Coverings of Discrete Quasiperiodic Sets (Springer Tracts in Modern Physics)

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Author: Peter Kramer | Zorka Papadopolos

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Springer Tracts in Modern Physics Volume 18o M a n a g i n g Editor: G. H6hler, K a r l s r u h e Editors: H. F u k u y a m a , Kashiwa J. K i i h n , K a r l s r u h e Th. Miiller, K a r l s r u h e A. Ruckenstein, New Jersey E Steiner, U l m J. Triimper, G a r c h i n g P. W61fle, K a r l s r u h e H o n o r a r y Editor: E. A. Niekisch, J/ilich

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Peter Kramer ZorkaPapadopolos(Eds.)

Coverings of Discrete Quasiperiodic Sets Theoryand Applicationsto Quasicrystals With 128 Figures, Including 5 in Color

~

Springer

Professor Peter Kramer

Dr. Zorka Papadopolos Universit/it TObingen Institut ffir Theoretische Physik Auf der Morgenstelle 14 72076 Tfibingen, G e r m a n y E-mail: [email protected] zorka.pap a d o p o l o s @ u n i - t u e b i n g e n . d e

congress Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek - CIP-Einheitsaufnahme Coverings of discrete quasiperiodic sets : theory and applications to quasicrystals / Peter Kramer; Zorka Papadopolos (ed.). - Berlin ; Heidelberg ; New York ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2003 (Springer tracts in modern physics ; Vol. 18o) (Physics and astronomy online library) ISBN 3-540-43241-8

Physics and Astronomy 68.37.Lp

Classification

Scheme

(PACS): 61.44.Br, 68.35.Bs, 68.37.Ef,

M a t h e m a t i c a l S u b j e c t C l a s s i f i c a t i o n ( 2 0 0 0 ) : 52C23, 0 5 B 4 o

I S S N p r i n t edition: o o 8 1 - 3 8 6 9 I S S N e l e c t r o n i c e d i t i o n : 1615-o43o ISBN 3-540-43241-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg ~oo3 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by Da-TEX, Gerd Blumenstein, Leipzig Cover design: design &production GmbH, Heidelberg Printed on acid-free paper

SPIN:10842438

56/3141/YL 5 4 3 21 o

i/i I,VIIIIlllllA

Fig. Two rhombic polyhedra from Johannes Kepler, Harmonices Mundi V, Linz 1619 anticipate concepts in crystals and quasicrystals. The rhombic dodecahedron on the left is the Voronoi or Wigner-Seitz cell from the cubic F-lattice. It tiles 3space E 3 periodically and forms a fundamental domain for this crystal lattice. The rhombic triacontahedron on the right is the icosahedral projection of the Voronoi cell from the hypercubic F- or D6-1attice in E 6. It covers E ~ quasiperiodically but incompletely. It forms a fundamental domain compatible with a quasiperiodic icosahedral tiling (T, D6), used as a model for quasicrystals

Preface

This book deals with the covering of discrete point sets in Euclidean space E ~ by congruent, overlapping polytopes. The polytopes in the covering are called the covering clusters. The discrete point sets analyzed are quasiperiodic. They originate from points in positions of high symmetry in an n-dimensional lattice A E E n. Subsets of these points, bounded by windows, are projected onto a subspace E "~ embedded irrationally into E n. These point sets have the Delone property, characterized by two finite positive constants (r, R), r 3. Conway and Sloane, in [6] pp. 11–12, list references for applications of sphere packings in geometry and number theory, in digital communication, in chemistry and physics, in numerical approximations, and in superstring theory in mathematical physics. Tilings are, in a sense, optimal packings, leaving no space between the bodies. Their applications range from practical tilings or tessellations of walls and areas of ground, through structure determination in crystallography [5] and the physics of crystalline matter, to aperiodic tilings [14] and to the mathematical analysis of topological manifolds [40] and their applications in cosmology [37]. In many applications, a local motif is uniquely related to a body or geometric object. The geometric arrangement then generates a pattern with this motif. In covering, one allows the overlap of the geometric objects. Any point is still covered by at least one geometric object. Therefore local motifs attached to geometric objects again generate a pattern. We turn to a more precise description of these notions. Consider Euclidean space E m and a set S of fixed, compact, convex, geometric, m-dimensional objects K1 , K2 , . . . in it. We shall also refer to these objects as “clusters”. For the moment, assume that all these objects are congruent and of m-dimensional volume |K1 | = |K2 | = . . . =: |K| as in the case of polytopes or spheres. In this case S can be described by a corresponding set of Euclidean operations, composed of relative positions and relative orientations. In packing problems we consider the objects Ki as impenetrable, so that in the set S, any pair Ki , Kj , i = j, shares at most points from the boundaries and so (Ki ∩ Kj ) = ∅. Now take a finite subset Sp ∈ S, formed by selecting p objects. We can form the convex hull conv (Sp ) [56] and call Sp ∈ S a packing of conv (Sp ). We follow [6] and quantify this finite packing by the ratio ∆ between the volumes of the set Sp ∈ S, |Sp | = p|K|, and its convex hull, i.e. ∆=

p|K| . |conv (Sp )|

(1.1)

This ratio always obeys ∆ ≤ 1 and is called the density of the packing. Efficient packings should have large values of this ratio. Conway and Sloane [6] define in addition the center density, P. Kramer (Ed.): Convering of Discrete Quasiperiodic Sets, STMP 180, 1–21 (2002) c Springer-Verlag Berlin Heidelberg 2002

2

Peter Kramer

δ = ∆/|K| =

p . |conv (Sp )|

(1.2)

Wills in [56] gives a brief introduction to finite sphere packings. Infinite packings arise if one considers packings in balls of radius R in E m and takes the limit R → ∞. For applications of sphere packings to crystals and quasicrystals, compare [46]. In sphere packings, the objects Ki are congruent spheres. In most sphere packings, the midpoints of the spheres are periodically distributed on all the points of a lattice. Sphere packings in various dimensions and for various lattices have been studied in great detail by Conway and Sloane [6]. The values of ∆ for the densest sphere packings on lattices, as a function of the dimension m ≤ 48, are given in [6] p. 14. In tiling problems [14], one sharpens the condition for a packing by demanding that there is no m-dimensional volume uncovered by the set of objects Ki , now called the tiles, which implies that ∆ = 1 in (1.1). For more general tilings, one admits a finite number of classes of mutually congruent tiles. Periodic tilings arise naturally from lattices, with tiles formed by the unit cells of the lattice. In covering problems, one relaxes the condition of impenetrability and admits p overlapping objects Ki in a finite or infinite part Sp of En . From tilings, one retains the conditionthat there should be no points uncovered by the set Sp of p objects Ki ∈ Sp Ki . A covering can again be quantified by the ratio Θ of the total volume |Sp | = p|K| of the objects to the volume | Sp Ki | covered by the objects, i.e. p|K| . Θ= | Sp K i |

(1.3)

This quantity Θ, where Θ ≥ 1, or its limit for infinite arrangements, is, in analogy to (1.1) called the thickness of the covering. In [6] p. 37 the thicknesses of sphere coverings on lattices are given as a function of the dimension m ≤ 24. In Table 1.1, we give the values of the thickness for a planar periodic covering and three quasiperiodic coverings. The spheres of the covering for the lattice A2 have the circumradius of the Voronoi hexagons shown in Fig. 1.2 ([6] p. 32). A simple computation yields the thickness (1.3), given in the first row of Table 1.1. The quasiperiodic module A4 with 5-fold point symmetry is the projection of the root lattice A4 . A derivation of the thickness of the triangle tiling is given in Chap. 4. So far, the packings and, in particular, coverings organize compact point sets in E n . Given a discrete point set on E m , one may also ask if its points belong to a certain packing, tiling, or covering. If, moreover, these given points by themselves form a tiling, one can ask if a covering contains complete tiles or only parts of tiles. These questions arise in the theory of shelling and also in the covering of quasiperiodic point sets, to be discussed in later chapters. In all three cases considered, one encounters the following notions:

1 Covering of Discrete Quasiperiodic Sets: Concepts and Theory

3

Table 1.1. Thickness for three planar coverings Tiling

module

cluster(s)

thickness Θ

Hexagonal

A2

1 sphere

√ 2π/(3 3)

= 1.209 . . .

1

Penrose

A4

1 decagon

5/(2τ − 1)

= 2.236 . . .

4

Triangle

A4

2 pentagons

5/(τ + 2)

= 1.382 . . .

4

A4

2 pentagons

2/τ

= 1.236 . . .

5

C

s

contribution

(ptc1): Convex geometric objects or clusters in a finite number of congruence classes under Euclidean transformations, which form the geometric objects of the packing, tiling or covering. Similar problems may be formulated in non-Euclidean spaces such as hyperbolic space ([39], [6] p. 29). (ptc2): Local conditions which restrict the pairwise positions of the objects for packing, tiling, or covering. (ptc3): Arrangements of these geometric objects in E m . If the center positions of equal spheres are restricted to an infinite lattice, one has an infinite periodic arrangement. In this case the quantification of packings and coverings can be derived from the unit cells of such lattices. At the other extreme one might think of nonlattice or random arrangements for packings ([6] pp. 7–8), tilings, and coverings. In Fig. 1.1 we illustrate the notions of sphere packings, tilings, and coverings. Periodic arrangements of geometric objects form a particular field of interest. A lattice Λ ∈ E m can be viewed in two ways. (i) It determines a discrete, countable infinite subset of points q ∈ E m . (ii) The differences q − q0 with respect to a fixed lattice point q0 form a commutative discrete translation group, which we denote again by Λ. Taken as a transformation group acting on q0 ∈ E m , Λ produces all the lattice points. But we can let this transformation group act on any other point x ∈ E m . This action decomposes E m into orbits under Λ. A lattice Λ will, in many cases, admit symmetries other than translations. In particular, it can have symmetry under point transformations, reflections, or rotations which preserve a fixed point. The maximal set of point symmetry transformations with respect to a fixed lattice point q ∈ Λ is called the holohedry of Λ. The translation group, combined with reflections or rotations which preserve fixed points, forms a space group of the kind studied in the crystallography of m dimensions [5, 47]. From a lattice Λ ∈ E m and the Euclidean metric, one can construct a number of geometric objects which form tools for the packing, tiling, and covering problems associated with Λ. Of particular importance are the root lattices [16, 6], which appear in any dimension m. Around each point q of a lattice Λ ∈ E m , its Voronoi polytope V (q) is defined as the set of points x ∈ E m that are at least as close to q as to any other lattice point q , i.e.

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Peter Kramer

P acking

T iling

Covering

Fig. 1.1. Sphere packing, hexagon tiling and dodecagon covering of the root lattice A2

V (q) = x| |x − q| ≤ |x − q |, x ∈ E m , q = q , q, q ∈ Λ .

(1.4)

By constructing the Voronoi polytopes centered at all lattice points ([6] pp. 33–35 and 449–475), one obtains the Voronoi (cell) complex. The Voronoi complex determines a Λ-periodic Voronoi tiling of E m . In Fig. 1.2 we illustrate the Voronoi and Delone complexes for the root lattice A2 . Comparing Fig. 1.2 and Fig. 1.1, one may note that the radius in a packing of spheres on a lattice is bounded by the inradius of its Voronoi polytopes. Similarly, the radius for the thinnest covering by spheres on a lattice is bounded by the circumradius of the Voronoi polytopes ([6] p. 32). Voronoi polytopes may also be constructed on more general discrete point sets. Under the geometric group action of q ∈ Λ, x, x ∈ E m , (q, x) → x = x + q, a fundamental domain is a subset F ∈ E m which has exactly one point from each translational orbit. This notion may be extended to elementary functional analysis on E m . The geometric group action yields for functions f on E m the group operators Tq : f (x) → (Tq f )(x) := f (x − q). Suppose now that f is periodic on E m modulo Λ. Then its domain of definition, which determines all its values on E m , can be identified as a fundamental domain under Λ. On a lattice, the Voronoi polytopes are all congruent. Each one forms a fundamental domain. The shape of the fundamental domain is by no means unique; there are alternatives, for example the usual unit cell or an appropriate combination of Delone polytopes as defined below. In Fig. 1.2, each Voronoi hexagon and also any pair of Delone triangles with a black and a white center form a fundamental domain of A2 . The boundaries X ∈ V (q) of dimension p, called p-boundaries, are shared by the Voronoi polytopes for a finite sets of lattice points. A second construction of geometric objects obtained from a lattice are the Delone polytopes ([6] pp. 35–36). The Delone polytopes are centered at the vertices h of the Voronoi polytopes. They are defined as the convex hull of the lattice points q whose Voronoi polytopes V (q ) share the vertex h. Since these vertices fall into finitely many distinct translational orbits h = a, b, c, . . . under the action of Λ, there are in general several distinct orbits of Delone polytopes


5

Fig. 1.2. For the root lattice A2 , the Voronoi complex (upper part) consists of one translational orbit of hexagons centered at lattice points (black squares). The dual Delone complex (lower part) has two translational orbits of triangular Delone cells, centered at two orbits of holes (black and white circles). Lattice points and holes are marked only if they belong to the ten selected Delone cells

Da , Db , Dc , . . .. The Delone polytopes form another Λ-periodic (cell) complex. They form a periodic tiling with finitely many classes of congruent tiles. The p-boundaries X ∗ of the Delone cell complex have a local dual relation to (m − p)-boundaries X of the Voronoi complex. Tilings on lattices such as the Voronoi and Delone complexes play an important part in n-dimensional crystallography [5, 47]. Their use in sphere packings and coverings is studied in [6]. As an example of their use, note that in Fig. 1.1 the spheres of the packing on the left-hand side have the inradius of the hexagonal Voronoi hexagons. In crystallography and physics

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Peter Kramer

the Voronoi construction is known as the Wigner–Seitz cell of the lattice. The fundamental-domain property implies for a crystal that, given a finite set of atomic positions in the Voronoi cell, the atomic positions are fixed everywhere by translations. So the atomic positions in the Voronoi polytope play the role of a motif for the entire crystal. In the present book we shall study the theory and applications of infinite aperiodic systems, and their tilings and coverings. These aperiodic arrangements lack the periodicity of lattices but differ from random arrangements in their long-range order. We shall see that aperiodic tilings and coverings are characterized by additional local conditions. For the quasiperiodic sets found in quasicrystals, the geometric notions of lattices and crystallography reappear in n-dimensional versions, where n > 3, and with new interpretations.

1.2 Aperiodic and Quasiperiodic Systems with Long-Range Order The discovery of quasicrystals in 1984 has stimulated the theory of discrete aperiodic and quasiperiodic systems. Aperiodic systems lack the periodicity found in crystals. For various mathematical aspects of these systems, we refer to [19, 42]. Lagarias [37] characterizes the simplest aperiodic point sets by the following properties, shared with periodic crystals: (ap1): (ap2): (ap3): (ap4):

volume-bounded number of inequivalent patches pure point diffraction linear repetitivity of patches self-similarity.

