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Classification of Local Configurations in Quasicrystals

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Classification of Local Configurations in Quasicrystals 553 RESEARCH PAPERS Acta Cryst. (1994). AS0, 553-566 Classification of Local Configurations in Quasicrystals BY M. BAAKE Institut ffi'r Theoretische Physik, Universitiit Tiibingen, Auf der Morgenstelle 14, D-72076 Tiibingen, Germany S. I. BEN-ABRAHAM Department of Physics, Ben-Gurion University, POB 653, IL-84105 Beer-Sheba, Israel AND R. KLITZING, P. KRAMERAND M. SCHLOTrMAN Institut ffir Theoretische Physik, Universit~it Tiibingen, Auf der Morgenstelte 14, D-72076 Tiibingen, Germany (Received 23 September 1993; accepted 7 February 1994) Abstract promptly and for very good reasons dubbed quasicrystals Aperiodic crystalline structures, commonly called quasi- (Levine & Steinhardt, 1984). crystals, display a great variety of combinatorially possi- Since then, a plethora of quasicrystalline systems has ble local configurations. The local configurations of first been found (Jari6, 1990; Nissen & Beeli, 1990; DiVi- order are the vertex configurations. This paper investi- cenzo & Steinhardt, 1991; Bancel, 1991). The micro- gates, catalogues and classifies in detail the latter in the scopic structure of these phases has ever since been a following important two-dimensional cases: the Penrose subject of intensive and extensive research, both exper- tiling, the decagonal triangle tiling and some twelve- imental and theoretical. Various structural models have fold tilings, including the patterns of Stampfli, GS.hler, been suggested in different instances and by different Niizeki and Socolar, as well as the square-triangle and authors. The models fall roughly into three classes: (1) the shield patterns. The main result is a comprehensive the quasiperiodic tiling model (Kramer & Neff, 1984; study of the three-dimensional primitive icosahedral Levine & Steinhardt, 1984; Duneau & Katz, 1985; tiling in its random version. All its 10 527 combinato- DiVicenzo & Steinhardt, 1991; Katz & Duneau, 1986; rially possible noncongruent vertex configurations are Socolar & Steinhardt, 1986); (2) the orientationally constructed, coded, listed and classified. Methods for ordered polyhedral glass model (Jari6, 1990; Nissen coding and representation of local configurations by & Beeli, 1990), commonly and imprecisely called the formulae and diagrams, in particular those of Schlegel, 'icosahedral glass' model; (3) the random tiling model are discussed. The paper also describes the algorithm (Elser, 1985; Widom, Strandburg & Swendsen, 1987; used to generate them. The formal classification of local Jari6, 1990; Henley, 1990, 1991; Nissen & Beeli, 1990; configurations by the characteristic integers rank, degree Tang, 1990; Bancel, 1991; DiVicenzo & Steinhardt, and order is also discussed. 1991). It seems that phases with the primitive icosahedral structure, represented by A1-Cu-Li, may be adequately described by the random tiling model, whereas the face- 1. Introduction centred icosahedral phases, represented by A1-Cu-Fe, Local configurations in random tilings should be of are apparently fairly good realizations of a quasiperiodic interest to quasicrystallographers, as well as to crystallo- tiling. Important evidence for the relevance of quasiperi- graphers and quasicrystallographers. In this Introduction, odic tiling models for the octagonal, decagonal and do- we try to give some convincing arguments in favour of decagonal phases comes from high-resolution transmis- this statement and we attempt to support them by the sion electron microscopy (Nissen & Beeli, 1990, 1993; content of this paper. Hiraga, 1991), X-ray diffraction (Steurer & Kuo, 1990) Almost a decade ago, Shechtman, Blech, Gratias & and scanning tunnelling microscopy (Kortan, Becker, Calm (1984) and almost simultaneously Ishimasa, Nissen Thiel &Chen, 1990; Becker & Kortan, 1991). & Fukano (1984) discovered, recognized and announced The wealth of known quasicrystalline phases and the existence of novel aperiodic crystalline phases in those yet to be discovered opened up a new and im- the A1-Mn and Cr-Ni systems, respectively. These were mensely rich world of aperiodic ordered solids. We © 1994 International Union of Crystallography Acta Crystallographica Section A Printed in Great Britain - all fights reserved ISSN 0108-7673 © 1994 554 CLASSIFICATION OF LOCAL CONFIGURATIONS IN QUASICRYSTALS must expect to find an extremely wide, intricate and partition of a natural number, the count s, into a set fairly dense spectrum of ordering in this realm. One of of given natural numbers S = {sl,..., sk}, sometimes its unifying features is the established physical reality subject to certain obvious constraints. The numbers si of symmetries forbidden by the dogma of classical are the internal angles of the tiles in