Methods for Calculating Empires in Quasicrystals

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.

Starting from variable p-veetors half-stars whíeh verify Hadwiger's theorem, the cut-projeetion method is used here. The strip ofprojeetion is projeeted on a rotatory subspaee and a variable tiling is obtained. Two out standing examples are developed. The first, a eontinuous evolution from a two-dímensional octagonal quasilattiee to two square lattiees 45° rotated in between. The seeond is a eontinuous evolution from a three-dimensional Penrose tiling to an f.e.e. vertex lattiee. Physieal applieations to quasierystal-<:rystal transitions are poínted out. After quasicrystalline phases were discovered (Shechtman, Blech, Gratias and Cahn 1984), some theoretical (El ser and Henley 1985, Kramer 1987) and experimental works (Guyot and Audier 1985, Urban, Moser and Kronmüller 1985, Audier and Guyot 1986a, b, Guyot, Audier and Lequette 1986) began to pay attention to the close and systematic relationship between quasicrystals and crystals. RecentIy, many works have pointed in the same direction (Poon, Dmowski, Egami, Shen and Shiflet 1987, Zhou, Li, Ye and Kuo 1987, Yamamoto and Hiraga 1988, Zhang, Wang and Kuo 1988, Sadananda, Singh and Imam 1988, Yu-Zhang, Bigot, Chevalier, Gratias, Martin and Portier 1988, Fitz Gerald, Withers, Stewart and Calka 1988, Yang 1988, Henley 1988, Chandra and Suryanarayana 1988, Cahn, Gratias and Mozer 1988). Some authors even state that the transition from quasicrystalline to crystalline phases is continuous over a range ofintermediate phases (Reyes-Gasga, Avalos-Borja and José-Yacamán 1988, Zhou, Ye, Li and Kuo 1988). We present here a geometric model to describe simple and plausible continuous evolutions from quasilattices to lattices. Our method is a version ofthe well known cutprojection method (Kramer and Neri 1984, Duneau and Katz 1985, EIser 1986). In the above mentioned work, EIser and Henley (1985) modified the cut-projection method to allow study of the connection between crystal and quasicrystal structures. These authors tilted the strip of projection with respect to the hypercubic lattice (defined in the hyperspace EP) but they fixed the projection hyperplane (or projection subspace P, p> n). So, different hypercubic roofs were projected in such a way that the quasicrystal structure was the limit of a discontinuous sequence of periodic structures. In this work, we develop the contrary strategy and we describe a lattice as an atrophical quasilattice. We fix the particular strip (in the p-dimensional hypercubic lattice of EP) which generates the standard quasiperiodic tiling but we rotate the projection hyperplane (or 0950--0839/

Acta Crystallographica Section A Foundations of Crystallography, 1994

Aperiodic crystalline structures, commonly called quasicrystals, display a great variety of combinatorially possible local configurations. The local configurations of first order are the vertex configurations. This paper investigates, catalogues and classifies in detail the latter in the following important two-dimensional cases: the Penrose tiling, the decagonal triangle tiling and some twelvefold tilings, including the patterns of Stampfli, GS.hler, Niizeki and Socolar, as well as the square-triangle and the shield patterns. The main result is a comprehensive study of the three-dimensional primitive icosahedral tiling in its random version. All its 10 527 combinatorially possible noncongruent vertex configurations are constructed, coded, listed and classified. Methods for coding and representation of local configurations by formulae and diagrams, in particular those of Schlegel, are discussed. The paper also describes the algorithm used to generate them. The formal classification of local configurations by the characteristic integers rank, degree and order is also discussed. appearing in CBED patterns are given for all the (3 + 1)-dimensional space groups of the incommensurately modulated crystal.

Journal De Physique, 1989

2014 Au moyen de trois tuiles, nous construisons un pavage quasipériodique du plan, que nous relions au quasicristal octogonal. Ainsi, nous montrons que les coordonnées des n0153uds peuvent être obtenues de deux manières différentes. Le facteur de structure est calculé exactement. Ce pavage qui possède « presque » une symétrie d'ordre huit, soulève la difficulté de la détermination pratique de la symétrie d'un quasicristal. Finalement, nous montrons comment construire une large classe de pavage du type de l'octogonal, à partir de ce nouveau pavage.

Acta Crystallographica Section A Foundations of Crystallography, 1991

Conventionally, Penrose tilings with fivefold symmetry are constructed with the aid of two characteristic rhombic tiles and sets of rules based on either matching of markings on the tiles or their subdivision. Both these procedures involve decision making when tiling is to be done extensively. In the present communication, a foolproof method of producing Penrose tilings using a set of operations that can be repeated ad infinitum is described. The steps in the present procedure are akin to conventional crystallographic operations and can be expressed in simple mathematical terms which bring out some interesting aspects of Penrose tilings.

Physical review, 1990

Inspired by phason dynamics in tiling models of quasicrystals, we investigate a class of constrained Ising models. Phason shifts in the Penrose tiling model of quasicrystals appear as flips of rows of tiles, known as "worms. " When worms cross one another, a hierarchy is established in which some of the worms cannot flip until others have. A complex set of constraints on worm flips is thereby introduced by the intricate pattern of worm crossings in quasicrystalline tilings. We introduce a simple model of interacting sets of one-dimensional Ising chains that mimics this set of constraints and study the possible consequences of these constraints for phason dynamics and the relaxation of phason strain in 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 $tt^{\mathrn{3}}$ 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.

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

This article presents a novel Hamiltonian architecture based on vertex types and empires for demonstrating the emergence of aperiodic order (quasicrystal growth) in one dimension by a suitable prescription for breaking translation symmetry. At the outset, the paper presents different algorithmic, geometrical, and algebraic methods of constructing empires of vertex configurations of a given quasi-lattice. These empires have non-local scope and form the building blocks of the new lattice model. This model is tested via Monte Carlo simulations beginning with randomly arranged N tiles. The simulations clearly establish the Fibonacci configuration, which is a one dimensional quasicrystal of length N , as the final relaxed state of the system. The Hamiltonian is promoted to a matrix operator form by performing dyadic tensor products of pairs of interacting empire vectors followed by a summation over all permissible configurations. A spectral analysis of the Hamiltonian matrix is performed and a theoretical method is presented to find the exact solution of the attractor configuration that is given by the Fibonacci chain as predicted by the simulations. Finally, a precise theoretical explanation is provided which shows that the Fibonacci chain is the most probable ground state. The proposed Hamiltonian is a one dimensional model of quasicrystal growth.