Simple tiling model and phason kinetics for decagonal Al-Cu-Co (original) (raw)

Tilings, coverings, clusters and quasicrystals

2000

A quasiperiodic covering of the plane by regular decagons and an analogous structure in three dimensions are described. The 3D pattern consists of interpenetrating triacontahedral clusters, related to the ttmathrn3tt^{\mathrn{3}}ttmathrn3 inflation rule for the 3D Penrose tiling patterns. The overlap regions are triacontahedron faces, rhombic dodecahedra and rhombic icosahedra, The structure leads to a plausible model for the T2 icosahedral quasicrystalline phases.

The Equivalence Between Unit-Cell Twinning and Tiling in Icosahedral Quasicrystals

Scientific Reports

It is shown that tiling in icosahedral quasicrystals can also be properly described by cyclic twinning at the unit cell level. The twinning operation is applied on the primitive prolate golden rhombohedra, which can be considered a result of a distorted face-centered cubic parent structure. The shape of the rhombohedra is determined by an exact space filling, resembling the forbidden five-fold rotational symmetry. Stacking of clusters, formed around multiply twinned rhombic hexecontahedra, keeps the rhombohedra of adjacent clusters in discrete relationships. Thus periodicities, interrelated as members of a Fibonacci series, are formed. The intergrown twins form no obvious twin boundaries and fill the space in combination with the oblate golden rhombohedra, formed between clusters in contact. Simulated diffraction patterns of the multiply twinned rhombohedra and the Fourier transform of an extended model structure are in full accord with the experimental diffraction patterns and can be indexed by means of three-dimensional crystallography. The alternative approach is fully compatible to the rather complicated descriptions in a hyper-space. Ever since quasicrystals (QCs) were first reported 1 they attracted great interest, because they apparently contradicted some basic concepts of crystallography 2-4. Contrary to fully disordered solids and perfectly grown single crystals, characterized by their rotational and translational symmetries, QCs with their forbidden five-fold rotational symmetry and with the apparently lost translational order represented something in between the two categories. However, some of their properties contradict this distinction. Their shapes can be well developed and the corresponding diffraction patterns (DPs) show exceptionally sharp reflections, without any diffuse scattering, characteristic of short-range order, modulation, or any other deviation from an ideal crystalline structure. Pauling was convinced that none of the existing crystallographic rules was violated in the newly discovered materials 5-10. He believed these crystals were composed of twinned cubic domains with huge unit cells, whose basic building elements were composed of one Mn atom linked to twelve Al atoms. Although the existing experiments seemingly supported his model, he after all run into problems. Another major problem with Pauling's approach was, that no twin boundaries were ever detected in QCs 11. Contrary to Pauling, a number of researchers 12-18 considered QCs an exception to the known solid state structures, which required a novel approach. Their explanation was based on the so-called Amman tiling 19 , the three-dimensional equivalent of the two-dimensional Penrose tiling. Likewise to two Penrose rhombic tiles filling a plane, their three-dimensional equivalents, the prolate and the oblate golden rhombohedra, will fill the space and form the QC structure. It is shown in the present work that tiling in the icosahedral QC structure can also be properly explained by cyclic unit cell twinning 20,21 , applied on primitive golden rhombohedra, forming thus intergrown twins without explicit twin-boundaries. Results Procedure. Instead of describing a QC structure in a hyper-space, the present description is based on twinning of the basic building units. Dependent on the sizes and the composition of the constituent atoms a hypothetical parent face-centered cubic structure of an alloy may collapse into a primitive rhombohedral one along four equivalent directions. If the resulting rhombohedral angle is close to 63.43°, i.e. the angle of a prolate golden rhombohedron, it will lock

Quantum dynamics in high codimension tilings: From quasiperiodicity to disorder

Physical Review B, 2003

We analyze the spreading of wavepackets in two-dimensional quasiperiodic and random tilings as a function of their codimension, i.e. of their topological complexity. In the quasiperiodic case, we show that the diffusion exponent that characterizes the propagation decreases when the codimension increases and goes to 1/2 in the high codimension limit. By constrast, the exponent for the random tilings is independent of their codimension and also equals 1/2. This shows that, in high codimension, the quasiperiodicity is irrelevant and that the topological disorder leads in every case, to a diffusive regime, at least in the time scale investigated here. PACS numbers: 61.44.Br, 71.23.Ft It is now well established that quasiperiodic order has a strong influence on the quantum dynamics of wavepackets. Indeed, the nature of the eigenstates in quasiperiodic systems, which are neither spatially extended (as in periodic systems) nor localized (as in disordered systems) but critical, is often responsible for a sub-ballistic motion. Although most of the studies about this anomalous diffusion concern one-dimensional systems such as the Fibonacci or the Harper chain, there has also been a great interest for the, more physical, higher-dimensional ones . However, the parameters that determine the characteristics of the long time dynamics, such as the diffusion exponent β, remains misunderstood.

