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Star-shaped (exhibiting fivefold symmetry) cluster B (diameter 24 A ˚ ) along the periodic direction after refinement. (a) x 5 = 0, (b) 1 4 or 3 4 , (c) 1 2 and (d) projection of all layers. This cluster is better ordered than cluster A with one Mg split position on the x 5 = 0 layer. The blue lines indicate CN 16 polyhedra and gold lines and dashed lines indicate two different CN 12 polyhedra types, one ideal (at the center) and five distorted (surrounding the ideal one). Dy: blue; Zn: red; Mg: green.
Source publication
A single-crystal X-ray diffraction structure analysis of decagonal Zn–Mg–Dy, a Frank–Kasper-type quasicrystal, was performed using the higher-dimensional approach. For this first Frank–Kasper (F–K) decagonal quasicrystal studied so far, significant differences to the decagonal Al–TM-based (TM: transition metal) phases were found. A new type of twof...
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Citations
... It is claimed that due to a correlation of parameters in the refinement, calculated values are much more reliable than the observed trends in the intensities of the peaks. Similar studies of stability were performed in a d-ZnMgDy Frank-Kasper-type quasicrystal [151]. The Frank-Kasper ordering has also been recently shown to be present in the Socolar tiling known for dodecagonal quasicrystals [152]. ...
From the discovery of quasicrystals, exhibiting rotational symmetries forbidden by the classical crystallography, the idea of a quasicrystalline geometry spreads out to a great variety of scientific fields, including soft matter physics, optics, or nanotechnology. However, the atomic structure of quasicrystals is still unclear and many questions are unanswered, including the most peculiar one asked by Per Bak: ‘Where are the atoms?’ [Icosahedral crystals: where are the atoms? Phys Rev Lett. 1986;56:861–864]. To answer those questions, more detailed structural analysis needs to be done. Two approaches to the structural description and refinement are used for aperiodic crystals: the higher dimensional method, well established in the community, and the statistical method. We introduce the essence of the crystallography of quasicrystals, giving the historical background and fundamentals of the higher dimensional modelling, but focus is put on the details and application of the statistical approach. We present also a case study of the decagonal Al–Cu–{Co,Ir,Rh} phase, which is an example of the crystal structure solved using the statistical method. In the last chapter, a brief discussion of the recently developed approach of a phasonic corrective factor, alongside the effect of phonons on the atomic distribution function, is given.
... It is a very hot topic in crystallography of quasicrystals and still not a solved problem-are quasicrystals stabilized by energy or entropy? Recent high temperature diffraction measurements and other structural analysis, also for crystalline approximants, opened a new discussion on the problem with preference of entropic stabilization mechanism [50][51][52][53][54]. The origin of phason modes acquires a better understanding in superspace description. ...
In this paper, we show the fundamentals of statistical method of structure analysis. Basic concept of a method is the average unit cell, which is a probability distribution of atomic positions with respect to some reference lattices. The distribution carries complete structural information required for structure determination via diffraction experiment regardless of the inner symmetry of diffracting medium. The shape of envelope function that connects all diffraction maxima can be derived as the Fourier transform of a distribution function. Moreover, distributions are sensitive to any disorder introduced to ideal structure—phonons and phasons. The latter are particularly important in case of quasicrystals. The statistical method deals very well with phason flips and may be used to redefine phasonic Debye-Waller correction factor. The statistical approach can be also successfully applied to the peak’s profile interpretation. It will be shown that the average unit cell can be equally well applied to a description of Bragg peaks as well as other components of diffraction pattern, namely continuous and singular continuous components. Calculations performed within statistical method are equivalent to the ones from multidimensional analysis. The atomic surface, also called occupation domain, which is the basic concept behind multidimensional models, acquires physical interpretation if compared to average unit cell. The statistical method applied to diffraction analysis is now a complete theory, which deals equally well with periodic and non-periodic crystals, including quasicrystals. The method easily meets also any structural disorder.