Lagarias works with Delone sets. A Delone set is any infinite discrete set in E m for which there are positive constants (r, R) such that each ball of radius r contains at most one point, and each ball of radius R contains at least one point of the set. Almost- and quasi-periodicity have a long history in mathematics and mathematical physics. Quasiperiodic systems were characterized by Bohr [4] in 1925 in terms of their diffraction properties (ap2). Consider, for example, point scatterers in E m and take their Fourier integral transform as a function on a space E (m,R) . The label R stands for reciprocal space in the terminology of crystallography. In the case of a quasiperiodic pure point spectrum, the Fourier integral transform is carried by an integral and countable linear combinations of a minimal finite set of r > m vectors in the space E (m,R) . The set of vectors is called a basis of a Z-module M R of rank r. The condition r < ∞ distinguishes the Fourier transforms of quasiperiodic systems from those of almost-periodic functions. In a quasiperiodic tiling, the reference points of the tiles belong to another Z-module M of rank r, whose reciprocal module is M R . These relations can be made clearer through an argument due to Bohr [4] for quasiperiodic functions. The space E m =: E may be considered


7

as the irrational section of a larger space E n = E ⊕ E⊥ , E⊥ ⊥ E , such that there is a lattice Λ ∈ E n whose projection on E m is the module M = Λ . Conversely, we call Λ the lattice lifted from the module M . Consider now a Λ-periodic function f in E n . The Fourier coefficients of f are associated with the points of the lattice ΛR reciprocal to Λ. The lattice ΛR is part of a second Fourier space E (n,R) . Its lattice points carry the Fourier transforms of Λ-periodic functions in E n . The restriction f |E m of f to the irrational section E m is a quasiperiodic function. By implementing the restriction f |E m in the Fourier integrals, one finds that its Fourier amplitudes may be attached (n,R) (m,R) = E . When these Fourier coefficients to the points M R := ΛR ∈ E

are lifted to their preimages on ΛR ∈ E (n,R) , they determine the Fourier series for a Λ-periodic function on E n . Quasicrystals are quasiperiodic systems and hence are characterized by modules (M, M R ). By lifting these modules into lattices (Λ, ΛR ), one arrives at an n-dimensional crystallography for quasicrystals [18, 19]. Examples of these modules and lattices will be given in the present chapter and in Chap. 4. A distinction will arise between quasiperiodic systems of points and of tiles in tilings, and also between functions associated with them. In the case of tilings, the position space is organized by the tiles with the purpose of finding a correspondence between the atomic structures on equal tiles shifted only by translations. The implications for functional analysis of a position space with a tiling, and a notion of compatibility, will be illuminated in Sect. 1.3 for the Fibonacci case. In the description of quasicrystals as quasiperiodic systems in the sense of Bohr [4], point symmetries and their representations play a dominant role [18]. These symmetries and representations select the irrational subspace which becomes the configuration space of the quasicrystal. In fact, the observation of point group symmetries in diffraction patterns of 5-fold, 12-fold, and icosahedral type, all of them forbidden in 3D crystallography, was the starting point for the discovery of long-range quasiperiodic order in quasicrystals. When the module used for the description of a quasicrystal is lifted from the configuration space E m into a lattice Λ ∈ E n , its forbidden point group H is lifted into a proper point symmetry group of Λ ∈ E n . The point symmetry group of the quasicrystal forms a subgroup H ∈ G of the holohedry G of Λ. Moreover, the n-dimensional representation of H in E n must transform E m and also its orthogonal complement into itself and so must decompose into two orthogonal representations. In typical quasicrystals, the representation of H of dimension m is irreducible. Therefore the configuration space E m of a quasicrystal can be characterized as an irreducible irrational subspace under H, and the lattice Λ, with holohedry G, must admit H < G such that these conditions on the representations are fulfilled. For quasicrystals with 5-fold symmetry, the holohedry of the lattice Λ = A4 is G = S5 , and H = I2 (5) is a Coxeter group in the notation of Humphreys [16]. For icosahedral quasicrystals with the lattice Λ = D6 , H = H3 is the icosahedral Coxeter

8

Peter Kramer

group [16]. The triacontahedron with icosahedral symmetry, found by Kepler in 1611, was recognized by Kowalewski [24] in 1938 as the icosahedral projection of the 6-dimensional hypercube. Today it plays an important part in the description of the structure of icosahedral quasicrystals [25, 29, 30]. The lattice Λ ∈ E n and the geometry of its dual Voronoi and Delone cell complexes play a fundamental role in the theory of quasiperiodic canonical tilings of E . This is illustrated in Sects. 1.3 and 1.4 and described in more detail in Chap. 4. The connection of these cell complexes to quasiperiodicity is made as follows. Parallel and perpendicular projections of dual boundaries of dimensions (m, n−m) on to two complementary orthogonal, irrational subspaces, again of dimensions (m, n − m), determine in E n a Λ-periodic klotz polytope construction. The parallel sections of this klotz polytope construction yield two dual quasiperiodic tilings (T , Λ), (T ∗ , Λ). Their tiles are the projections X, dim(X) = m and X ∗ , dim(X ∗ ) = n − m of m- and (n − m)boundaries, respectively. An example of these constructions for the root lattice A4 is studied in [2]. The dual projections will be described in more detail in Chap. 4. The ideas of packing, tiling, and covering will be reconsidered in this book and applied to aperiodic and quasiperiodic systems. In the next two sections we introduce and illustrate various notions about the quasiperiodic Fibonacci and Penrose tilings and demonstrate, in these examples, a covering construction. We anticipate several notions which are treated with more rigor and detail in Chap. 4.

1.3 The Quasiperiodic Fibonacci Tiling and its Covering by Delone Clusters 1.3.1 Fibonacci Tiling and Klotz Construction Consider the Fibonacci tiling constructed from the square lattice Λ = Z × Z of edge length s in E 2 by duality [26]. Its Voronoi cells V (q) are squares centered on all lattice points q. Its dual Delone cells D (see Sect. 4.2) are similar squares centered at all vertices of the Voronoi cells. All Delone cells belong to a single translational orbit. A 2D fundamental domain F in E 2 under the action of Λ is provided by a single Voronoi cell V or, equivalently, by a single Delone cell D. The 1-boundaries P of a Voronoi cell are its four edge lines. The dual 1-boundaries X ∗ of a Delone cell are its four edge lines. Pairs X, X ∗ of dual 1-boundaries intersect in midpoints of the edges of the squares. Define the decomposition E 2 = E + E⊥ in the following fashion: x runs, with respect to a densest lattice plane of Λ, along lines of irrational √ slope τ −1 and τ = (1 + 5)/2, respectively. The klotz construction [28] for the Fibonacci tiling [26] arises as follows. For each intersecting dual pair X, X ∗ of 1-boundaries, form at its intersection point the convex klotz cells X⊥ ⊕ X∗ . The two klotz cells (A, B) (see Fig. 1.3) are two squares (A, B)


9

√ of edge length |L| = τ |S| and |S| = s/ τ + 2, respectively, with boundaries perpendicular or parallel to E . Any line with points x = x + c⊥ , −∞ < x < ∞ intersects the klotz construction in a Fibonacci tiling T ∗ with the tiles (L := A , S := B ). Owing to the periodicity in E 2 , all different intersections can be described within a so-called window (compare Chap. 4). A window for the full local isomorphism class of all Fibonacci tilings T ∗ may be taken as a perpendicular interval of length |L| + |S| centered on a lattice point q. This interval is the perpendicular projection V⊥ (q) of the Voronoi cell and appears in the klotz construction at all positions of the lattice points q. 1.3.2 Alternative Fundamental Domains Consider elementary functional analysis on E 2 for a periodic function. A natural fundamental domain of the square lattice for a periodic function would be one of its Voronoi polygons, a unit square. We can find an alternative adapted to an irrational decomposition as follows. Fundamental domain of the square lattice: Two klotz cells (A, B) form a fundamental domain (F , Z × Z) for functions f on E 2 periodic modulo Λ = Z × Z. Proof : The pairs of dual boundaries underlying the cells (A, B) are representatives of different translation orbits under Λ. The cells do not overlap and together have the same volume as the Voronoi square. 1.3.3 Quasiperiodic functions on a parallel line section of E 2 Consider a function f , defined by its values on the two cells (A, B) (or on any other equivalent fundamental domain) (F , Z × Z), and repeated periodically on E 2 modulo Λ. The restriction of f to its values on a line x = x + c⊥ , −∞ < x < ∞ gives rise to a quasiperiodic function on this line. The domain of definition of a quasiperiodic function whose value is determined everywhere on the 1D horizontal line is seen in the embedding space E 2 , as a 2D fundamental domain with respect to the action of Λ. Quasiperiodic functions of this general type do not take the same values on different passages of the line through A or through B, and so they are not compatible with the Fibonacci tiling T ∗ . The class of quasiperiodic functions f compatible with the tiling T ∗ on the line E must have the following restricted property, as discussed for example in [26]: on each of the two chosen klotz cells (A, B), its values must be independent of x⊥ . These values by repetition under Λ, generate on any parallel line section a quasiperiodic function which takes the same values on each passage through A or B. We refer to Arnold [1] for a discussion of quasiperiodic functions along similar lines.

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Peter Kramer

Fig. 1.3. The square lattice Λ of edge length s in E 2 has Voronoi squares V (q), centered on lattice points q (full squares), and Delone squares D, centered on holes (open circles). The lattice admits a periodic tiling into two new squares (A, B) of edge lengths |L| = τ |S|, |S|, called klotz cells, shown on the left-hand side. The edges of these squares run along directions x (horizontal ), and x⊥ (vertical ), of slopes τ −1 , τ with respect to a densest lattice line. A pair (A, B) of two such squares forms a fundamental domain F of the lattice. The intersection of a parallel line with the two squares (A, B) forms a Fibonacci tiling T ∗ with tiles L = A , S = B . The window of the tiling is V⊥ (q), centered on lattice points q. Its size is indicated by a vertical bar on the left-hand side. The Delone projections D on position space E centered on Voronoi vertices (heavy lines with open circles) provide fundamental domains for functions compatible with the Fibonacci tiling. They bound pairs A∪B and B∪A from below and above. On parallel line sections they give rise to D-clusters (LS) or (SL). A second periodic tiling in E 2 with two rectangles (A , B ) is shown on the lower right-hand side (dashed lines). Its intersection with a horizontal line x = x + c⊥ , −∞ < x < ∞, yields a deflated Fibonacci tiling τ −1 T ∗ with tiles (L , S ) of lengths scaled by the factor τ −1 . The union of the two tilings is shown in the middle part. In the parallel subtiling from this union, any cluster (LS), (SL) of T ∗ has the symmetric subdivision (L S L ), and consecutive clusters are disjoint or are linked by a tile L

1.3.4 Fundamental Domain Compatible with a Tiling A domain of definition of a quasiperiodic function f on the line E compatible with the tiling T ∗ will be denoted by F (T ∗ , Λ), and called a fundamental domain for the tiling T ∗ , the lattice Λ, and the projection E 2 = E + E⊥ .


11

An object with this property could be considered as a generalization of the unit cell in crystals. For the Fibonacci tiling, we find that the fundamental domain F (T ∗ , Λ), i.e. the domain of definition of a function f compatible with the Fibonacci tiling T ∗ , can be taken as a line interval in E of length |L|+|S|, consisting of a short and a long interval of the tiling T ∗ . The values of f on the intervals are then extended on each klotz cell (A, B) to a 2D function independent of x⊥ . By repetition modulo Λ and intersection with a parallel line, they give rise to a particular quasiperiodic function. 1.3.5 Linked Delone Clusters The parallel projections D of the Delone squares are line sections of length |L|+|S| (compare Fig. 1.3). These projections appear in the klotz tiling at the Delone centers. Each one separates a pair (B, A) on top from a pair (A, B) of klotz cells at the bottom. For uniqueness, we assign the boundary line itself to the top pair of klotz cells. If a horizontal intersection line passes through the top or bottom pair, any one of the two cuts provides a fundamental domain F (T ∗ , Λ). Both the (SL) and the (LS) combination are termed Delone D-clusters. 1.3.6 Delone Covering of the Fibonacci Tiling The Fibonacci tiling (T ∗ , (L, S)) is equivalent to a chain of linked D-clusters of type (LS), (SL). These clusters cover the tiling. They are compatible with the tiling, since they respect the tiling both in their interior asymmetric composition and in their overlap. Each one is equivalent to the parallel projection D of a Delone cell and forms a fundamental domain F (T ∗ , Λ). Consecutive clusters are disjoint or are linked by a tile S in the form (L(S)L). For the proof, compare Fig. 1.3. In the tiling T ∗ , form disjoint clusters from all consecutive strings (LS) except for the string (LSLLS). This string is interpreted with three clusters, as (L(S)L)(LS), with the first two clusters linked by the tile S.

1.4 Decagonal Voronoi Clusters and Covering of the Penrose Tiling The best-known paradigm for a discrete quasiperiodic system is the quasiperiodic rhombus tiling due to Penrose (1974) [44]. It results from the root lattice Λ = A4 ∈ E 4 , as the tiling (T , A4 ) [2]. Its two rhombus tiles are the projections of the 2-dimensional boundaries, or 2-boundaries, of the Voronoi complex. Part of this tiling is shown in Fig. 1.4. We emphasize that, seen in terms of the geometry of the lattice A4 and its projection, not all pairs

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Fig. 1.4. Part of the quasiperiodic Penrose rhombus tiling (T , A4 ). Dotted lines indicate the thick rhombus tiles of the τ -inflated tiling. Projected lattice points on these thick rhombus tiles are marked by black squares

of opposite vertices of a rhombus tile are on the same footing, compare the analysis of the dual triangle tiling given in Chap. 4. The dual tiling (T ∗ , A4 ) is the triangle tiling discussed in detail in Chap. 4. We first summarize the construction of the decagon covering due to Gummelt [15]. It uses the Penrose tiling T with rhombus edge length s, its vertex configurations, and its inflation. The vertex configurations of the Penrose tiling around a fixed point fall into eight types. Among them is the “king”, shown in Fig. 1.5. An analysis of the tiling around a king vertex configuration shows that the king enforces a decagon of edge length s. The Penrose rhombus tiling (T , A4 ) admits operations of τ -inflation and τ -deflation. These operations transform a Penrose rhombus tiling of edge length s into Penrose tilings of edge length τ s, τ −1 s which we denote by (τ T , A4 ), (τ −1 T , A4 ), respectively. Consider the deflation sequence of tilings (τ T , A4 ) → (T , A4 ) → (τ −1 T , A4 ) and start with a thick rhombus. On it we mark a point by a full square (compare Fig. 1.5). The first deflation yields, in T , a vertex configuration called “jack”. The next deflation yields, in Tτ −1 , a vertex configuration called “king”. This king enforces in (τ −1 T , A4 ) the “cartwheel” decagon of edge length s. The marked point is maintained in the three steps. It follows from this sequence of deflations that the decagon centers are fixed at the marked points on all the thick rhombus tiles of Tτ . The filled decagons have a mirror symmetry. Otherwise they break the local symmetry given by the outer decagonal shape. The dual tiling theory applied to the Penrose tiling [2] yields the following results [32, 33]. The decagons can be interpreted as Voronoi clusters, or V -clusters, taken as parallel projections of Voronoi polytopes of the root lat-


13

Fig. 1.5. τ -Deflation of a thick rhombus tile (dotted lines) of the Penrose tiling leads in two steps to the jack (full lines) and to the king (broken lines). The king enforces the cartwheel decagons

Fig. 1.6. Decagonal V -clusters covering the Penrose tiling shown in Fig. 1.4. The centers of the decagons are located on the thick rhombus tiles of the τ -inflated Penrose tiling. Their centers are marked by black squares. Mirror pairs form a fundamental domain F(T , A4 )

tice A4 . Their centers are projections of lattice points. They appear in 10 orientations, cover the tiles of the Penrose tiling, and have the thickness [32] given in Table 1.1. The covering is illustrated in Fig. 1.6 on the same patch as chosen in Fig. 1.4. We can now take up the discussion of a fundamental domain previously given for the Fibonacci tiling. A fundamental domain F (T , A4 ) for functions compatible with the Penrose tiling should consist of thick and thin rhombus

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tiles, each in ten possible orientations. Comparing now the V -clusters, one finds that a mirror pair of filled asymmetric decagonal V -clusters does provide such a fundamental domain. With this qualification, one can verify the claim of Steinhardt [51] that the decagons form quasi-unit cells. The links between V -clusters are clearly related to the sharing of (parallelprojected) dual boundaries. The window technique allows us to characterize the links and their relative frequencies [33] and to compare them with the results of [15]. The linkage of pairs of decagons is illustrated in Fig. 1.7.

a

d

e

c

Fig. 1.7. The four possible linkages (a, d, e, c) according to Gummelt [15], upper row, for pairs of decagonal V -clusters in terms of the thick rhombus tiles of the τ -inflated Penrose tiling, lower row. The thick-rhombus vertex on the edge of a decagon is marked by a black circle

1.5 Coverings of Aperiodic and Quasiperiodic Sets In this section, we briefly summarize a variety of concepts and results in this field. Some of these topics will be taken up in more detail in the following chapters. 1.5.1 Covering and Cluster Density in 2D Systems The decagon covering of the Penrose tiling was introduced in 1996 by Gummelt [15]. The decagons considered by Gummelt have an internal structure which breaks the 5-fold symmetry and yields geometric matching rules for overlapping clusters (see also Fig. 1.7). Steinhardt and collaborators, beginning in 1996 [48, 49], were the first to develop the decagon covering as a tool for the description of atomic positions


