The physical space model of the Tsai-type quasicrystal.

The binary CdYb 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 Å and 3m internal point symmetry.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.