Rod-shaped precipitates with their long axis parallel to [0001]α commonly form in Mg-Zn-Al alloys, providing strengthening effects. These precipitates are metastable and are generally accepted to be the MgZn2 and/or Mg4Zn7 phases. In this paper, the cross-sectional structure of precipitate rods in an aged ZA84 alloy is examined at the atomic scale using high-angle annular dark-field scanning transmission electron microscopy. It is found that these precipitates are neither pure MgZn2 nor Mg4Zn7. The structures of the rod-shaped precipitates do not have long-range translational symmetry nor long-range 5-fold rotational symmetry in the two-dimensional projection plane perpendicular to the rod long axis. However, these structures can be described by the tiling of different shapes, with four such shapes being most frequently observed. The structural details revealed in this work may help rationalize the many complex precipitate structures in Mg-Zn-Al and Mg-Zn alloys.
A set of X-ray data collected on a fragment of decagonite, Al 71 Ni 24 Fe 5 , the only known natural decagonal quasicrystal found in a meteorite formed at the beginning of the Solar System, allowed us to determine the first structural model for a natural quasicrystal. It is a two-layer structure with decagonal columnar clusters arranged according to the pentagonal Penrose tiling. The structural model showed peculiarities and slight differences with respect to those obtained for other synthetic decagonal quasicrystals. Interestingly, decagonite is found to exhibit low linear phason strain and a high degree of perfection despite the fact it was formed under conditions very far from those used in the laboratory.
The structure of a decagonal quasicrystal in the Zn 58 Mg 40 Y 2 (at.%) alloy was studied using electron diffraction and atomic resolution Z -contrast imaging techniques. This stable Frank–Kasper Zn–Mg–Y decagonal quasicrystal has an atomic structure which can be modeled with a rhombic/hexagonal tiling decorated with icosahedral units at each vertex. No perfect decagonal clusters were observed in the Zn–Mg–Y decagonal quasicrystal, which differs from the Zn–Mg–Dy decagonal crystal with the same space group P 10/ mmm . Y atoms occupy the center of `dented decagon' motifs consisting of three fat rhombic and two flattened hexagonal tiles. About 75% of fat rhombic tiles are arranged in groups of five forming star motifs, while the others connect with each other in a `zigzag' configuration. This decagonal quasicrystal has a composition of Zn 68.3 Mg 29.1 Y 2.6 (at.%) with a valence electron concentration ( e / a ) of about 2.03, which is in accord with the Hume–Rothery criterion for the formation of the Zn-based quasicrystal phase ( e / a = 2.0–2.15).
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.
More than 35 years and 11 000 publications after the discovery of quasicrystals by Dan Shechtman, quite a bit is known about their occurrence, formation, stability, structures and physical properties. It has also been discovered that quasiperiodic self-assembly is not restricted to intermetallics, but can take place in systems on the meso- and macroscales. However, there are some blank areas, even in the centre of the big picture. For instance, it has still not been fully clarified whether quasicrystals are just entropy-stabilized high-temperature phases or whether they can be thermodynamically stable at 0 K as well. More studies are needed for developing a generally accepted model of quasicrystal growth. The state of the art of quasicrystal research is briefly reviewed and the main as-yet unanswered questions are addressed, as well as the experimental limitations to finding answers to them. The focus of this discussion is on quasicrystal structure analysis as well as on quasicrystal stability and growth mechanisms.
Where are we now in quasicrystal (QC) research more than three decades after Dan Shechtman's discovery? Do we fully understand the origin of quasiperiodicity, the formation, growth, thermodynamic stability, structure and properties of quasicrystals? First, I will shortly present the status quo, then I will address the still open questions, and identify potential focus areas for future research. Because of the limited space, I will focus on decagonal quasicrystals (DQCs); the status quo for research on icosahedral quasicrystals (IQCs) is comparable.
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.