15

in decagonal quasicrystals. Once the atomic positions are fixed locally on the decagon, the matching rules of the decagon are expected to produce the global quasiperiodic structure. Steinhardt et al. and G¨ ahler [10, 11, 12] have introduced a principle of maximum density for the clusters. The overlap rules enforce local configurations with a minimum distance and definite orientations between decagons. Given a general covering obeying these local rules for the clusters, the claim is that quasiperiodicity of the covering will be enforced by a maximum-density principle. This theory and its applications are presented in Chap. 3 by G¨ ahler et al. It seems at first sight that a principle of maximum density is in conflict with the principle of minimum thickness Θ of a covering, as described by (1.3) in Sect. 1.1. To resolve this conflict, note that one measures the geometric overlap of clusters by the thickness. In contrast, matching rules respect the internal structure of the clusters and from it select specific distances and relative orientations. 1.5.2 Shelling of Quasicrystals In the shelling theory of periodic lattices, one computes the number of lattice points covered by spherical shells of increasing radius around a fixed lattice point. For example, in a hypercubic lattice Z n the number of lattice points on a shell of integer radius n R = m > 0 is given by the solutions of the Diophantine equation 1 x2i = m, x1 , . . . , xn ∈ Z n . This problem belongs to number theory and can be analyzed by means of the theta series of a lattice (compare [6] pp. 44–47). Moody and Weiss (1994) [41] consider the quasiperiodic variant of shelling. They use methods from number theory and the theta series. For quasicrystals derived from the root lattice E8 ∈ E 8 by projection onto the irrational subspace irreducible under the point group H4 ∈ E 4 , they develop an algorithm for counting the number of projected lattice points. This analysis has been extended by Weiss [55] to quasicrystals projected from the root lattice D6 ∈ E 6 to the subspace irreducible under the icosahedral point group H3 ∈ E 3 . Although the aim of the theory of shelling is not a covering, there is a clear correspondence which may be used for coverings of discrete periodic and quasiperiodic point sets by spheres. 1.5.3 Covering of Atomic Positions in Icosahedral Quasicrystals Atomic positions in a quasiperiodic tiling form a discrete quasiperiodic set. Given such a set, for example the vertex set of the Penrose tiling, one can ask about clusters of maximal size such that all point positions are fixed in a skeleton with respect to the cluster center. This problem was analyzed in 1996 by

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Duneau [7, 8]. In a similar spirit, Gratias and collaborators [13] have analyzed mainly icosahedral quasicrystals. The latter authors start out from a complete set of atomic positions in an icosahedral quasicrystal. These positions arise as projections from points of highest local symmetry, called Wyckoff positions, in the embedding 6-dimensional lattice. The structural quasicrystal models contain atomic clusters of the Bergman and Mackay types. These authors then explore maximal shells around typical positions formed by atoms in fixed mutual positions, which can serve as covering clusters. An analysis of this kind is presented in Chap. 2 by Duneau and Gratias. 1.5.4 Fundamental Domains and Unit Cells for Quasiperiodic Tilings The decagon covering of the Penrose tiling led Steinhardt et al. [48, 49, 51] to introduce the notion of a quasi-unit cell. The idea is to prescribe the local atomic positions on a single decagon and also the global structure by the quasiperiodic matching of the decagons. This approach is developed in full detail and compared with experiment in [50]. For a comment on the history of the subject, see Urban [54]. An application to atomic positions in octagonal quasicystals is given by Ben-Abraham and G¨ ahler [3]. The notion of a fundamental domain under translations for a quasiperiodic tiling can be formulated in terms of the tiling and its module without reference to clusters [26, 31, 34, 35]. Given this notion and a basic (set of) covering cluster(s), one may ask if this set provides a fundamental domain and therefore qualifies as a quasi-unit cell or cells. Specific answers to these questions are given in Chap. 4 for Voronoi and Delone clusters in canonical tilings. 1.5.5 Clusters in Quasicrystals A theoretical structural analysis of clusters in quasicrystals was given by Verger-Gaugry and Cotfas in [52, 53]. The clusters have a prescribed local point symmetry group G and are therefore called G-clusters. The cases analyzed by these authors cover the pentagonal Penrose, octagonal, dodecagonal and primitive icosahedral modules of quasicrystals [53]. The clusters cover a discrete set of quasiperiodic points. Clusters were considered as part of atomic structure models by Elser [9], Katz and Gratias [22, 23], and Kramer, Papadopolos and Kasner [20, 21, 43]. Clusters in icosahedral quasicrystals with covering properties were also studied by Lord, Ranganathan and Kulkani [38]. For dual projected canonical tilings, projections of Voronoi and Delone polytopes form the Voronoi and Delone clusters. These clusters are compatible with the tiling in that they are exactly filled by tiles of the tiling. The filling is asymmetric and unique up to orientation. These clusters and their cov-


17

ering of discrete point sets and of tiles, and their fundamental domain properties are analyzed in Chap. 4 on the basis of [25, 29, 30, 31, 33, 34, 35, 45]. An approach to clusters in quasicrystals derived from theoretical physics is given by Janot in [17]. Here the cluster structure is interpreted directly in terms of the quantum chemistry of quasicrystals. For perfect icosahedral quasicrystals, a structural skeleton based on hierarchical packing of atomic clusters is considered. The related inflation rules are claimed to constrain both the composition and the atomic valencies to have strictly defined values. Stability of the skeleton should then require that bonding electrons are recurrently localized at sites forming self-similar isomorphic subsets of the structure and that they are distributed into magic cluster states. 1.5.6 Surfaces of Quasicrystals The surface structure of quasicrystals can in theory be analyzed by studying the planar quasiperiodic sections of 3D quasicrystals. A mathematical study of systems of parallel lines and planes in terms of modules is presented in Chap. 6 by Pleasants. A structural analysis of planar quasiperiodic sections perpendicular to 5-fold axes of icosahedral quasicrystals (see also [20, 21, 43, 36]), is presented in Chap. 5 by Papadopolos and Kasner. Experimental evidence related to quasicrystal surfaces is described in Chaps. 7 and 8 by McGrath et al. and by Edagawa et al.

1.6 Perspectives on the Theory of Covering for Discrete Quasiperiodic Sets This survey of approaches to covering shows the variety of pathways opened up in this new field. In the theory of covering there arise a number of distinctions, some of which will be taken up in the other contributions to this book, as follows: (1) Objects to be covered. To describe a covering one must choose what geometric objects are to be covered – are they points or are they tiles of a tiling? If points are considered, are they the reference points for a class of atomic configurations which may differ in detail, or is the goal to cover the atomic positions as found in a concrete model of a quasicrystal? (2) Quality of covering. The quality of covering needs clarification. What is meant by full, partial, or average covering? Should coverings be quantified by their thickness and/or by principles of admissible overlap and density? (3) Shape and internal structure of clusters. For the covering cluster or clusters, one would like to have some general views about their shape, their internal structure, and their symmetry or symmetry breaking. A window theory can address global properties such as the frequency of overlaps.

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(4) Quasiperiodicity and windows. Quasiperiodicity allows one to extend classical Fourier theory and hence kinematical diffraction theory from crystals to quasicystals. An essential part of this theory is the lifting of the structure into a high-dimensional lattice and the determination of its windows. The collection of windows can be encoded in the high-dimensional unit cell of the embedding lattice. The concepts of clusters and covering should therefore be phrased in terms of windows. (5) Fundamental domain. The notion of a quasi-unit cell or of a fundamental domain defined exclusively in the tiling space differs from the highdimensional unit cell just mentioned and needs clarification. It expresses the view that the clusters in a covering describe a class of correlated atomic positions. One wishes to use the clusters as the building blocks of correlated atomic positions. The matching of clusters should then provide the long-range aperiodic or quasiperiodic structure. The structural theory must provide the tools and a possible basis for these claims. (6) Fourier theory and covering. So far, it has not been possible to give a simple Fourier theory for quasiperiodic coverings. Ultimately it will be necessary to approach their Fourier theory from the window side. The naive approach to a Fourier theory of clusters fails: owing to overlap, the Fourier transform of a set of overlapping clusters cannot be expressed in terms of the Fourier transform of a single cluster together with the transform of the quasiperiodic distribution of the centers of the clusters.

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2 Covering Clusters in Icosahedral Quasicrystals Michel Duneau and Denis Gratias

Summary. The structural analysis of various approximant phases of icosahedral quasicrystals shows local environments with icosahedral symmetry: icosahedra, Mackay clusters, and Bergman clusters. For the icosahedral phases i-AlCuFe and i-AlPdMn, these clusters have been proposed as complementary building blocks centered on particular nodes. However, computations have shown that these 2-shell or 3-shell clusters do not cover all atomic positions given by 6D models. On the other hand, the recent concept of a unique covering cluster has been shown to apply to 2D Penrose tilings and Ammann–Beenker tilings. In this chapter we examine the local environments in i-AlCuFe and i-AlPdMn models about Wyckoff positions of the 6D lattice. We consider extended Bergman clusters of 6 shells that appear naturally in the Katz–Gratias model. We discuss the cell decomposition of the atomic surfaces and the variable occupation number of some of the shells. We show that a fixed extended Bergman cluster of 6 shells and 106 atoms covers about 98% of atomic positions. We also prove that a variable extended Bergman cluster of 6 shells, which contains the previous fixed cluster, covers all atomic positions of the theoretical model.

2.1 Introduction Quasiperiodic tilings involve at least two prototiles: two Penrose rhombs and two rhombohedra for the 2D Penrose tiling (PT) and the 3D simple (or primitive) icosahedral tiling, respectively. Although no proof is available, it is believed that aperiodicity cannot be enforced by a unique prototile. In the 1970s Conway showed [1] that a Penrose tiling, in the version of Robinson’s triangles, could be covered with a unique decorated decagon (the so-called cartwheel), made of 82 triangles, with partial overlap between neighboring decagons. Since the discovery of quasicrystals [2] several hundred articles were more or less concerned with clusters in quasicrystals and approximant phases from either a theoretical or an experimental point of view have been published. Until 1995, however, very few papers were dedicated to the more specific question of covering a quasiperiodic structure with a unique cluster with possible overlaps. Burkov [3] proposed a 2D random model of the decagonal AlCuCo phase, and Janot et al. [4] discussed a hierarchical model of i-phases based on the repetition of a symmetric cluster of 50 atoms similar to the Mackay cluster. Jeong and Steinhardt [5] and G¨ ahler [6] examined the frequency of particular clusters from the point of view of their energy. P. Kramer (Ed.): Convering of Discrete Quasiperiodic Sets, STMP 180, 23–62 (2002) c Springer-Verlag Berlin Heidelberg 2002

24

Michel Duneau and Denis Gratias

Two papers specfically devoted to the question of unique covering cluster were presented at ICQ’5 in 1995. Gummelt [7] showed that Conway’s decagon, with its asymmetric decoration, would generate all 2D Penrose tilings provided overlapping rules were satisfied. This result was based on a proof that these rules were equivalent to the usual matching rules of the underlying Robinson triangles. Duneau discussed in [8] the constraints that must be applied to the atomic surface (AS) in order that a fully symmetric covering cluster could exist for the octagonal tiling and for the 2D PT. In these cases, infinite sequences of solutions show up, with larger and larger clusters located on subsets of the tilings with decreasing density. A large symmetric cluster, enforcing quasiperiodicity, was also shown to exist for the 2D Penrose tilings. Afterwards, several papers discussed more theoretical and physical implications of this new point of view (G¨ ahler [9], G¨ ahler and Jeong [10], Jeong and Steinhardt [11], Ben-Abraham and G¨ ahler [12]), showing that quasiperiodic tilings were ground states of random tilings in which a particular cluster is assumed to have a minimal energy. One must observe that the initial question of the existence of a covering cluster makes sense only once a deterministic model is given. The question then splits into the following ones: 1. Does there exist a covering cluster? 2. If yes, can a fully symmetric cluster be found? 3. In either case, are there overlapping rules by which the initial model can be recovered? It should be noticed that the requirement for a fully symmetric covering cluster is not without significance since the symmetric cluster for the 2D Penrose tiling presented in [8] is much larger than the cartwheel studied by Gummelt in [7]. In this chapter we shall be mainly concerned with the existence of covering clusters (symmetric or not) for the structural models of the i-phases proposed by Elser [13] and Katz and Gratias [14]. In Sect. 2.2 we recall the original descriptions of the Bergman (B) and Mackay (M ) clusters, discovered in the (Al, Zn)49 Mg32 phase, and in the α-AlMnSi phase respectively. In Sect. 2.3 we discuss the polyhedral atomic models for the i-AlCuFe and i-AlPdMn structures, especially the deterministic models of Elser and of Katz–Gratias (KG), with emphasis on the B and M types of clusters. Section 2.4 is devoted to an extended discussion of the very basis of the cell decomposition by union and intersection of polyhedra in E⊥ . The definition domains of B and M clusters are studied and compared in the KG model. In Sect. 2.5 we give a simple example of an application in the case of the atomic structure of i-AlPdMn, with the recently proposed model of Shramchenko et al. In Sect. 2.6 the existence of a covering cluster is examined and an extended Bergman cluster of six shells is considered: the B cluster is completed by two partial shells of 60 atoms, one dodecahedron

2 Covering Clusters in Icosahedral Quasicrystals

25

and one icosahedron that together form a triacontahedron. The cell decomposition of the atomic surfaces is discussed. We show that this cluster, with two variable inner shells, covers all atomic positions. It is also pointed out that a fixed cluster of 106 atoms, which is always contained in the variable extended Bergman clusters, accounts for about 98% of the structure.

2.2 Bergman and Mackay Clusters The Bergman cluster (B) (see Fig. 2.1a) was first identified in the (Al, Zn)49 Mg32 phase by Bergman et al. [16]. Afterwards it was found in the closely related R-Al5 CuLi bcc phase by Audier et al. [17]. The original (B) cluster is composed of the following shells denoted by n(r), where n is the number of points in the shell and r is its radius in nanometers: 12(0.248), 20(0.450), 12(0.493), 60∗ (0.678), 20(0.762) and 12(0.840), where 60∗ refers to a deformed truncated icosahedron. The 12-shell and 20-shell correspond to an icosahedron and a dodecahedron, respectively. If reduced to the first four shells, a similar cluster centered on the bcc lattice of the R-phase completely fills the structure, as shown in [17]. It was then argued that the same cluster could be involved in the structure of the stable i-AlCuLi phase [27] and possibly in other icosahedral phases such as i-AlCuFe and i-AlPdMn. The Mackay cluster (M ) (see Fig. 2.1(b)) was identified by Cooper and Robinson in a study of the α-AlMnSi phase [18]. It is composed of the following shells: 12(0.243), 30(0.470), 12(0.487), 60∗ (0.672) and 12(0.733), where the 30-shell corresponds to an icosidodecahedron and the 60∗ -shell refers to

Fig. 2.1. The first three shells of the (a) Bergman and (b) Mackay clusters

26


a rhombicosidodecahedron. A tentative model of the i-AlMnSi phase was proposed by Duneau and Oguey in [19], where the theoretical structure was partly filled by M clusters. A cluster analysis of the ξ -phase, an orthorhombic approximant phase of i-AlPdMn, was carried out by Beraha et al. [20]. It was shown that 87.5% of the atoms were covered by overlapping (pseudo) M s containing a Mn atom at the center, a partial Al dodecahedron (7/20 occupied sites), a Pd icosahedron, and an Al icosidodecahedron. In [21] these authors presented a similar analysis of two other approximant phases, T AlPdMn and R-AlPdMn, of the decagonal d-AlPdMn phase. It was shown that the structural models of these phases could be covered by overlapping Bs and “symmetric” M s, the Bs accounting for 90% of the atoms and the others for 92% of the atomic positions. More recently, Sugiyama et al. [22] proposed a structural analysis of two cubic approximants, the 2/1 and 1/1 approximants of the i-AlPdMn phase, of respective compositions Al70 Pd23 Mn6 Si and Al67.4 Pd11.4 Mn14.4 Si6.8 . These authors showed that a cluster of nine shells centered on the nodes fills all atomic positions of the 2/1 approximant together with a smaller cluster on the body centers, while a cluster of five shells can reproduce the 1/1 approximant.

2.3 The Al–Cu–Fe/Al–Pd–Mn Models Icosahedral phases in the Al–Cu–Fe and Al–Pd–Mn systems have been extensively studied since their discovery by Tsai et al. [23]. Both phases can be obtained as large, macroscopic single grains such as the one shown in Fig. 2.2 that are suitable for both X-ray and neutron diffraction studies. They can be described in a 6D space with an F-lattice, as ordered structures of a 6D primitive structure analogous to that of i-AlMnSi. Patterson analyses (see Fig. 2.3) of the two structures show clear maxima elongated along the perpendicular space and centered on the high-symmetry Wyckoff positions of the 6D F-lattice. This suggests modeling these structures with a maximum of four ASs located at the four Wyckoff positions, denoted n, n , bc, and bc , of the 6D cubic lattice a6D Z6 defined by

Fig. 2.2. Centimeter-size single grain of i-AlPdMn, obtained by slowly extracting the solid from the melt starting from a tiny seed (by courtesy of Yvonne Calvayrac and Annick Quivy)


27

Fig. 2.3. Examples of Patterson maps of i-AlPdMn in the 6D 5-fold plane, reconstructed from neutron (after N. Shramchenko et al. [39]) and X-ray (after Boudard et al. [34]) diffraction data

n = (0, 0, 0, 0, 0, 0) , n = (1, 0, 0, 0, 0, 0) , bc = (1/2, 1/2, 1/2, 1/2, 1/2, 1/2) , bc = (3/2, 1/2, 1/2, 1/2, 1/2, 1/2) .