units of w, so crystallography in spite of past and present attempts to they are convenient labels for the possible corners. It explain away experimental evidence (Hardy & Silcock, is expedient to list the numbers si in descending order, 1955/56; Sastry, Suryanarayana, Van Sande & Van Ten- that is, to have si >_ sj if i < j. deloo, 1978; Pauling, 1985). In order to understand these In our case, w = 7r/5, s = 10, k = 4, and S = aperiodic phases, we usually surmise the existence of 14,3,2,1]. First, we perform the partition without order- some ideal underlying structure and try to investigate ing; thus we get k-tuples of integers tr = {al,..., ak }, the deviations therefrom. This is one avenue leading to satisfying the study of defective local configurations. k From a quite different but by no means contradictory s = ~ ~risi, a~ e No. point of view, we assume only the existence of a few i=1 building blocks and try to assemble them in every The vertices are classified by the k-tuples a into classes possible way. Thus, we are again led to the study of of the same summary composition, ni I ... %o- k , a term all combinatorially possible local configurations. In this borrowed from chemistry. In the present instance, we approach, there are no defective configurations. Rather, have every possible local configuration is part of the global []=2 × × []+ 1 × N+0 × m structure. Since there is no obvious limitation to the symmetry of the building blocks, the resulting structure = ... etc., is expected to inherit at least some features of this whence we get the composition formulae 422, ... etc. symmetry. For the study of aperiodic ordered structures, Sometimes it is more expedient to write out the k-tuples the investigation, cataloguing and classification of local o" themselves, {2,0,1,0 I, ... etc. configurations should be of interest. Next, we permute the elements within the composition In § 2, we explain our intentions using as example the classes with repetition and without a fixed starting point. well known Penrose tiling. §§ 3 and 4 deal with instances Thus, we gain all isomers of the same composition. The of planar tilings, in particular the decagonal triangle result is a complete list of all vertex configurations of tiling and some twelvefold tilings. [The octagonal case the given tiling. For the random rhombic Penrose tiling, is left for diligent readers as an easy exercise. They it is shown in Fig. 1 and listed in Table 1. may check their result for one interesting example, the Each configuration has a multiplicity, i.e. it may occur Ammann-Beenker tiling, with the published literature within the pattern in a number of different orientations, (Baake & Joseph, 1990).] § 5 is the core of this paper. determined by the fundamental symmetry of the tiling. It is devoted to the three-dimensional primitive icosa- Some of these orientations may be indistinguishable. The hedral tiling. § 6 clarifies our methods of coding and vertex list includes the symmetry and multiplicity of representation of local configurations while § 7 explains the vertices (cfi Table 1). In particular, some vertices the algorithm used to generate them. § 8 deals with the have mirror symmetry, while others do not. The latter, formal classification of local configurations. Some minor in chemical parlance, show enantiomorphy, i.e. have technical topics are deferred to the Appendix to avoid different left and fight enantiomers. In our example, we distraction. have, up to rotations, 75 vertex configurations. Among these, there are 21 enantiomorphic pairs. Thus, ignoring rotations and reflections, we are left with 54 configura- 2. The random rhombic Penrose tiling tions. We start with the random rhombic Penrose tiling. This Out of these vertex configurations, only 16 occur in well known example will serve as a warm-up exercise, the class of perfect generalized Penrose tilings although as well as a paradigm of things to come. never do all 16 appear in one single tiling. Only a The tiles are the two bare (i.e. undecorated) Penrose single one of these, the 'cocktail', has different enan- rhombi with equal sides a: a large (or thick or fat) kind L tiomers. We are naturally led to classify all possible with acute angle 27r/5 and a small (or thin or skinny) kind configurations with respect to their deviation from the S with acute angle 7r[5. We construct all combinatorially perfect tiling. This is done by means of two characteristic possible local configurations of order 1, in other words, integers: degree and rank (Ben-Abraham, Baake, Kramer all vertex configurations. That amounts to partitioning & Schlottmann, 1993). The degree g is the maximum the full plane angle around a node into integer multiples dimension of a facet shared by such subconfiguratious of an angle unit w, determined by the point symmetry that cannot occur in the ideal tiling.
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