Tile Hamiltonian for decagonal AlCoCu derived from first principles

Physical Review B, 2003

A tile Hamiltonian ͑TH͒ replaces the actual atomic interactions in a quasicrystal with effective interactions, between and within tiles. We studied Al-Co-Cu decagonal quasicrystals described as decorated hexagon-boatstar ͑HBS͒ tiles using ab initio methods. The dominant term in the TH counts the number of H, B, and S tiles. The phason flips that replace an HS pair with a BB pair lower the energy. In Penrose tilings, quasiperiodicity is forced by arrow matching rules on the tile edges. The edge arrow orientation in our model of AlCoCu is due to Co/Cu chemical ordering. The tile edges meet in vertices with 72°or 144°angles. We find strong interactions between edge orientations at 72°vertices that force a type of matching rule. Interactions at 144°vertices are somewhat weaker.

Imaging quasiperiodic electronic states in a synthetic Penrose tiling

Nature communications, 2017

Quasicrystals possess long-range order but lack the translational symmetry of crystalline solids. In solid state physics, periodicity is one of the fundamental properties that prescribes the electronic band structure in crystals. In the absence of periodicity and the presence of quasicrystalline order, the ways that electronic states change remain a mystery. Scanning tunnelling microscopy and atomic manipulation can be used to assemble a two-dimensional quasicrystalline structure mapped upon the Penrose tiling. Here, carbon monoxide molecules are arranged on the surface of Cu(111) one at a time to form the potential landscape that mimics the ionic potential of atoms in natural materials by constraining the electrons in the two-dimensional surface state of Cu(111). The real-space images reveal the presence of the quasiperiodic order in the electronic wave functions and the Fourier analysis of our results links the energy of the resonant states to the local vertex structure of the qua...

Equilibrium quasicrystal phase of a Penrose tiling model

Physical Review B, 1990

A two-dimensional rhombus tiling model with a matching-rule-based energy is analyzed using real-space renormalization-group methods and Monte Carlo simulations. The model spans a range from T=O quasiperiodic crystal (Penrose tiling) to a random-tiling quasicrystal at high temperatures. A heuristic picture for the disordering of the ground-state quasiperiodicity at low temperatures is proposed and corroborated with exact and renormalization-group calculations of the phason elastic energy, which shows a linear dependence on the strain at T=O but changes to a quadratic behavior at T&0 and sufficiently small strain. This is further supported by the Monte Carlo result that phason fluctuations diverge logarithmically with system size for all T &0, which indicates the presence of quasi-long-range translational order in the system, meaning algebraically decaying correlations. A close connection between the rhombus tiling model and the general surface-roughening phenomena is established. Extension of the results to three dimensions and their possible implication to experimental systems is also addressed.

Tile Hamiltonian for Decagonal AlCoCu

2002

A tile Hamiltonian (TH) replaces the actual atomic interactions in a quasicrystal with effective interactions between and within tiles. We studied Al-Co-Cu decagonal quasicrystals described as decorated Hexagon-Boat-Star (HBS) tiles using ab-initio methods. The dominant term in the TH counts the number of H, B and S tiles. Phason flips that replace an HS pair with a BB pair lower the energy. In Penrose tilings, quasiperiodicity is forced by arrow matching rules on tile edges. The edge arrow orientation in our model of AlCoCu is due to Co/Cu chemical ordering. Tile edges meet in vertices with 72 • or 144 • angles. We find strong interactions between edge orientations at 72 • vertices that force a type of matching rule. Interactions at 144 • vertices are somewhat weaker.

A new method to generate quasicrystalline structures : examples in 2D tilings

Journal De Physique, 1990

2014 Nous présentons un nouvel algorithme pour la génération des structures quasicristallines. Il est relié à la méthode de coupe et projection, mais il permet une génération directement dans l'espace « physique » E de la structure. La sélection des sites dans l'espace orthogonal est remplacée par un test directement dans une grille de domaines d'acceptance dans l'espace E. Cette méthode montre qu'il y a une sorte de réseau cristallin sous-jacent au quasicristal. Nous illustrons la construction dans le cas 4D-2D avec les symétries d'ordre 5, 8, 10 et 12 qui sont obtenues par projection de 4D à 2D. Par la même méthode d'autres types de quasicristaux avec une symétrie plus basse, ayant un réseau moyen, sont construits. Nous présentons un exemple de symétrie 4. Les points de ce quasi-cristal sont un sous-ensemble des points du quasi-cristal ayant la symétrie complète d'ordre 8.

Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices Bn

Acta crystallographica. Section A, Foundations and advances, 2015

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A3), W(H2) × W(A1) and W(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B4 onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B5 lattice is used to describe both fivefold and tenfold symmetries. The lattice B6 can describe aperiodic tilings with 12-fold symmetry as well as a...