(2.1)

Each AS is copied on the nodes of the 6D F-lattice Λα (α = n, n , bc, bc ), defined, up to the a6D scaling, by Λn = {x ∈ Z6 , xi ∈ 2Z} , Λn = Λn + n , Λbc = Λn + bc , Λbc = Λn + bc .

(2.2)

We designate, for brevity, the sites of Λn as even n’s, those of Λn as odd n’s, those of Λbc as odd bc’s and those of Λbc as even bc’s. We designate by An , An , Abc , and Abc the prototypic polyhedra in perpendicular space, which are copied at the nodes of the corresponding lattices Λα , and designate by Aα the complete set of ASs of type α (α = n, n , bc, bc ) obtained under the action of the translations of Λα : Aα = Aα+t . (2.3) t∈Λα

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Simple geometry shows that the simplest set of prototypic simply connected polyhedra bounded by mirror planes and consistent with physically acceptable first-neighbor distances is a set of three polyhedra, as shown in Fig. 2.4. The polyhedron An is a triacontahedron obtained by a linear τ inflation of the canonical triacontahedron. The polyhedron An is the same object but truncated along the 5-fold directions in order to remove too short distances that are too short along these directions. The last polyhedron is Abc and is a triacontahedron obtained by a τ − 1 linear scaling of the canonical triacontahedron. The Wyckoff position bc is empty. A convenient way of showing what kind of order is generated in E by this set of ASs is to draw the rational 2D cuts corresponding to 5-, 3-, and 2-fold directions. In Fig. 2.5, each 2D map displays the trace of E as a horizontal line and E⊥ as an vertical line: the ASs appear as vertical segments. The horizontal segments join the external boundaries of the the ASs, showing the main phason jumps of the model. Cutting these drawings by horizontal lines at any vertical level leads to a 1D sequence of atom locations along the corresponding 5-, 3-, or 2-fold direction. To easily quantify the (3D) volumes and distances of the model, we choose the volume of Abc as the unit. With this choice, An has volume |An | = 8τ + 5, An has volume |An | = 6τ + 5, and of course, |Abc | = 1. The total volume of the ASs of the model is Vt = 14τ + 11. We characterize our polyhedra by a set of tetrahedra defined by the center and a triangular facet in the elementary sector of the icosahedral symmetry m35. Each triangle is defined by three vectors in E⊥ that are projections of rational 6D lattice nodes. For example, the canonical triacontahedron with volume 2τ +1 is defined by one single facet, defined by the perpendicular projection of the three 6D nodes a = (0, −1, 1, 0, 1, 1)/2, b = (−1, −1, 1, −1, 1, 1)/2, and c = (−1, −1, 1, 1, 1, 1)/2. The model generates a discrete set of interatomic distances. A few of the shortest distances are listed in Table 2.1 together with the symmetry of their orbits.

Fig. 2.4. The three prototypic basic ASs in E⊥ corresponding to the models derived by Cockayne et al. [15], Katz and Gratias [14], and Elser [13] (see Fig. 2.7): from left to right, Abc , An , and An . The ASs are projected along a 5-fold direction in E⊥ and fit exactly to avoid unphysical short distances


29

n’ n’ (1,0,0,0,0,0)

n

n bc

(0,0,0,1,1,1)

bc n (0,0,1,0,1,0)

(0,1,1,1,1,−1) (1,1,1,0,0,0)

(a)

(b)

(1,1,0,0,0,0)

(c)

Fig. 2.5. Principal rational cuts along 5-, 3-, and 2-fold directions: (a), (b), and (c), respectively; horizontal lines are parallel to E and vertical lines, parallel to E⊥ represent the traces of the ASs of the model

Our present set of ASs is chosen according to three main conditions: – they are large enough to generate an acceptable density of nodes with respect to real icosahedral phases; – they generate no unphysically short distances (see Table 2.2); – they are bounded by mirror planes of the centered group m35, thus inducing “symmetric” flips of atoms by translation in E⊥ . As an example, we show in Fig. 2.6 how An fits exactly with its closest neighboring An and An in E⊥ with no intersection. Our choice of ASs is, of course, not unique. It is always possible to transfer part of an AS to another as long as no overlap occurs that would generate excessively short distances in E . An example is shown in Fig. 2.7, which corresponds to Elser’s model in its deterministic version. The AS An is slightly enlarged by 2τ − 3, but correlatively, An is reduced by 2τ − 3 to avoid short distances along (1, 1, −2, 1, −1, 1) (the 5-fold direction). Another simple way of transferring volume between ASs is to manage the holes in their centers. Let us, for example, examine the process of transferring some volume from n to bc , which is originally empty. We first remove a poly-

Fig. 2.6. Typical connections between close ASs in E⊥ along 2fold (left) and 5-fold (right) directions. The ASs are adjacent to each other and do not overlap

30


p

Table 2.1. The first few interatomic distances and their corresponding translations in 6D space. The distances are given in units of a6D K, where K = 1/ 2(2 + τ ) (≈ 0.371748) is the geometric constant introduced by Cahn et al. [28]; for i-AlCuFe, a6D K ≈ 0.234744 nm and for i-AlPdMn a6D K ≈ 0.239815 nm Type

Symmetry(lattice)

6D vector

n − n

20(P )

(1, 0, 0, −1, −1, 0)

n − bc

12(I)

(1, 1, −1, 1, −1, 1)/2

n−n

30(F )

(0, −1, 1, 0, 1, 1)

n − n

60(P )

(1, 0, −1, 1, 0, 2)

n − bc

20(I)

(−1, 1, 1, 1, 1, 1)/2

n − n

12(P )

(0, 0, 1, 0, 0, 0)

n−n

30(F )

(0, 1, 0, 0, −1, 0)

n − bc

60(I)

(1, −1, 1, 1, 1, 3)/2

n−n

12(F )

(1, 1, −1, 1, −1, 1)

n−n

60(F )

(−1, 0, 2, 0, 1, 0)

n − bc

60(I)

(−1, 1, 3, −1, −1, −1)/2

n − n

60(P )

(0, 1, 0, 1, 0, 1)

n − bc

20(I)

(1, 1, 1, −1, −1, 1)/2

n−n

60(F )

(1, 0, 1, −1, −1, 0)

n − bc

12(I)

(1, 1, 1, 1, −1, 1)/2

n−n

30(F )

(0, 0, 1, 0, 0, 1)

n − bc

120(I)

(3, −1, 1, −1, −1, 3)/2

n − n

60(P )

(0, 0, 2, −1, 0, 0)

n − bc

60(I)

(−1, 1, 3, 1, 1, 1)/2

n−n

60(F )

(1, 0, 0, 1, 0, 2)

d (a6D K units) √ 6 − 3τ √ 3−τ √ 8 − 4τ √ 14 − 7τ √ 3 √ 2+τ 2 √ 7−τ √ 12 − 4τ √ 12 − 4τ √ 7 √ 6+τ √ 3 + 3τ √ 2 2 √ 3 + 4τ

d (a3D units) 0.5628 0.6180 0.6498 0.8597 0.9106 1 1.0515 1.2196 1.2361 1.2361 1.3910 1.4511 1.4734 1.4870 1.6180

n−n

20(F )

(−1, 1, 1, 1, 1, 1)

n − bc

60(I)

(−1, 3, 1, 1, −1, 1)/2

n − n

60(P )

(0, 1, 1, 0, −1, 0)

n−n

12(F )

(0, 0, 2, 0, 0, 0)

n−n

60(F )

(1, 1, 0, 0, −1, 1)

n − n

120(P )

(0, 0, 1, 1, 1, 2)

n − bc

60(I)

(3, 1, 1, −1, −3, 1)/2

n − bc

60(I)

(1, −1, 3, −1, 1, 3)/2

2τ √ 11 √ 10 + τ √ 7 + 3τ √ 2 3 √ 2 3 √ 7 + 4τ √ 6 + 5τ √ 2 2+τ √ 2 2+τ √ 14 + τ √ 11 + 3τ √ 11 + 3τ

n−n

120(F )

(1, −1, 2, −1, 0, 1)

4

2.1029

n−n

30(F )

(0, 2, 0, 0, −2, 0)

4 √ 11 + 4τ √ 10 + 5τ √ 10 + 5τ √ τ 7

2.1029

n − bc

60(I)

(1, −1, 3, 1, 1, 3)/2

n − n

60(P )

(1, 0, 1, −1, −1, 1)

n − n

12(P )

(1, 1, 0, 1, −1, 1)

n − bc

60(I)

(1, 1, 1, 1, −1, 3)/2

1.7013 1.7437 1.7920 1.8101 1.8212 1.8212 1.9297 1.9734 2 2 2.0777 2.0933 2.0933

2.1975 2.2301 2.2361 2.2506

hedron Abc in the center of An . This allows us to add, a priori, 12 such volumes at bc located at (−1, −1, 3, −1, 1, −1)/2 from n, as shown in Fig. 2.5. These additional pieces of ASs, however, overlap strongly (see Table 2.2 and Fig. 2.8) with An along the 5-fold direction by the translation (−1, −1, 3, −1, 1, −1)/2, along the 3-fold direction by the translation (−3, 1, 1, 3, 3, 1)/2, and with


31

Table 2.2. The first main unphysical interatomic distances in E and their corresponding translations in 6D space. The ASs should be chosen as large as possible for maximum compactness but must still present no intersections when projected into E⊥ after being displaced by these 6D translations. The notation is the same as in Table 2.1 Type

Symmetry(lattice)

6D vector

n − bc

12(I)

(−3, −3, 7, −3, 3, −3)/2

n − bc

20(I)

(5, −1, −1, −5, −5, −1)/2

n − n

12(P )

(1, 1, −2, 1, −1, 1)

n−n

30(F )

(0, −3, 2, 0, 3, 2)

n − n

60(P )

(1, −2, 3, −4, −1, −2)

n − bc

20(I)

(−3, 1, 1, 3, 3, 1)/2

n − bc

12(I)

(−1, −1, 3, −1, 1, −1)/2

n−n

30(F )

(0, 2, −1, 0, −2, −1)

n − bc

60(I)

(7, 1, −5, −3, −7, 1)/2

n − bc

60(I)

(5, −7, 3, −5, 1, 3)/2

n − n

60(P )

(1, −2, 0, 1, 2, 3)

n−n

60(F )

(−3, −1, 4, 0, 3, −1)

n−n

12(F )

(2, 2, −4, 2, −2, 2)

n − bc

60(I)

(−3, 1, 5, −3, −1, −5)/2

n − bc

60(I)

(−1, 3, −3, 5, 1, 3)/2

d (a6D K units) √ 47 − 29τ √ 39 − 24τ √ 18 − 11τ √ 52 − 32τ √ 70 − 43τ √ 15 − 9τ √ 7 − 4τ √ 20 − 12τ √ 67 − 41τ √ 59 − 36τ √ 38 − 23τ √ 72 − 44τ √ 72 − 44τ √ 35 − 21τ √ 27 − 16τ

d (a3D units) 0.1459 0.2150 0.2361 0.2482 0.3425 0.3478 0.3820 0.4016 0.4273 0.4555 0.4659 0.4721 0.4721 0.5313 0.5543

Fig. 2.7. Basic ASs projected into E⊥ along a 5-fold direction corresponding to Elser’s deterministic model: from left to right An , An , and Abc . The AS An is larger that the corresponding AS in Fig. 2.4 by 2τ − 3 but the AS An is reduced by the same amount

themselves along the 2-fold direction by the translation (0, 2, −1, 0, −2, −1). Removing all overlapping parts leads finally to transferring a global volume of only 4τ − 6 from An to bc in the process of making a hole of the same volume in An . A nonempty AS can be attached to bc if and only if we remove a part (here at the center) of the original An . The very same procedure can be applied to transfer volume from An to the site bc and applies also for transferring volume between n and n .

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(a)

(b)

(c)

(d) Fig. 2.8. Principle of volume transfer from one AS to another. (a) We excavate An by creating a central hole identical to Abc that we transfer to the closest bc Wyckoff position. (b) We obtain a set of 12 polyhedra around bc ; these polyehdra intersect (see the details in (c)) n along 5-fold directions and 3-fold directions and reduce finally to the star on the right (which can be seen to introduce no short distances along the 2-fold direction, as shown in (c), (right); (d) finally, the minimum excavation on the node is a polyhedron (left) of volume 4τ − 6 that has been split into a 12-branched star (right) of the same volume at bc

2.4 Local Environments From what we have said in the previous section, it is best to analyze the atomic local configurations directly in E⊥ where all the geometrical environments have a finite-size image that can be calculated exactly. The natural way of achieving this is the cell [30], or Kl¨ otze [31, 32], decomposition, which is


33

based on the simple idea that two atoms actually present in the structure are derived from two atomic surfaces whose projections in E⊥ have a nonempty intersection. Thus, studying how atomic surfaces projected into E⊥ intersect each other suffices to determine what kinds of clusters are present in the real structure. The work is considerably simplified by the fact that the first main interatomic distances (see Table 2.1) are along 3-, 5-, and 2-fold directions. Hence, we can draw the traces (vertical lines) of the ASs in the 5-, 3-, and 2-fold 2D planes of the 6D space as shown in Fig. 2.5 and directly visualize the basic intersections between neighboring atomic surfaces. The same method can be used to identify the existence domain of a given cluster: the set of points in E where such a cluster is present corresponds, in E⊥ , to a domain obtained from particular intersections of atomic surfaces of the model, projected into and translated in E⊥ . Now, when the base point of the cluster runs over the existence domain, each of the other points of the cluster covers a translated domain which is entirely contained in the atomic surfaces. The union of these domains represents the set of points covered by the cluster in E . This union need not fill all the atomic surfaces of the model. A cluster is a covering cluster if its existence domain is large enough for the atomic surfaces of the model to be entirely covered by the translations of this domain. Besides clusters of one point, which obviously cover any structural model, the edges of many tilings (2D and 3D Penrose tilings, the Ammann–Beenker tiling, etc.) provide covering clusters of two points. In this case, a covering is achieved with a cluster and all its images in the symmetry group of the tiling. In the following, we consider some more symmetric clusters, such as Mackay and Bergman clusters, which contain at least two shells about a center. 2.4.1 Computation of Environments Convex Polyhedra. Compilers use various basic types of numerical data, such as “float” and “double” in C and C++. Arrays of such type are used to represent vectors, matrices, etc. In the the present case, the coordinates of points derived from projection of the 6D lattice in a parallel or perpendicular space, or from the intersection of quasilattice planes belong to the field Q[τ ] = {q1 + q2 τ ; q1 , q2 ∈ Q}. It is very convenient to implement a new scalar type for coding elements of Q[τ ] together with the usual operations of the field. The advantage of such an implementation is that exact calculations can be carried out without rounding errors. A scalar type for Q[τ ] may be coded, for instance, by four long integers or four 64 bit integers (for reducing the risk of integer overflow during a computation). The atomic surfaces of the model described here are polyhedra in E⊥ with full icosahedral symmetry. The existence domains of particular environments or particular chemical species, are generally not convex polyhedra (see, for instance, An ) but they can still be defined as intersections or unions of finitely many convex polyhedra (here, all our polyhedra are defined as unions

34


of tetrahedra and are simplexes of the 3D space), and the decomposition problem reduces to a problem of intersection/union of convex polyhedra. A convex polyhedron may be defined as the intersection of finitely many half-spaces bounded by planes. It is convenient to code a plane as a pair p = (v, c), where v is a nonzero 3-dimensional vector and c is a real number. The 2-dimensional plane is the set of points Hp = {x : x.v = c} .

(2.4)

By convention, the corresponding closed half-space is the set Sp = {x : x.v ≤ c} ,

(2.5)

so that the vector v points outwards from Sp . The opposite half-space is coded as p = (−v, −c) and we have Sp ∩ Sp = Hp = Hp . Notice that this coding is not unique unless the vectors v are normalized to a given value. This must be kept in mind when comparisons of planes are required. Once again, it may be convenient to use a special scalar type in order to handle particular cases without rounding errors (checking for parallel planes, for instance). Now a sequence P = {p1 , . . . , pn } of planes codes for the convex closed polyhedron C which is the intersection of the corresponding half-spaces Spi : C = {x : x.vi ≤ ci , i = 1, . . . , n} =

n

Spi .

