Icosahedral quasicrystal decoration models. I. Geometrical principles

An atomic scale study of two-dimensional quasicrystal nucleation controlled by multiple length scale interactions

Formation of quasicrystal structures has always been mysterious since the discovery of these magic structures. In this work, the nucleation of decagonal, dodecagonal, heptagonal, and octagonal quasicrystal structures controlled by the coupling among multiple length scales is investigated using a dynamic phase-field crystal model. We observe that the nucleation of quasicrystals proceeds through local rearrangement of length scales, i.e., the generation, merging and stacking of 3-atom building blocks constructed by the length scales, and accordingly, propose a geometric model to describe the cooperation of length scales during structural transformation in quasicrystal nucleation. Essentially, such cooperation is crucial to quasicrystal formation, and controlled by the match and balance between length scales. These findings clarify the scenario and microscopic mechanism of the structural evolution during quasicrystal nucleation, and help us to understand the common rule for the formation of periodic crystal and quasicrystal structures with various symmetries.

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.

Looking for alternatives to the superspace description of icosahedral quasicrystals

A multiple-cell approach is discussed as a possible alternative to the higher dimensional crystallography of icosahedral quasicrystals. It is based on the Socolar-Steinhardt tiling combined with the quasi-unit cell model. Quasi-unit cells fill the space without gaps and overlappings similar to those in periodic crystals. Similarly, the atoms can occupy general and special positions. The alloy stoichiometry and the packing density can be calculated through the relative tile frequencies, which in turn are determined as the components of the Perron-Frobenius eigenvector of the corresponding substitution matrix. The calculation of the diffraction pattern reduces to the Perron projection of a special matrix, the entries of which reflect the contribution of each type of quasi-unit cell to the coherent scattering.

Atomic structure of icosahedral quasicrystals: stacking multiple quasi-unit cells

An effective tiling approach is proposed for the structural description of icosahedral quasicrystals based on the original substitution algorithm.

Medium range real atomic structure of face-centred icosahedral Ho9Mg26Zn65

A complementary approach to solve quasi crystalline atomic structures in 3-dimensional (3D) real space is presented: The atomic pair distribution function (PDF) of face centred icosahedral Ho9Mg26Zn65 [a(6D) = 2 × 5.18(3) Å] has been obtained from in-house powder X-ray diffraction data (MoKα 1). For the first time, full profile PDF refinements of a quasicrystal were performed: Starting with rational approximant models, derived from 1/1- and 2/1-Al—Mg—Zn, its local and medium range structure was refined (r < 27 Å; R = 12.9%) using the PDF data. 85% of all atoms show Frank-Kasper (FK) type coordinations. The basic structural unit is the 3-shell, 104-atom Bergman cluster (d ≈ 15 Å) comprising a void at its center. The clusters are interconnected sharing common edges and hexagonal faces of the 3rd shells. The remaining space is filled by some glue atoms (9% of all atoms), yielding an almost tetrahedrally close packed structure. All Ho atoms are surrounded by 16 neighbours (FK-polyhedron “P”). Most of them (89%) are situated in the 2nd shell and form a Ho8 cube (edge length 5.4 Å); they are completed by 12 Mg atoms to a pentagon dodecahedron. Cubes in neighbouring clusters are tilted with respect to each other; their superposition generates diffraction symmetry 2/m3̅5̅. The remaining Ho atoms act as glue atoms. As a result and as can be expected for real matter, local atomic coordinations in quasicrystals are similar when compared to common crystalline intermetallic compounds. From our results, the long range quasiperiodic structure of icosahedral Mg—Zn—RE (RE = Y and some rare earths) is anticipated to be a canonical cell tiling (CCT, after Henely) decorated with Bergman clusters.

Atomic structure of the binary icosahedral Yb-Cd quasicrystal

Icosahedral quasicrystals (i-QCs) are long-range ordered solids that show non-crystallographic symmetries such as five-fold rotations. Their detailed atomic structures are still far from completely understood, because most stable i-QCs form as ternary alloys suffering from chemical disorder. Here, we present the first detailed structure solution of i-YbCd(5.7), one of the very few stable binary i-QCs, by means of X-ray structure determination. Three building units with unique atomic decorations arrange quasiperiodically and fill the space. These also serve as building units in the periodic approximant crystals. The structure is not only chemically feasible, but also provides a seamless structural understanding of the i-YbCd(5.7) phase and its series of related i-QCs and approximant crystals, revealing hierarchic features that are of considerable physical interest.