structures are compared, with a discussion of the effect of melatonin on the co-crystallization process.

The atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann-Kramer-Neri tiling.

In this study, the atomic structure of the ternary icosahedral ZnMgTm quasicrystal (QC) is investigated by means of single-crystal X-ray diffraction. The structure is found to be a member of the Bergman QC family, frequently found in Zn-Mg-rare-earth systems. The ab initio structure solution was obtained by the use of the Superflip software. The infinite structure model was founded on the atomic decoration of two golden rhombohedra, with an edge length of 21.7 Å, constituting the Ammann-Kramer-Neri tiling. The refined structure converged well with the experimental diffraction diagram, with the crystallographic R factor equal to 9.8%. The Bergman clusters were found to be bonded by four possible linkages. Only two linkages, b and c, are detected in approximant crystals and are employed to model the icosahedral QCs in the cluster approach known for the CdYb Tsai-type QC. Additional short b and a linkages are found in this study. Short interatomic distances are not generated by those linkages due to the systematic absence of atoms and the formation of split atomic positions. The presence of four linkages allows the structure to be pictured as a complete covering by rhombic triacontahedral clusters and consequently there is no need to define the interstitial part of the structure (i.e. that outside the cluster). The 6D embedding of the solved structure is discussed for the final verification of the model.

Phason-flips refinement of and multiple-scattering correction for the d-AlCuRh quasicrystal.

The origin of the characteristic bias observed in a logarithmic plot of the calculated and measured intensities of diffraction peaks for quasicrystals has not yet been established. Structure refinement requires the inclusion of weak reflections; however, no structural model can properly describe their intensities. For this reason, detailed information about the atomic structure is not available. In this article, a possible cause for the characteristic bias, namely the lattice phason flip, is investigated. The derivation of the structure factor for a tiling with inherent phason flips is given and is tested for the AlCuRh decagonal quasicrystal. Although an improvement of the model is reported, the bias remains. A simple correction term involving a redistribution of the intensities of the peaks was tested, and successfully removed the bias from the diffraction data. This new correction is purely empirical and only mimics the effect of multiple scattering. A comprehensive study of multiple scattering requires detailed knowledge of the diffraction experiment geometry.

Pushing the limits of crystallography.

A very serious concern of scientists dealing with crystal structure refinement, including theoretical research, pertains to the characteristic bias in calculated versus measured diffraction intensities, observed particularly in the weak reflection regime. This bias is here attributed to corrective factors for phonons and, even more distinctly, phasons, and credible proof supporting this assumption is given. The lack of a consistent theory of phasons in quasicrystals significantly contributes to this characteristic bias. It is shown that the most commonly used exponential Debye-Waller factor for phasons fails in the case of quasicrystals, and a novel method of calculating the correction factor within a statistical approach is proposed. The results obtained for model quasiperiodic systems show that phasonic perturbations can be successfully described and refinement fits of high quality are achievable. The standard Debye-Waller factor for phonons works equally well for periodic and quasiperiodic crystals, and it is only in the last steps of a refinement that different correction functions need to be applied to improve the fit quality.

Structure factor for an icosahedral quasicrystal within a statistical approach.

This paper describes a detailed derivation of a structural model for an icosahedral quasicrystal based on a primitive icosahedral tiling (three-dimensional Penrose tiling) within a statistical approach. The average unit cell concept, where all calculations are performed in three-dimensional physical space, is used as an alternative to higher-dimensional analysis. Comprehensive analytical derivation of the structure factor for a primitive icosahedral lattice with monoatomic decoration (atoms placed in the nodes of the lattice only) presents in detail the idea of the statistical approach to icosahedral quasicrystal structure modelling and confirms its full agreement with the higher-dimensional description. The arbitrary decoration scheme is also discussed. The complete structure-factor formula for arbitrarily decorated icosahedral tiling is derived and its correctness is proved. This paper shows in detail the concept of a statistical approach applied to the problem of icosahedral quasicrystal modelling.

Generalized Penrose tiling as a quasilattice for decagonal quasicrystal structure analysis.