(2.6)

i=1

It may happen that planes of the sequence P are spurious in the sense that they do not belong to the boundary planes of the convex set C (exterior planes) or they occur more than once in the sequence P . In the latter case, duplicated planes may be eliminated by means of a comparison function. Alternatively, a closed convex set can be defined as the convex hull of a set of vertices. For any finite set of points X = {x1 , . . . , xm }, the corresponding closed convex polyhedron is m m λi .xi : 0 ≤ λi ≤ 1, λi = 1 . (2.7) C= i=1

i=1

The sequence X may also include spurious points that do not belong to the vertices of C (interior points) or that are duplicated. The latter case may be handled by using a simple comparison function. In many calculations it is convenient to have a mapping between both representations of a convex polyhedron. This can be achieved by simple geometric methods. 1. If C is defined by a sequence of planes P = {p1 , . . . , pn } the vertices are obtained as follows. For every triplet of planes {pi , pj , pk }, we compute the intersection xi,j,k , if any. If xi,j,k belongs to all other half-spaces, i.e. if xi,j,k .vl ≤ cl for all other indices l, then this point belongs to the vertices of C and may be recorded.


35

2. Conversely, if C is defined by a sequence of points X = {x1 , . . . , xm } the bounding planes are obtained as follows. For every triplet of points {xi , xj , xk } we compute the corresponding plane pi,j,k , if any, up to its orientation. If all other points xl of X belong to the same half-space of pi,j,k , then the orientation can be fixed and pi,j,k is a boundary plane of C. For the sake of efficiency of numerical computation, it is helpful to define a procedure that cleans up the representation of a convex polyhedron by elimination of spurious planes and spurious points. This is achieved by running one or both of the above procedures and dropping duplicated elements in P and X. A convex polyhedron may then be coded by its bounding planes and its vertices. The computation of the volume of a convex polyhedron C is an easy task. First an interior point x0 is calculated, for instance x0 = (1/m) xi . For each bounding plane p = (v, c), we determine the corresponding facet Fp as the set of vertices which satisfy xi .v = c. The set Fp = {y1 , . . . , yk } is then ordered so that for i = 2, . . . , k − 1 the triplets {yi − y1 , yi+1 − y1 , v} are right-handed. The volume of the pyramid of base Fp and top x0 can be easily calculated as a sum of tetrahedra. Intersections of Convex Polyhedra. The intersection of two or more convex polyhedra is a (possibly empty) convex polyhedron. If C and C are polyhedra of this kind then we construct the union of their bounding planes P = P ∪ P = p1 , . . . , pn , p1 , . . . , pn . The intersection C = C ∪ C is then calculated by reducing the sequence of planes P as explained above: first obtain the vertices of C from P and then obtain the bounding planes from the set of vertices. The calculation of intersections may be shortened if for each convex polyhedron we compute a bounding box, recorded together with the sets of planes and vertices. Many cases of non-intersection can be detected effectively at once without entering CPU-expensive loops. Unions of Convex Polyhedra. Since the union of two or more convex polyhedra is not in general a convex polyhedron, we must introduce a new structure for coding unions. This is achieved by using lists of convex polyhedra C1 , . . . , Cn . The calculation of the volume of a union of convex polyhedra requires particular attention since we must take care of possible intersections. In the case of two polyhedra C1 and C2 , the volume of the union is simply given by |C1 ∪ C2 | = |C1 | + |C2 | − |C1 ∩ C2 | .

(2.8)

We see that the above formula leads to a recursive volume function. Namely, the volume of the union of n convex polyhedra reads n n−1 n−1 (2.9) Ci + |Cn | − (Ci ∩ Cn ) , Ci = i=1

i=1

i=1

36


and recursion is possible since the last expression involves only n − 1 convex polyhedra. The representation of a union of convex polyhedra by a list may not be optimal. A polyhedron Ci may be incuded in some other one Cj , or a particular union Ci ∪ Cj may be convex. Handling such situations by reducing the representation of the union is possible but requires a lot more calculation. Computing Local Environments. The computation of environments requires calculations of intersections and unions of polyhedra associated with the atomic surfaces of the structural model. In the framework of cuts described earlier, the atomic positions are given by the intersection of E with the set of atomic surfaces. If β denotes a vector in E⊥ , the corresponding structure Xβ for our previous icosahedral structures is given by Xβ = x0 ∈ E : x0 + β ∈ (An ∪ An ∪ Abc ∪ Abc ) = (E + β) ∩ (An ∪ An ∪ Abc ∪ Abc ) − β . (2.10) A point x0 of E belongs to the set Xβ if x0 + β ∈ Aα0 + ξ0 for some index α0 in {n, n , bc, bc } and some ξ0 in Λα0 . This condition reads x0 + β = y0 + ξ0 , y0 ∈ Aα0 , ξ0 ∈ Λα0 , using the natural embedding of E and E⊥ in R6 . By projection onto E and E⊥ , one obtians the equivalent conditions ξ0, = x0 , ξ0,⊥ ∈ β − Aα0 ,

(2.11)

which state that x0 belongs to the parallel projection Λ of Λ = ∪α Λα and that the cut occurs within the atomic surface Aα0 translated by ξ0 . The relative density of points associated with the Aα0 atomic surface is simply |Aα | ρ1 (α0 ) = 0 . α |Aα |

(2.12)

More generally, if Ω is a regular subset of Aα0 , the relative density of points of Xβ associated with Ω is |Ω| . α |Aα |

ρ1 (α0 , Ω) =

(2.13)

Similarly, if x1 = ξ1, is another point of Xβ , we have ξ1 ∈ Λα1 for some index α1 ∈ n, n , bc, bc and ξ1,⊥ ∈ β − Aα1 . Thus x1 − x0 = (ξ1 − ξ0 ) with ξ1 − ξ0 ∈ Λα1 − Λα0 ⊂ Λ, which means that the relative positions belong to Λ . Now, if t ∈ Λ the condition for both x0 = ξ0, and x1 = x0 + t = ξ1, to belong to the structure Xβ reads


37

ξ0,⊥ ∈ β − Aα0 , ξ0,⊥ + t⊥ ∈ β − Aα1 , i.e. ξ0,⊥ ∈ β − [Aα0 ∩ (Aα1 + t⊥ )] ,

(2.14)

where α1 is uniquely specified as the index of Λα0 + t. The condition (2.14) defines a subset Aα0 ∩ (Aα1 + t⊥ ) of the atomic surface Aα0 which we may call the existence domain for the pairs (x0 , x0 +t ). It follows that the relative density of such pairs is given by ρ2 (α0 , t ) =

|Aα0 ∩ (Aα1 + t⊥ )| α |Aα |

(2.15)

when x0 is assumed to come from an Aα0 atomic surface. More generally, if x0 is limited to some subset Ω ⊂ Aα0 , the relative density of pairs of points (x0 , x0 + t ) is given by ρ2 (α0 , Ω, t ) =

|Ω ∩ (Aα1 + t⊥ )| . α |Aα |

(2.16)

As an example, we show in Fig. 2.9 the basic vertex decomposition of the standard 3D Penrose tiling obtained by the intersections of the canonical triacontahedron with itself under the 5-fold translation (1, 0, 0, 0, 0, 0). Superimposing all 12 equivalent translated triacontahedra on top of the central one leads to a decomposition into 9 different cells, building successive shells about the first, central shell. 2.4.2 Atomic Clusters Now we define a 6D cluster Γ as a finite set of sites in Λ: Γ = γ0 , γ1 , . . . , γn , where γ0 is considered as the base point of Γ . The cluster is actually defined by the relative positions ti = γi − γ0 (i = 1, . . . , n), which belong to the lattice Λ. We wish to answer several questions such as the following: 1. Find the existence domain of Γ , i.e. the subset Xβ,Γ of all points x0 such that the translated cluster Γ +x0 belongs to the structure Xβ . Assuming that x0 runs over a subset of Xβ associated with some domain Ω of an atomic surface Aα we also ask: 2. Find the average number of points of Γ + x0 which belong to the structure, 3. Compute the volume occupied by Γ + x0 in the atomic surfaces Aα (α = n, n , bc, bc ). The relative density of the cluster can be deduced from the above discussion. If x0 is assumed to come from Aα0 , we obtain ρn+1 (α0 , t1, , . . . , tn, ) =

|Aα0 ∩ (Aα1 + t1,⊥ ) ∩ . . . ∩ (Aαn + tn,⊥ )| . α |Aα |

(2.17)

38


−3+2τ

1

13−8τ

−16+10τ

−3+2τ

10−6τ

1

−3+2τ

1

Fig. 2.9. The cell decomposition of the intersections of basic triacontahedron of the 3D Penrose tiling with itself along the 5-fold direction (1, 0, 0, 0, 0, 0). We obtain 9 cells, corresponding to 9 different vertex configurations. Each configuration appears in the 3D pattern with a frequency proportional to the volume of the corresponding cell

where the indices αi are uniquely specified by the conditions Λα0 + ti = Λαi . The existence domain for the cluster is given by the intersection in the righthand side of (2.17). Therefore the cluster γ splits according to the disjoint union Γ = Γn ∪ Γn ∪ Γbc ∪ Γbc ,

(2.18)

where the different parts correspond to the relative positions ti belonging to Λn , Λn , Λbc , and Λbc respectively. We obtain a similar result if the base point x0 of the cluster belongs to some domain Ω ⊂ A0 : ρn+1 (α0 , Ω, t1, , . . . , tn, ) =

|Ω ∩ (Aα1 + t1,⊥ ) ∩ . . . ∩ (Aαn + tn,⊥ )| . (2.19) α |Aα |

2.4.3 B and B Clusters As can be seen from Fig. 2.5 and Table 2.1, Abc projects into E⊥ inside An , located at (1, 1, −1, 1, −1, 1)/2 in a 5-fold direction, and inside An , located at (−1, 1, 1, 1, 1, 1)/2 in a 3-fold direction. Thus, each time E passes through Abc , it necessarily passes through An and An and all other polyhedra of the same orbits around bc. This makes a total of 12 (icosahedron) +20 (dodecahedron) atomic sites around each bc, defining a 33-atom cluster, which we will designate for short as a B cluster because it is reminiscent of the Bergman clusters. The B cluster (see Fig. 2.10) is thus defined by: • a central bc atom; √ • a full icosahedron of radius 3√− τ (0.275 nm for i-AlCuFe); • a full dodecahedron of radius 3 (0.406 nm for i-AlCuFe).


39

The complete cell decomposition is shown in Fig. 2.11. Because the 12 small triacontahedra Abc fall inside An with no overlap, any point of An inside one of these generates a site in E that belongs to one and only one B icosahedron: the B icosahedra do not overlap. The fraction of An sites which are taken into account in the B icosahedron is given by volume of the twelve Abc divided by the volume of An , i.e. 12/(8τ + 5) ≈ 66.87%: two third of the An sites belong to a B icosahedron. In contrast, the 20 triacontahedra Abc overlap each other in pairs when projected into An , as shown in Fig. 2.11, so that a certain fraction of An sites belong simultaneously to two B dodecahedra. The B dodecahedra are connected by two vertices forming an edge of the B dodecahedron. We designate by B dodecahedron(1) (“B3 unshared” in Elser’s notation) the vertices of the dodecahedron that belong to one and only one dodecahedron and designate by B dodecahedron(2) (“B3 shared” in Elser’s notation) those that belong to two adjacent dodecahedra. The fraction of An sites that belong to (at least) one B dodecahedron is given by the volume of the union of the interpenetrating Abc , i.e. 20 − 4τ , so that (20−4τ )/(6τ +5) ≈ 91.97% of the An sites belong to a B dodecahedron. The fraction of An sites that form the connected edges between these B dodecahedra is given by the volume of the intersection all pairs of the Abc , which is 4τ . Therefore 4τ /(6τ + 5) ≈ 47.84% of the An sites are involved in the B dodecahedron-to-B dodecahedron connections. The B clusters are connected together along 2-fold directions (icosidodecahedron) by (1, 1, 0, 0, 0, 0) translations at distances R = 2τ (0.759 nm for i-AlCuFe) as shown in Fig. 2.12. The decomposition of Abc by itself under these translations leads to 15 cells with an average coordination number of Z¯B = 4τ ≈ 6.4721. As already noted by Elser [13] and Kramer et al. [36], the B clusters are distributed on the odd nodes of a τ -scaled canonical 3D Penrose tiling.

Fig. 2.10. The 33-atom B cluster, made up of a central atom, an icosahedron, and an external dodecahedron, describes almost 80% of the atoms of the structure

40


1

1

2

2

3

3

4

5

Fig. 2.11. Cell decomposition of An (top) and An (bottom) by means of Abc , located at (−1, 1, 1, 1, 1, −1)/2 and (−1, −1, −1, 1, 1, 1)/2, respectively, defining the B cluster. The corresponding volumes are (with the atomic fraction with respect to the total structure in percent in parenthesis): on An , #1, −2 + 2τ (3.67304), #2, 12(35.6586), #3, −5 + 6τ (13.9907); on An , #1, −6 + 4τ (1.40298), #2, 10 − 6τ (0.867086), #3, 4τ (19.2323), #4, 10 − 2τ (20.0994), #5, −9 + 6τ (2.10446)

(a)

(b)

Fig. 2.12. (a) The network of B clusters is a set of “flat” layers perpendicular to the 5-fold directions. (b) Observed along 5-fold directions, the B clusters are grouped in pentagonal “flowers” that are typical of high-resolution STM images

√ They form layers of three alternating thicknesses L = A 2(τ + 1)/(τ + 2), M = L/τ , and S = M/τ , following a quasiperiodic sequence. This sequence can be generated by copying Abc on the nodes of the 2D lattice defined by (5, −1, −1, −1, −1, 1)/5 and (0, 2, 2, 2, 2, −2)/5 that results from the projection onto the 5-fold 2D plane of the overall 6D structure. Each length ap-


41

pears with frequencies of 1/2 for M , (τ − 1)/2 (30.9%), for L and (2 − τ )/2 (19.1%) for S. This feature is of the greatest importance in understanding the sequence of the terrace steps observed in STM studies of quasicrystal surfaces [37]. The B clusters describe a remarkably large number of the atoms of the structure: the volumes that have been explored in constructing the cluster are the volume of Abc (1) plus a volume of 12 in An and a volume of 20−4τ in An . Hence the total fraction of sites explored is (1+12+20−4τ )/(11+14τ ) ≈ 78.83% of the total number of atomic sites in the structure. The second remarkable cluster is the B cluster, which is centered around the empty bc Wyckoff position. This cluster is similar to the B cluster, with the same kind of external dodecahedron but differs from it in the following ways: • it has an empty center; • the inner icosahedron has split into two shells, one with three atoms shrunk towards the center, generated by An at (−3, 1, 1, 1, 1, −1)/2, and one with nine atoms on the icosahedron, generated by An at (−1, 1, 1, 1, 1, −1)/2; • it has one to three additional atoms generated by Abc at (1, 0, 0, 0, 0, 0) on top of the three inner atoms. In contrast to the previous case, the B cluster has three different configurations shown in Fig. 2.13. The complete cell decomposition induced by the B cluster is shown in Fig. 2.14.

