The Curled Up Dimension in Quasicrystals (original) (raw)
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PERIODIC SURFACES IN THE DESCRIPTION OF QUASICRYSTALS
Le Journal de Physique Colloques, 1990
To give a full account of the atomic ordering in modulated crystals or in quasiclystals amounts to find suitable periodic surfaces living in a euclidean space of dimension higher than that of the physical 3 0 space. In a first part, I summarize the main features of quasiperiodic order and show that it is naturally described by the so-called section or cut method. Next, the problem of short range order will be raised :for quasicrystals, the solution consists in building surfacetin higher D whose intersections with the physical space (imbedded in RD as a vector subspace) provide the atomic sites. As an example, the "atomic surface" of the octagonal quasiperiodic tiling is analysed in more details in the last section.
Geometric models for continuous transitions from quasicrystals to crystals
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/
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.
Physical Review Letters, 1986
It is shown that the icosahedral quasicrystal and the recently observed T phase are closely related to each other, The latter is a periodic stacking of two-dimensional quasilattices with mirror symrnetry. Their diffraction patterns, though appearing very different, can be indexed by a set of primary vectors that are only small deformations of each other. Ho~ever, because of the mirror symmetry, their quasilattices are not related by small deformations. A calculation based on the model free energy of Kalugin et ai. shows that this periodic quasicrystal is very competitive with (and in fact energetically more favorable than) the icosahedral quasicrystal.
Structural transformations in quasicrystals induced by higher dimensional lattice transitions
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.
A highly symmetric four-dimensional quasicrystal
Journal of Physics A: Mathematical and General, 1987
A quasiperiodic pattern (or quasicrystal) is constructed in real four-dimensional Euclidean space, having the noncrystallographic reflection group [3,3,5] of order 14400 as its point group. It is obtained as a projection of the eight-dimensional lattice E 8 , and has as a cross-section a three-dimensional quasicrystal with icosahedral symmetry.
Quasicrystals: tiling versus clustering
Philosophical Magazine A, 2001
A quasiperiodic covering of a plane by regular decagons is described, and an analogous structure in three dimensions is deduced. This consists of a pattern of interpenetrating congruent triacontahedral clusters, related to the ½ 3 in¯ation rule for quasiperiodic Ammann tiling patterns. The overlap regions are triacontrahedron faces, oblate hexahedra, rhombic dodecahedra and rhombic icosahedra. The structure leads to a plausible model for T2 icosahedral quasicrystalline phases.
Periodic Diffraction Patterns for 1D Quasicrystals
Acta Physica Polonica B, 2005
A simple model of 1D structure based on a Fibonacci sequence with variable atomic spacings is proposed. The model allows for observation of the continuous transition between periodic and non-periodic diffraction patterns. The diffraction patterns are calculated analytically both using ``cut and project'' and ``average unit cell'' method, taking advantage of the physical space properties of the structure.