The generalized Penrose tiling is, in fact, an infinite set of decagonal tilings. It is constructed with the same rhombs (thick and thin) as the conventional Penrose tiling, but its long-range order depends on the so-called shift parameter (s ∈ 〈0; 1)). The structure factor is derived for the arbitrarily decorated generalized Penrose tiling within the average unit cell approach. The final formula works in physical space only and is directly dependent on the s parameter. It allows one to straightforwardly change the long-range order of the refined structure just by changing the s parameter and keeping the tile decoration unchanged. This gives a great advantage over the higher-dimensional method, where every change of the tiling (change in the s parameter) requires the structure model to be built from scratch, i.e. the fine division of the atomic surfaces has to be redone.

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

The structure of decagonal 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. The experiments were performed at 293, 1223, 1153, 1083 and 1013 K. A common subset of 1460 unique reflections was used for the comparative structure refinements at each temperature. A comparison of the high-temperature datasets suggests that the best quasiperiodic ordering should exist between 1083 and 1153 K. However, neither the refined structures nor the phasonic displacement parameter vary significantly with temperature. This indicates that the phasonic contribution to entropy does not seem to play a major role in the stability of this decagonal phase in contrast to other kinds of structural disorder, which suggests that, in this respect, this decagonal phase would be similar to other complex intermetallic high-temperature phases.

What periodicities can be found in diffraction patterns of quasicrystals?

The structure of quasicrystals is aperiodic. Their diffraction patterns, however, can be considered periodic. They are composed solely of series of peaks which exhibit a fully periodic arrangement in reciprocal space. Furthermore, the peak intensities in each series define the so-called `envelope function'. A Fourier transform of the envelope function gives an average unit cell, whose definition is based on the statistical distribution of atomic coordinates in physical space. If such a distribution is lifted to higher-dimensional space, it becomes the so-called atomic surface - the most fundamental feature of higher-dimensional analysis.

Comparative structural study of decagonal quasicrystals in the systems Al-Cu-Me (Me = Co, Rh, Ir).

A comparative single-crystal X-ray diffraction structure analysis of the family of Al-Cu-Me (Me = Co, Rh and Ir) decagonal quasicrystals is presented. In contrast to decagonal Al-Cu-Co, the other two decagonal phases do not show any structured disorder diffuse scattering indicating a higher degree of order. Furthermore, the atomic sites of Rh and Ir can be clearly identified, while Cu and Co cannot be distinguished because of their too similar atomic scattering factors. The structure models, derived from charge-flipping/low-density elimination results, were refined within the tiling-decoration method but also discussed in the five-dimensional embedding approach. The basic structural building units of the closely related structures are decagonal clusters with 33 Å diameter, which are consistent with the available electron-microscopic images. The refined structure models agree very well with the experimental data.

Structure factor for decorated Penrose tiling in physical space.

The structure factor for an arbitrarily decorated Penrose tiling has been calculated in the average unit cell description. The obtained formula uses only the physical coordinates of the atoms decorating a structure. The final equation can be easily extended so that it can describe the other physical properties of a structure. Its usefulness is demonstrated by its use in the Al-Ni-Co alloy structure-refinement process.

Statistical approach in cluster analysis of two-dimensional quasicrystals.

An analytical formula for the structure factor of Penrose tiling in the cluster approach was derived and tested. Probability distributions obtained for each Penrose position allow the number of different atoms that can decorate the cluster to be found. Calculations were performed in the average-unit-cell approach for Gummelt's cluster of 33 atoms, divided into three independent groups of atoms, and a kite cluster of 17 atoms, divided into seven independent groups.

Average unit cell for Penrose tiling and its Gaussian approximation.

In this paper, the average unit cell for a quasicrystal is constructed by a statistical approach. For the Penrose tiling, it is shown that such a unit cell is fully equivalent to the oblique projection of the atomic surface onto physical space. The obtained statistical distributions can be easily extended to imperfect structures by using a Gaussian approximation. This leads to simple analytical expressions for diffraction intensities, which can be very useful in structure refinement.