Fig. 2.13. The B cluster has three different configurations containing (left to right) 35, 34, and 33 atoms. These configurations appear with frequencies −25+16τ (≈ 0.888544) for the first configuration with 35 atoms and 13 − 8τ (≈ 0.0557281) for each of the two others

2.4.4 M and M Clusters We define as M clusters and M clusters [35] the atomic clusters centered on n and n , respectively, that are surrounded by a complete icosidodecahedron

42


Fig. 2.14. The cell decomposition of An , An , and Abc generated by the B cluster. The corresponding volumes are (with the atomic fraction with respect to the total structure in percent in parenthesis): for An (top) #1, −6 + 4τ (1.40298), #2, 10 − 6τ (0.867086), #3, 4τ (19.2323), #4, 10 − 2τ (20.0994), #5, 3(8.91465); #6, −12 + 8τ (2.80595); for An (bottom left) #1, −2 + 2τ (3.67304), #2, 9(26.7439), #3, −2 + 4τ (13.2892); for Abc (bottom right) #1, −25 + 16τ (2.64035), #2, 13 − 8τ (0.165599), #3, 13 − 8τ (0.165599)

of radius 2 (≈ 1.051 a3D ). As can be seen from Fig. 2.5, these can be obtained from n and n centers, respectively, generated by a small triacontahedron of volume −3 + 2τ , which we designate as A0 . Hence, the centers of the M and M clusters are distributed on the nodes of a canonical 3D Penrose tiling scaled by τ + 1 (even nodes for M and odd nodes for M ). Both the M and the M clusters consist of (see Fig. 2.15): • a central n atom for M (n atom for M ); √ • a partially occupied (7 atoms out of 20) dodecahedron of radius 6 − 3τ generated by An or An for M or M , respectively, translated by (0, 0, 0, 1, 1, 1); √ • a full icosahedron of radius 2 + τ generated by An or An for M or M , respectively, translated by (1, 0, 0, 0, 0, 0, 0); • a full icosidodecahedron of radius 2 generated by An or An for M or M , respectively, translated by (0, 0, 1, 0, 1, 0). The inner dodecahedron is generated by intersection of A0 with An translated by (0, 0, 0, 1, 1, 1). As shown in Fig. 2.16a, this intersection is only partial, the intersection volume being 7(2τ − 3) instead of 20(2τ − 3), the value it would have if A0 were fully embedded in An . Thus the inner dodecahedron is occupied by 7 atoms only, out of the 20 vertices of the dodecahedron. This is consistent with the fact that the edges of this dodecahedron have a too short a length to be physically acceptable as interatomic distances (0.175 nm for i-AlCuFe). The 7 atoms are distributed on the dodecahedron such that they


(a)

43

Fig. 2.15. (a) The M cluster contains 50 atoms distributed as one atom in the center, 7 out of 20 atoms (out of 20 positions) on an inner dodecahedron, 12 atoms on an icosahedron, and 30 atoms on an icosidodecahedron. (b) Detailed view of the partially occupied inner dodecahedron

(b)

never occupy first-neighbor sites and opposite sites simultaneously. As shown by Lyonnard et al. [33], there are 100 possibilities, which can be grouped into two prototypes with respect to icosahedral symmetry, one with local symmetry 3 of multiplicity 40√and one with a mirror of multiplicity 60. The large icosahedron of radius 2 + τ (0.4465 nm for i-AlCuFe) is generated by intersecting A0 with An translated by (1, 0, 0, 0, 0, 0). As shown in Fig. 2.16b, we have a full immersion of A0 in An , identical to the case of the full icosahedron of the B clusters but deflated by a factor τ . Node sites generated by A0 have a full icosahedral shell originating from n sites. Atoms on this M icosahedron belong to one and only one such shell. Finally, the outer icosidodecahedron of radius 2 (0.469 nm for i-AlCuFe) is obtained when An is translated by (0, 1, 0, 0, −1, 0). Here again (see Fig. 2.16c), A0 is entirely contained in the projection of An , thus leading to a fully occupied icosidodecahedron. The fraction of atoms belonging to an M cluster can be calculated by summing the volumes of the atomic surfaces that have been explored: 2τ − 3 for the central atom, 7(2τ − 3) for the atoms of the partial dodecahedron,

(c)

(b)

(a)

Fig. 2.16. Cell decomposition for the M cluster characterized by a small triacontahedron (left) ...centered on n: (a) cell on An corresponding to the partial inner dodecahedron, (b) cell on An corresponding to the large icosahedron, (c) cell on An corresponding to the outer icosidodecahedron. The corresponding volumes are A0 − 3 + 2τ (0.701488), (a) −21 + 14τ (4.91041), (b) −36 + 24τ (8.41785), (c) −90 + 60τ (21.0446). The very same decomposition applies for M clusters if n and n are exchanged

44


12(2τ − 3) for the icosahedron, and 30(2τ − 3) for the icosidodecahedron: 50(2τ − 3) in total. This represents a fraction of 50(2τ − 3)/(14τ + 11) ≈ 35.0744% of the atoms of the structure. The M clusters are disconnected from each other but they significantly intersect with B clusters (see Fig. 2.17). This can be quantify by examining the intersections in E⊥ between the cells of the B clusters and those of the M clusters: all seven atoms of their inner dodecahedra are common to B dodecahedra, 11 atoms over 12 of the M -icosahedra belong to B dodecahedra and 21 atoms of the M -icosidodecahedra belong to B-icosahedra. Each of the two families of M and M clusters, taken alone, is a set of disconnected clusters. Together, they have a few intersections that correspond to a small fraction, 2.78%, of the atoms of the structure being common to M and M clusters. The crucial difference between the M and M clusters is the way they intersect with the B clusters [13]. The cells corresponding to the partially occupied inner dodecahedron (Fig. 2.16a) have an empty intersection with the cells of the B clusters on n: the atoms of the M dodecahedron do not belong to B clusters. In contrast, 8 atoms out of 12 of the M icosahedra belong to B clusters. The atoms of the M icosidodecahedron are distributed so that: 16−2τ (≈ 12.7639) are common to a B dodecahedron on sites that do not link two B clusters, and 19 − 2τ (≈ 15.7639) are on sites that connect two B clusters. Finally only −5 + 4τ (≈ 1.47214) sites of the M icosidodecahedra do not belong to B clusters. Hence, most atoms of the M icosidodecahedra are atoms of the B dodecahedra. Loosly speaking, the M clusters can be seen as “complementary” to the B clusters. The B clusters intersect M clusters along the 3-fold directions with four different configurations. This result can be obtained by means of decomposing Abc by A0 located at (1, 1, 1, 1, 1, 1)/2, as shown in Fig. 2.18. B The average number of intersecting M clusters is Z¯M = 5−2τ (1.764) with a high frequency for the configuration in which two M clusters intersect a B-

M’

M M’

M

M’ M

M

M

M’

M

M’

Fig. 2.17. 5-fold cut of the structure showing B, M , and M clusters. The M clusters intersect the B clusters substantially, whereas the M clusters share complete facets with B clusters. Loosely speaking, B and M clusters are “complementary” to each other (the same is true for M and B clusters)


45

Fig. 2.18. B/M connections: #1, 26 − 16τ , #2, −42 + 26τ , #3, 4 − 2τ , #4, 13 − 8τ

cluster. The B clusters are connected to M clusters (along 5-fold directions) and share a full pentagonal face. The average number of adjacent M clusters B is Z¯M = −148 + 92τ (0.859) and 36 − 22τ (40.325%) of B clusters have no adjacent M clusters. A similar analysis leads to the conclusion that M clusters have unique conM = 7 intersecting B clusters distributed in the same way figuration, with Z¯B as the atoms of the M dodecahedron with the configuration of multiplicity 60 (mirror symmetry) shown in Fig. 2.15b. This configuration corresponds to 5 B clusters distributed on a pentagon and two out of the plane (like a “stone thrower used as weapon in Roman times”). The M clusters are distributed M among six configurations with an average number of Z¯B = 12 − 2τ (8.764) adjacent B clusters. There are two major configurations with same frequency, one with 12 neighboring B clusters and the other with 8. We finally come to the analysis of the three kinds of clusters B, M , and M together. This is achieved by computing the mutual intersections between all the cells discussed previously. At that stage, by regrouping the cells associated with B, M , and M cluster configurations, we were able to describe roughly 95% of the whole atomic structure. The results are given in Figs. 2.19 and 2.20, associated with Tables 2.3 and 2.4, respectively. The first column in these tables defines the cell number, the second column gives its volume, which, divided by the total volume of the atomic surfaces, gives, in the third column, the global concentration in at% of the atoms generated by the cell. The subsequent columns give the geometrical characteristics of the atoms generated by the cell with respect to the three clusters. For example, the cell #7 on An generates atoms that simultaneously belong to a B icosahedron, an M icosidodecahedron, and an M icosahedron; similarly, the cell #8 on An generates atoms that belong to two B dodecahedra (i.e. on the vertices of the pairs that link two B clusters), an M icosahedron, and an M icosidodecahedron. Both kinds of atomic sites represent a concentration of 0.6325% of the atoms of the structure.

46


1

2

3

4

9

5

6

10

7

11

8

12

13

Fig. 2.19. The complete decomposition of An , defining the local configurations with respect to both B and M clusters (see Table 2.3)

1

2

9

13

3

4

10

14

5

6

7

11

15

8

12

16

Fig. 2.20. The complete decomposition of An , defining the local configurations with respect to both B and M clusters (see Table 2.4)

2.5 Atomic Clusters and Chemical Decoration of i-AlPdMn The quite detailed cell decomposition discussed in the previous section leads to a possible tailoring of the cristal chemistry of the icosahedral structures based on the three kinds of clusters. We can decorate each cell with the


47

Table 2.3. Cell decomposition of An with respect to B and M clusters. The cells labeled with an asterisk correspond to “glue” atoms. Notations for the cluster shells , M icosahedron; Bico , B icosahedron; Micosi , M are: Mctr centers of M ; Mico icosidodecahedron; and Mdodeca , M dodecahedron (see Fig. 2.19) Cell #

Volume

at%

Mctr

1

−71 + 44τ

0.575

•

2

68 − 42τ

0.1265

•

∗

Mico

Bico

3

81 − 50τ

0.2921

−64 + 40τ

2.1435

5

−16 + 10τ

0.5359

6

−380 + 236τ

5.5153

•

•

7

340 − 210τ

0.6325

•

•

8

455 − 278τ

15.42

• •

9

−403 + 252τ

14.1

10

−367 + 228τ

5.681

11∗

383 − 236τ

3.399

12

340 − 210τ

0.6325

13

−361 + 224τ

4.278

Mdodeca

•

4 ∗

Micosi

•

• • • •

• •

various atomic species in any way we wish to obtain a chemical ordering in the cluster that is consistent with quasiperiodicity and all the overlaps between the clusters considered. We exemplify this process with the case of the atomic structure of i-AlPdMn. The icosahedral AlPdMn phase has been the subject of extensive structural studies on high-quality samples using both X-ray and neutron diffractions. The main atomic model that is currently accepted is to due M . Boudard et al. [34] and is based on three spherical ASs located at n, n , and bc of volumes close to those of the main triacontahedra discussed in the previous sections. The chemical ordering is taken into account by decomposing the spheres into concentric spherical shells, each decorated with a given atomic species. The radii of the shells are fitted to reproduce the diffraction data and the stoichiometry as well as possible. Although it gives quite satisfactory results with respect to diffraction, the model inherits from the overlaps of spherical ASs in E⊥ the generation of a few percent of atoms at distances from one another that are too short. Also, the number of local configurations of the atomic clusters increases dramatically with the cluster size because of the many overlaps of spherical shells that occur in superimposing neighboring ASs. Finally, because of the isotropy of the ASs, it is difficult to understand how collective phason flips can appear to lead to the

48


Table 2.4. Cell decomposition of An , with respect to B and M clusters associated with Fig. 2.20. The notation is the same as in Table 2.3 Cell # Volume

at%

Mctr Mico Bdodeca (1) Bdodeca (2) Micosi Mdodeca

•

1

−71 + 44τ

0.575

2

68 − 42τ

0.1265 •

∗

•

3

68 − 42τ

0.1265

4

13 − 8τ

0.1656

5

−71 + 44τ

0.5750

•

6

−3 + 2τ

0.7015

•

7

−370 + 230τ 6.382

•

8

340 − 210τ

0.6325

•

9

457 − 282τ

2.123

10

−427 + 266τ 10.09

11

397 − 244τ

6.5368

•

12

−366 + 228τ 8.6524

•

13

23 − 14τ

1.033

14

340 − 210τ

0.6325

15

−361 + 224τ 4.278

16∗

−32 + 20τ

• • • •

•

• •

• • •

• •

•

• •

1.072

various approximant crystal structures that are present in the equilibrium phase diagram of the (Al, Pd, Mn) system. A very interesting attempt to mimic these spherical shells with polyhedra has been proposed by Yamamoto et al. [29], and is based on a τ 3 inflation of the primitive Penrose 3D tiling. The atomic surfaces An and Abc of the model are shown in Fig. 2.21. The AS An (not shown here) is designed to be adjusted exactly with Abc and An , to provide the shortest allowed distances along the 5-fold directions. However, overlaps occur along the 2-fold directions that generate short interatomic distances in the model. An alternative solution has been recently presented by Shramchenko et al. [39], based on the former cell decomposition with respect to B and M clusters. The idea is to find a natural chemical decoration of the cells that fits the diffraction data reasonably well. Several models have been considered, of which the simplest is the following. Because of the absence of a magnetic moment in this alloy, we expect manganese atoms to be distributed at relatively large distances from one another. This implies, that the Mn ASs are confined inside the canonical triacontahedron. From the qualitative study of Trambly and Mayou [38], who obtain localized states for manganese atoms located on an icosahedron with edge 0.48 nm, we distribute Mn on one of


(a)

49

(b)

Fig. 2.21. left: Atomic surfaces An and Abc (volume 1) used in the model of Yamamoto et al. of i-AlPdMn (left) compared with the basic An and Abc (right)

the M or M icosahedron orbits. Observing that the corresponding cells have a volume of 12(2τ − 3), i.e. a fraction 8.4178% of the total structure, we have then exhausted the Mn distribution. We now notice that the Pd content of the nominal alloy is close to a 30/12 ratio with respect to Mn. The simplest decoration is to distribute Pd on one of the icosidodecahedra of M or M ; this exhausts the Pd content. All remaining cells of the basic ASs are filled with Al. Comparing this starting atomic structure with the diffraction data leads to the conclusion that Mn should be distributed on the M icosahedra and Pd on the M icosidodecahedra. It is also very clear from Fourier difference calculations that Pd atoms are also distributed on Abc and that Mn should have a fraction of atoms on the M centers (also possibly on the M center, in another variant of Shramchenko’s models). This is achieved by considering the chemical decomposition presented in Fig. 2.22. This basic model has

n

bc

Pd Al

n’

Mn

Al

Mn Al

Pd

Al

Fig. 2.22. Chemical cell decomposition of An , Abc , and An proposed by Shramchenko et al. [39] as a plausible starting-point structure for i-AlPdMn. The central cell (c1 in Table 2.3) of An can be filled by Mn as described

50


(a)

(b)

(c)

Fig. 2.23. Chemical decoration of the basic Shramchenko model for the three kinds of clusters. The most probable configurations are: (a) B-cluster, made of a central Pd, a complete Al icosahedron and a mixture of all species on the external dodecahedron; (b) M cluster, made of a central Mn, 7 Al on the partial dodecahedron, a large Mn icosahedron, and an Al icosidodecahedron; (c) M cluster, made of a central Mn, Al on both the partial dodecahedron and the large icosahedron, and Pd on the icosidodecahedron

a composition Al69.92 Pd21.72 Mn08.36 , and a density of 4.98◦ / cm3 , and fits the neutron diffraction spectrum for 217 independent reflections with R-factors, Rf = 0.079597 (sphere 0.075532) and Ri = 0.028279 (sphere 0.024794) that are comparable – slightly higher, but with no fitting parameters – to those of the reference sphere model of Boudard et al. By comparison with Boudard’s X-ray spectrum, the model leads to Rf = 0.087338 (sphere 0.056901) and Ri = 0.146648 (sphere 0.111858). More sophisticated models based on increasing the Pd content on Abc are in progress and will not be discussed here. The atomic structure in physical space can be directly described directly by simple examination of Tables 2.3 and 2.4: • Mn: from An , possibly cell 1, M center; from An , cell 1, M center, and cell 2, 5, 6, and 7 M icosahedron sites that do not belong to an M icosidodecahedron • Pd: from Abc , B center; from An , cells 10 and 12, M icosidodecahedron sites that are shared with a B dodecahedron. • Al: all other sites. Typical chemical configurations of the model are given in Fig. 2.23. The manganese atoms that are not centers of M and M clusters are distributed on the large M icosahedra, with a fraction (−376 + 234τ )/(−36 + 24τ ) = 92.4858% of these icosahedra being fully occupied by Mn and (340 − 210τ )/(−36 + 24τ ) = 7.51416% with only 7 atoms of Mn; this leads to an average Mn occupancy of Z¯ = 11.6243 Mn atoms on these icosahedra (see Fig. 2.24). This might be an important feature of the structure with respect to electronic structure and localization.


51

Fig. 2.24. The elementary Mn cluster has two configurations: (a) the complete M icosahedron, representing 92.5% of the configurations, (b) a 7 atom configuration made of a centered pentagon and one isolated Mn atom opposite to the center of the pentagon; this last configuration has a much smaller frequency of only 7.5%

(a)

(b)

Fig. 2.25. (a) Mn and (b) Pd networks in the Shramchenko model: Mn atoms are distributed on large icosahedra connected by squares; Pd atoms are distributed on large icosidodecahedra that are shared with B dodecahedra

Disregarding M , M , and B centers, Mn and Pd atoms are distributed on relatively simple networks, as shown in Fig. 2.25. Mn icosahedra are connected by square bridges, thus introducing no other basic interatomic distance. Pd atoms form a network of icosidodecahedra with elementary edges of length 0.29 nm connected by additionnal edges with the same length.

2.6 Covering Clusters: the XB Cluster The previous results show that genuine B and M clusters fail to provide unique covering clusters for deterministic 6D models of i-AlPdMn or i-AlCuFe, although the covering is almost perfect in the case of approximant phases [20, 21]. The domains of their centers cannot, however, be extended without introducing incomplete shells. It therefore seems reasonable to con-

52


sider extensions of B or M clusters, obtained by adding exterior shells in order to cover more points of the structures. From the various shells given in Table 2.1, we shall consider extended B clusters (XB) of 6 shells centered on bc sites. This extension is obtained from the B cluster by adding the next four distances around the bc sites, leading to a cluster, denoted by XB for short, with 6 shells. This cluster is shown in Fig. 2.26 and is defined by (see Tables 2.5 and 2.6): • • • • •

icosahedron: bc − n at (1, 1, −1, 1, −1, 1)/2; dodecahedron: bc − n at (−1, 1, 1, 1, 1, 1)/2; truncated icosahedron 1 (T I) (*): bc − n at (1, −1, 1, 1, 1, 3)/2; truncated icosahedron 2 (T I ) (*): bc − n at (−1, 1, 3, −1, −1, −1)/2; triacontahedron (T r): T r5: bc − n at (1, 1, 1, −1, −1, 1)/2, T r3: bc − n at (1, 1, 1, 1, −1, 1)/2.

T I refers to the first incomplete truncated icosahedron of n sites, and T I is a second incomplete truncated icosahedron of n sites. The last two shells T r3 and T r5 form a fully occupied triacontahedron. The mean occupation number of incomplete shells can be calculated for XB clusters. We obtain Z¯T I = −15+24τ (23.83) for T I and Z¯T I = 181−98τ (22.43) for T I (see Tables 2.5 and 2.6). These shells have close radii, and one can check that the occupancy of a site on one shell excludes the occupancy of the nearest site on the other shell. The noninteger occupation numbers of the shells indicate that the filling depends on the location of the cluster in the structure. This precludes the existence of a unique covering cluster. Nevertheless, we can evaluate the covering ratio of these variable clusters by considering all points belonging to the variable shells. This amounts to calculating the filling of the three atomic surfaces An , An , and Abc in perpendicular space, which is obtained by convenient translations of the polyhedra defining the centers (τ −1 T R). The covering is complete with the 6 shells defined above.

Fig. 2.26. The XB cluster proposed by one of us (MD) is an extension around the B cluster, including two partially occupied truncated icosahedra and a fully occupied external triacontahedron. Atom distribution (shell radius): 1 (0), 12 (at 0.618 a3D ), 20 (at 0.911 a3D ), 60∗1 (at 1.219 a3D ), 60∗2 (at 1.391 a3D ), 20 (at 1.473 a3D ), 12 (at 1.618 a3D )


53

Table 2.5. XB cluster about bc nodes: occupation of n nodes Sa 12 60∗1 20

r/a6D K √ 3−τ √ 7−τ √ 3 + 3τ

r/a3D N b

nc

Σnd

N e nf

0.618

12

12 (8τ +5)

12 (8τ +5)

12

12 (8τ +5)

12 (8τ +5)

1.22

23.83 0.698

0.931

22

0.669

0.902

1.473

20

1

20

0.393

0.971

(≈ 0.669)

0.393

Σng

a

The type of shell with radius r. N is the mean occupation number in variable clusters. c Relative filling of An for a variable cluster. d Cumulative filling of An for a variable cluster. e N is the occupation number in a fixed cluster. f Relative filling of An for a fixed cluster. g Cumulative filling of An for a fixed cluster. b

Table 2.6. XB cluster about bc nodes: occupation of n nodes Sa 20 60∗2 12

r/a6D K √ 3 √ 7 √ 3 + 4τ

r/a3D N

b

n

c

Σn

d

N

e

n

f

Σn

g

(20−4τ ) (6τ +5)

20

(20−4τ ) (6τ +5)

(20−4τ ) (6τ +5)

23.36 0.692

0.968

19

0.672

0.952

12

1

12

0.276

0.984

0.911

20

1.391 1.618

(20−4τ ) (≈ (6τ +5)

0.276

0.920)

a

The type of shell with radius r. N is the mean occupation number in variable clusters. c Relative filling of An for a variable cluster. d Cumulative filling of An for a variable cluster. e N is the occupation number in a fixed cluster. f Relative filling of An for a fixed cluster. g Cumulative filling of An for a fixed cluster. b

Considering the variable occupation numbers of particular shells, one may ask if this not merely a consequence of different overlaps between neighboring clusters according to their positions. These variations of the occupation number occur because the translations of the τ −1 T R domain, which defines the centers in Abc , are not entirely contained in the ASs. Thus we may search, for each variable shell, a constant subset of the shell (up to icosahedral rotations) which is always fully occupied by “sure” atoms. In order to take advantage of the icosahedral symmetry, we consider clusters the centers of which fall in a given fundamental domain of Abc . Then, for each shell, we can determine which translations of this tetrahedron are entirely contained in An or An . This defines a subset of the shell which is fully occupied when the center runs over the tetrahedron. All other configurations of sure atoms in this shell follow from rotations of the icosahedral group. This leads to fixed minimal clusters with the following properties.

54


The T I shell contains 22 sure atoms and the T I shell contains 19 sure atoms. With this unique cluster (up to icosahedral symmetry) of 106 atoms, the filling of An is 98.39% and that of An is 97.06%, which leads to a total covering ratio of 97.73%, including Abc (Tables 2.5 and 2.6). Considering the variable clusters again, we observe that the T I and T I shells have strong overlaps with the neighboring B clusters. Indeed, the orbit T I contains a fraction 65.164% ((12 − 5τ )/6) of the atoms of the B icosahedra, and T I contains 70.02% ((3 + 4τ )/(20 − 4τ )) of the atoms of the B dodecahedra. The orbit T I gives 3 different configurations, containing from 22 to 24 atoms. The configuration corresponding to the maximum number of atoms (24) has, by far, the highest frequency (88.85% of the XB clusters). The atoms of the T I of a given XB cluster belong, on average, simultaneously to 1.88 others ((−25 + 30τ )/(19 − 4τ )). They represent a fraction 69.81% ((19 − 4τ )/(5 + 8τ )) of the atoms generated by An . The orbit T I gives 6 different configurations, with 20 to 23 atoms. Here also, the configuration C1 , corresponding to the maximum number of atoms, has the highest frequency (65.25%). Atoms of T I of a given XB cluster belong, on average, to 2.19 others ((−10 + 20τ )/(−6 + 10τ )). They represent a fraction 69.21% ((−6 + 10τ )/(5 + 6τ )) of the atoms generated by An . The last shell – containing a 5-fold and a 3-fold orbit – is the canonical triacontahedron of the primitive Penrose 3D tiling and is fully occupied (32 atoms). It overlaps with the neighboring B clusters in the following way. The 3-fold shell of the triacontahedron contains 46.06% ((3 − τ )/3) of the atoms of the B icosahedra, and the 5-fold orbit contains 26.49% ((23 − 12τ )/(20−4τ )) of those of the B dodecahedra. The surface An is decomposed into 12 cells by the 3-fold orbit, the largest one corresponding to atoms that do not belong to any triacontahedron of the XB cluster. Hence a fraction 39.32% ((20 − 8τ )/(5 + 8τ )) of the atoms generated by An belong to at least one triacontahedron. Each of these atoms belongs to 2.736 ((−26 + 28τ )/(20 − 8τ ))) XB clusters on average. Also, 10.45% ((31 − 18τ )/(5 + 6τ )) of the atoms generated by An belong simultaneously to a T I and a 3-fold orbit of the triacontahedron. The 5-fold shell splits An into 13 cells, the largest one corresponding to atoms that do not belong to any triacontahedron of the XB cluster. The atoms of the 5-fold orbit represent 27.57% ((17 − 18τ )/(5 + 6τ )) of the atoms generated by An . Each of these atoms belongs to 3.02 ((9 + 2τ )/(17 − 8τ )) XB clusters on average. Atoms generated by Tn never belong simultaneously to a T I and a 5-fold orbit of the triacontahedron. The T I orbit is distributed inside the triacontahedron along 3-fold directions of the closest atoms of the 5-fold orbit of the triacontahedron. The T I orbit is distributed on the main diagonal facets of the triacontahedron in the standard ratio τ between the two opposite vertices of the facets. The whole XB cluster can be decomposed into the standard set of prolate and oblate


1

7

2

8

3

4

5

6

9

10

11

12

55

Fig. 2.27. The AS Abc decomposes into 12 different cells corresponding to the 12 different configurations of the XB cluster: cells #1, 20 − 12τ (0.583592); #2, −42 + 26τ (0.0688837); #3, 68 − 42τ (0.0425725); #4, 68 − 42τ (0.0425725); #5, −42 + 26τ (0.0688837); #6, −165 + 102τ (0.0394669); #7, −110 + 68τ (0.0263112); #8, 178−110τ (0.0162612); #9, −110+68τ (0.0263112); #10, 123−76τ (0.0294169); #11, 68 − 42τ (0.0425725); #12, −55 + 34τ (0.0131556)

rhombohedra of the canonical 3D Penrose tiling with additional atoms decorating some of the facets and 3-fold axes. There are 6 different decorations for the oblate rhombohedron and 14 for the prolate. As previously mentioned, regrouping the cells generated by all six shells of the XB cluster leads to a full covering of the basic atomic surfaces: the XB cluster defines a template cluster with an average number of 111.265 (231 − 74τ ) atoms (ranging from 109 to 112). Any atom of the structure belongs to one at least such template, centered on a bc site or sites. Performing the complete cell decomposition, projecting all six orbits of atomic surfaces properly located in 6D space onto Abc , leads to the 12 cells shown in Fig. 2.27. The XB cluster therefore has 12 different configurations (irrespective of the point symmetry operations), where the most important configuration (with almost 60% of such clusters), associated with cell #1, contains the maximum number of 112 atoms. Here too, we can perform the complete cell decomposition of both An , shown in Fig. 2.28) and Table 2.7 and An , shown in Fig. 2.29 and Table 2.8. Because it has several configurations, the XB cluster is not a covering cluster sensu stricto; since its local atomic decoration varies (although these configurations share 106 atoms) from site to site on the two partially T I and T I orbits. It is not to be compared with the covering cluster discussed by Gummelt [7] for Penrose tilings. This latter is unique and satisfies specific overlap rules – equivalent to matching rules – that ensure that the tiling is quasiperiodic if they are satisfied everywhere. In our present case, the template cluster is not unique, and no covering rules, if any, can be deduced from our simple geometrical analysis.

56


1

2

3

4

5

8

9

10

13

14

15

18

22

19

6

11

16

20

23

7

12

17

21

24

Fig. 2.28. The AS An decomposes into 24 different cells, corresponding to the various locations of the generated atomic sites with respect to XB (extracted from F. Puyraimond, PhD thesis). Cells #1, 36−22τ (0.403252); #2, −16+10τ (0.18034); #3, −42 + 26τ (0.0688837); #4, 26 − 16τ (0.111456); #5, −6 + 4τ (0.472136); #6, −16 + 10τ (0.18034); #7, 46 − 28τ (0.695048); #8, −110 + 68τ (0.0263112); #9, 124 − 76τ (1.02942); #10, −66 + 42τ (1.95743); #11, −3 + 2τ (0.236068); #12, 21 − 12τ (1.58359); #13, 13 − 8τ (0.0557281); #14, −3 + 2τ (0.236068); #15, 13 − 8τ (0.0557281); #16, −3 + 2τ (0.236068); #17, 30 − 18τ (0.875388); #18, −24 + 18τ (5.12461); #19, −11 + 8τ (1.94427); #20, 33 − 20τ (0.63932); #21, −38 + 24τ (0.832816); #22, −3 + 2τ (0.236068); #23, 7 − 4τ (0.527864); #24, −3 + 2τ (0.236068)

2 Covering Clusters in Icosahedral Quasicrystals Table 2.7. An cell decomposition for the XB cluster shown in Fig. 2.28 Cell #

Volume

Total at%

Bico

TI

T ria3

Total number of atoms

1

36 − 22τ

1.198

0

0

7

7

2

−16 + 10τ

0.536

0

0

6

6

3

−42 + 26τ

0.2047

0

0

6

6

4

26 − 16τ

0.3312

0

0

5

5

5

−6 + 4τ

1.403

0

0

4

4

6

−16 + 10τ

0.536

1

0

4

5

7

46 − 28τ

2.065

1

0

5

6

8

−110 + 68τ

0.0782

1

0

4

5

9

124 − 76τ

3.059

1

0

3

4

10

−66 + 42τ

5.816

1

0

2

3

11

−3 + 2τ

0.7015

1

0

0

1

12

21 − 12τ

4.7057

1

1

1

3

13

13 − 8τ

0.1656

1

0

1

2

14

−3 + 2τ

0.7015

0

2

1

3

15

13 − 8τ

0.1656

0

3

1

4

16

−3 + 2τ

0.7015

1

1

0

2

17

30 − 18τ

2.601

1

1

0

2

18

−24 + 18τ

15.228

1

2

0

3

19

−11 + 8τ

5.777

0

2

0

2

20

33 − 20τ

1.033

0

3

0

3

21

−38 + 24τ

2.4747

0

3

0

3

22

−3 + 2τ

0.7015

0

2

0

2

23

7 − 4τ

1.568

0

1

0

1

24

−3 + 2τ

0.7015

0

4

0

4

57

58


1

2

3

4

9

5

10

13

11

14

17

15

18

21

24

6

19

7

8

12

16

20

22

23

25

26

Fig. 2.29. The AS An decomposes into 26 different cells, corresponding to the various locations of the generated atomic sites with respect to XB (extracted from F. Puyraimond, PhD thesis). Cells #1: 13−8τ (0.0557281); #2, −55+34τ (0.0131556); #3, 68 − 42τ (0.0425725); #4, −42 + 26τ (0.0688837); #5, 13 − 8τ (0.0557281); #6, −3 + 2τ (0.236068); #7, 13 − 8τ (0.0557281); #8, −3 + 2τ (0.236068); #9, −6 + 4τ (0.472136); #10, 33 − 20τ (0.63932); #11, 13 − 8τ (0.0557281); #12, −38 + 24τ (0.832816); #13, −55 + 34τ (0.0131556); #14, −2 + 2τ (1.23607); #15, −2 + 2τ (1.23607); #16, 68 − 42τ (0.0425725); #17, 26 − 16τ (0.111456); #18, −32 + 20τ (0.36068); #19, 26 − 16τ (0.111456); #20, −11 + 8τ (1.94427); #21, −16 + 10τ (0.18034); #22, −66 + 42τ (1.95743); #23, 42 − 24τ (3.16718); #24, −16 + 10τ (0.18034); #25, 30 − 18τ (0.875388); #26, 7 − 4τ (0.527864)


59

Table 2.8. An cell decomposition for the XB cluster shown in Fig. 2.29. The notation is the same in Table 2.7 Cell #

Volume Total at% Bdodeca T I T ria5 Total number of atoms

1

13 − 8τ

2 3

0.1656

0

0

12

12

−55 + 34τ

0.039

0

0

10

10

68 − 42τ

0.1265

0

0

9

9

4

−42 + 26τ

0.2047

0

0

8

8

5

13 − 8τ

0.1656

0

0

7

7

6

−3 + 2τ

0.7015

0

0

6

6

7

13 − 8τ

0.1656

1

0

6

7

8

−3 + 2τ

0.7015

0

0

6

6

9

−6 + 4τ

1.403

2

0

4

6

10

33 − 20τ

1.033

2

0

3

5

11

13 − 8τ

0.1656

1

0

3

4

12

−38 + 24τ

2.4747

2

0

2

4

13

−55 + 34τ

0.039

1

0

2

3

14

−2 + 2τ

3.673

2

0

1

3

15

−2 + 2τ

3.673

2

1

0

3

16

68 − 42τ

0.1265

1

0

1

2

17

26 − 16τ

0.3312

2

0

0

2

18

−32 + 20τ

1.0718

1

0

0

1

19

26 − 16τ

0.3312

1

1

0

2

20

−11 + 8τ

5.7775

2

2

0

4

21

−16 + 10τ

0.5359

1

2

0

3

22

−66 + 42τ

5.8166

1

2

0

2

23

42 − 24τ

9.41145

1

3

0

4

24

−16 + 10τ

0.536

0

3

0

3

25

30 − 18τ

2.6013

1

2

0

3

26

7 − 4τ

1.5686

0

2

0

2

60


2.7 Conclusion We have performed here an exhaustive study of the basic clusters encountered in the F-type icosahedral phases i-AlCuFe and i-AlPdMn, described by the simplest three main ASs that reasonably fit the diffraction data. We have used the cell decomposition (intersection/union of polyehdra in E⊥ ) method to list all the local configurations (r-atlas) of these structures up to the extended B cluster previously introduced by one of us (MD). It turned out that, with respect solely to geometry, the XB cluster is not a covering cluster although most of the atoms of the structure can be described by only one of its 12 configurations. Considering that the chemical ordering – an example of which has been given here with a tentative model for i-AlPdMn – introduces far more different local environments, we can conclude that a unique chemically decorated covering cluster is unlikely to be found in real icosahedral phases.

References 1. B. Gr¨ unbaum, G. C. Shephard: Tilings and Patterns (Freeman, New York 1987) 23 2. D. Shechtman, I. Blech, D. Gratias, J. W. Cahn: Phys. Rev. Lett. 53, 1951 (1984) 23 3. S. E. Burkov: J. Phys. I France 2, 695 (1992); Phys. Rev. Lett. 67, 614 (1991) 23 4. C. Janot, M. de Boissieu: Phys. Rev. Lett. 72, 1674 (1994); C. Janot: Phys. Rev. B 53, 181 (1996); C. Janot: J. Phys. Cond. Matter 9, 1493 (1997); C. Janot, J. Patera: J. Non Cryst. Solids 234, 234 (1998) 23 5. H. C. Jeong, P. J. Steinhardt: Phys. Rev. Lett. 73, 1943 (1994) 23 6. F. G¨ ahler: Phys. Rev. Lett. 74, 334 (1995) 23 7. P. Gummelt, Construction of Penrose Tilings by a Single Aperiodic Set in: Proceedings of the 5th International Conference on Quasicrystals, C. Janot, R. Mosseri (Eds.), World Scientific, Singapore 1995, pp. 84; Geometriae Dedicata 62 (1996) 1 24, 55 8. M. Duneau, Quasiperiodic Structures with a Unique Covering Cluster in: Proceedings of the 5th International Conference on Quasicrystals, C. Janot, R. Mosseri (Eds.), World Scientific, Singapore 1995, pp. 116 24 9. F. G¨ ahler, Cluster Interactions for Quasiperiodic Tilings in: Proceedings of the 6th International Conference on Quasicrystals, S. Takeuchi, T. Fujiwara (Eds.), World Scientific, Singapore 1998, pp. 95 24 10. F. G¨ ahler, H. C. Jeong: J. Phys. A: Math. Gen. 28, 1807 (1995) 24 11. H. C. Jeong, P. J. Steinhardt: Phys. Rev. B 553, 1493 (1997) 24 12. S. I. Ben-Abraham, F. G¨ ahler: Phys. Rev. B 60, 860 (1999) 24 13. V. Elser, Phil. Mag. B73 (1996) 641 and Random Tiling Structure of Icosahedral Quasicrystals in: Proceedings of the 6th International Conference on Quasicrystals, S. Takeuchi, T. Fujiwara (Eds.), World Scientific, Singapore 1998, pp. 19 24, 28, 39, 44


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14. A. Katz, D. Gratias: J. Non-Crystalline Solids 153–154, 187–195 (1993); and Chemical Order and Local Configurations in AlCuFe-type Icosahedral Phase In:Proceedings of the 5th International Conference on Quasicrystals, C. Janot, R. Mosseri (Eds.), World Scientific, Singapore 1995, pp. 164 24, 28 15. E. Cockayne, R. Phillips, X. B. Kan, S. C. Moss, J. L. Robertson, T. Iskimaza, M. Mori: J. Non-Crystal. Solids 153–154, 140 (1993) 28 16. G. Bergman, L. T. Waugh, L. Pauling: Acta Cryst. 10, 254 (1957) 25 17. M. Audier, J. Pannetier, M. Leblanc, C. Janot, J. Lang, B. Dubost: Physica B 153, 136 (1988) 25 18. M. Cooper, K. Robinson: Acta Cryst. 20, 614 (1966) 25 19. M. Duneau, C. Oguey: Journal de Phys. 50, 135 (1989) 26 20. L. Beraha, M. Duneau, K. Klein, M. Audier: Phil. Mag. A 76, 587 (1997) 26, 51 21. L. Beraha, M. Duneau, K. Klein, M. Audier: Phil. Mag. A 78, 345 (1998) 26, 51 22. K. Sugiyama, N. Kaji, K. Hiraga and T.Ishimasa, Zeitschr. f¨ ur Krist. 213 (1998) 90; Zeitschr. f¨ ur Krist. 213 (1998) 168 26 23. A.-P. Tsai, A. Inoue, T. Masumoto: Jpn. J. Appl. Phys. 26, L1505–L1507 (1987); A.-P. Tsai, A. Inoue, Y. Yokoyama, T. Masumoto: Mater. Trans. Jpn. Inst. Met. 31, 98 (1990) 26 24. M. Boudard, M. de Boissieu, C. Janot, G. Heger, C. Beeli, H.-U. Nissen, H. Vincent, R. Ibberson, M. Audier, J. M. Dubois: J. Phys.: Condens. Matter 4, 10149 (1992) 25. M. Cornier-Quiquandon, A. Quivy, S. Lefebvre, G. Elkaim, D. Gratias: Phys. Rev. B 44, 2071 (1991) 26. M. Boudard, H. Klein, M. de Boissieu, M. Audier, H. Vincent: Phil. Mag. A 74, 939 (1996) 27. M. Mihalkovic, P. Mrafko: Phil. Mag. Lett. 69, 85 (1994) 25 28. J. W. Cahn, D. Shechtman, D. Gratias: J. Mater. Res. 1, 13–26 (1986) 30 29. A. Yamamoto, A. Sato, K. Kato, A.P. Tsai, T. Masumoto: Mater. Sci. Forum 150–151 211–222 (1994) 48 30. C. Oguey, M. Duneau, A. Katz: Commun. Math. Phys. 118, 99–118 (1988) 32 31. P. Kramer: J. Math. Phys. 29, 516–524 (1988) 32 32. V. I. Arnol’d: Physica D 33, 21–24 (1988) 32 33. S. Lyonnard, G. Coddens, Y. Calvayrac, D. Gratias: Phys. Rev. B 53, 3150– 3160 (1996) 43 34. M. Boudard, M. de Boissieu, C. Janot, J. M. Dubois, C. Dong: Phil. Mag. Lett. 64, 197–206 (1991); M. Boudard, M. de Boissieu, C. Janot, G. Heger, C. Beeli, H. U. Nissen, H. Vincent, M. Audier, J. M. Dubois, J. Non-Crystal. Solids 153, 5–9 (1993) 27, 47 35. C. Janot, M. de Boissieu: Phys. Rev. Lett. 72, 1674 (1994) 41 36. P. Kramer, Z. Papadopolos, W. Liebermeister, Atomic Positions in Icosahedral Quasicrystals In: Proceedings of the 6th International Conference on Quasicrystals, S. Takeuchi, T. Fujiwara (Eds.), World Scientific, Singapore 1998, pp. 71, 76 39 37. Z. Papadopolos, P. Kramer, W. Liebermeister, Atomic Positions for the Icosahedral F-Phase Tiling In: Proceedings of the International Conference on Aperiodic Crystals, M. de Boissieu, J.-L. Verger-Gaugry, R. Currat (Eds.), World

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Scientific, Singapore, 1997, pp. 173–181; Z. Papadopolos, P. Kramer, G. Kasner, D. E. B¨ urgler, Mater. Res. Soc. Symp. Proc., 553 (Materials Research Society), 231–236 (1999) 41 38. G. Trambly and D. Mayou, Phys. Rev. Lett. 85, 3273–3276 (2000) 48 39. N. Shramchenko et al. , unpublished 27, 48, 49

3 Generation of Quasiperiodic Order by Maximal Cluster Covering Franz G¨ ahler, Petra Gummelt, and Shelomo I. Ben-Abraham

3.1 Introduction In quasicrystals, certain structural motifs occur very frequently, and sometimes even cover the entire structure. This property is particularly visible in high-resolution electron micrographs (HREMs) of decagonal quasicrystals. In this chapter, these important structural motifs will be called clusters.1 On the basis of the observation that at least some quasicrystals can be regarded as being covered by a single kind of cluster, Burkov [1] was one of the first to propose a structure model which was explicitly given as a covering with overlapping copies of a single cluster. The Burkov model remained mostly at the descriptive level. However, if a cluster occurs so frequently that it covers the entire structure, it must certainly be an energetically favorable local configuration. Moreover, the many overlaps, which necessarily occur if the entire structure is covered, impose constraints on the possible relative positions and orientations of the clusters. Overlapping clusters will share atoms in the overlap, and thus have to agree in the overlap region. This can severely limit the number of possible overlaps. Generally speaking, the constraints on the possible overlaps create order. On the basis of such ideas, Jeong and Steinhardt [2] have formulated a simple ordering principle which can produce a perfectly ordered quasicrystal. They suggested that a quasicrystal tries to maximize the density of the most favorable local configurations (clusters), and thus effectively expels all the other local configurations which do not occur in the perfect structure. These authors could show by Monte Carlo simulations that favoring just the few most important local motifs can produce a perfectly ordered Penrose tiling. This ordering principle will be called the cluster density maximization principle. Having such a simple model for the energetics of a quasicrystal, which is still able to produce a perfectly ordered structure as its ground state, is a clear advantage over earlier models based on matching rules [3], or local rules [4], as they are sometimes called. Matching rules consist of an atlas of all those local configurations, up to a radius R, which can occur in the ordered quasicrystal structure. This atlas must have the additional property that every structure containing only local configurations from the atlas is perfectly ordered. The minimal radius R for which an atlas with these properties exists 1

Note that these clusters are embedded in the surrounding quasicrystal, and should not be confused with finite clusters in a vacuum.

P. Kramer (Ed.): Convering of Discrete Quasiperiodic Sets, STMP 180, 63–95 (2002) c Springer-Verlag Berlin Heidelberg 2002

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is called the range of the matching rules. An atlas with these properties (and thus matching rules) need not always exist for any R, but there are many examples of quasiperiodic tilings which do have matching rules [5, 6, 7, 8]. Quasicrystals can often be regarded as being an atomic decoration of such a tiling. There is an obvious way to use matching rules to construct simple energetics for the corresponding tiling model of a quasicrystal. One just has to make sure that all allowed local configurations (those occuring in the atlas) are energetically more favorable than all other possible local configurations. However, such an atlas usually contains very many local configurations, and trying to realize such interactions for the atoms which decorate the tiles is a hopelessly complicated task. The cluster model approach of Jeong and Steinhardt [2] suggests that it is not necessary that all allowed local configurations have lower energy than all disallowed local configurations. Preferring just the most important motifs and ignoring all the rest can be enough to produce an ordered quasicrystal. This is a tremendous simplification over the more naive matching-rule approach. Nevertheless, it is still a non trivial task to select the right clusters, which must produce an ordered ground state. Such clusters must be sufficiently restrictive on the overlaps they admit. In this respect, asymmetric clusters work much better, and it is interesting to note that asymmetric clusters seem to be preferred by the electronic structure in decagonal quasicrystals [9]. Moreover, the clusters must be big enough. In fact, it has been shown [10] that their size can be no smaller than the range of the matching rules. While Jeong and Steinhardt suggested that one should maximize the density of certain well-chosen clusters, a slightly different approach was proposed by Gummelt [11, 12], who presented a decagon with associated overlap rules having the property that every covering of the plane with this decagon obeying the overlap rules is isomorphic to a perfectly ordered Penrose tiling. This decagon, together with its overlap rules, will in the following be called the aperiodic decagon. This result was an important advance in two respects. Firstly, a single cluster (the decagon) with overlap rules was sufficient to produce a perfectly ordered structure, and secondly, the cluster density maximization principle was replaced by a cluster-covering principle, which, from a mathematical point of view, is much easier to handle. Indeed, the properties of the aperiodic decagon can all be proved mathematically, whereas the results of Jeong and Steinhardt [2] were supported only by simulations. The fact that the properties of the aperiodic decagon are mathematically well established probably had a further effect, perhaps together with the extraordinary beauty of the model. It has encouraged many researchers [13, 14, 15, 16, 17] to interpret their HREM images using models based on the aperiodic decagon, thus providing an extraordinary impetus to the field. The aperiodic decagon can also be used together with the cluster density maximization principle. Jeong and Steinhardt [18] could show that the perfect Penrose tiling has a decagon cluster density no smaller than any other tiling, whether it is covered or not.

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As we have already indicated above, the cluster density maximization principle is a particularly efficient realization of the matching rules of a tiling or quasicrystal. The same is obviously the case for the cluster-covering principle also. This idea can be pushed a little further. Many tilings have very complicated matching rules. This is particularly so for octagonal and dodecagonal tilings [6, 7, 8]. The matching rules of the tiling require decorations of the tiling which are not locally derivable [19] from the undecorated tiling [8]. There is, however, a simple subset of the matching rules for these tilings, the so-called alternation condition [20], which is also local for the undecorated tiling, and which enforces perfectly ordered tilings, albeit ones which in general have lower symmetry [21]. In this situation, a combination of the cluster density maximization principle and the cluster-covering principle can be used. In the first step, the alternation condition is enforced by a suitable cluster-covering principle. In the second step, the fully symmetric octagonal or dodecagonal tiling can then be selected as the one with the highest cluster density, but only out of the tilings which are completely covered by the cluster. This will be called the maximal cluster-covering principle. It can use relatively simply clusters at the cost of a more refined ordering principle. The maximal covering principle was used implicitly for octagonal tilings in [22], and more explicitly in [23]. Recently, it was applied also to dodecagonal tilings [24]. We therefore have three simple ordering principles for quasicrystals, which can produce a perfectly ordered quasicrystal. The structure of the quasicrystal is selected according to one of the following rules: • require that the structure maximizes the density of some well-chosen cluster(s); • require that the structure is covered by some well-chosen cluster(s); • require that the structure is covered by the cluster(s), and of such coverings take the ones with the highest cluster density. These ordering principles are clearly closely related, but conceptually distinct. In some cases, all three may be applied, but in other cases only the third one will work. Conceptually, they all have their advantages and disadvantages. If an energy function simply counts the number of (overlapping) clusters, there is clearly some double counting, which needs to be avoided or compensated. The covering principle may be better suited in this respect. It basically requires that every atom is contained in some energetically favorable local configuration. The maximal cluster-covering principle is a further refinement: it requires a cluster covering, but prefers among those the ones with the highest cluster density. In a covering with a higher cluster density, it is more likely that atoms are well inside the interior of some cluster, which might be better than just on the cluster surface. We should emphasize that the notion of a covering may have different definitions, depending on the context. In the context of a tiling, by a covering

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of the tiling we usually mean that all tiles of the tiling are covered by copies of a tile cluster, but there are other possibilities. For instance, a covering of a tiling could also mean that all vertices of the tiling are covered by a vertex cluster. The general concepts are very flexible, and one has to define in each particular case what exactly shall be the meaning of a covering. In the case of an atomic structure, a cluster covering usually means that every atom in the structure is part of at least one of the atom clusters. In other words, it is the set of atoms which is to be covered. Even with this understanding of a covering, there may still be variants of the covering concept. In this chapter, we require that two atom clusters completely agree in the overlap region. In other words, any atom in the overlap region must be part of both clusters involved. Of course, this requires a suitable definition of the overlap region. Other authors use a looser notion of matching in the overlap region of two clusters. In [9, 17, 18], for instance, cluster overlaps are admissible if not all atoms in the overlap region of two clusters are part of both clusters. This may make it difficult to compare different models, and statements that the overlap rules imposed by an atomic cluster are equivalent to, for example, those of the aperiodic decagon, have to be taken with a pinch of salt if it is not clearly defined under what conditions two overlapping clusters are compatible with one another. We should also note that our ordering principles do not specify which cluster is the right one. The emphasis of this article will be precisely on this question: which clusters can produce, by means of the above ordering principles, a perfectly ordered quasicrystal, or at least some other “interesting” class of structures, and which tilings (and related quasicrystals) admit such clusters. Our goal is therefore not merely to describe a given quasicrystal structure in terms of a cluster covering, but rather to determine the class of structures admitted by a given set of overlap rules, which is imposed by the internal structure of the clusters that have been chosen for the covering. It is clear that not every cluster we may choose can lead to a perfectly ordered quasicrystal. Much more typical is the converse. Very often, a cluster either cannot cover any interesting structure completely or can cover too many. The latter case can still be interesting. In such cases, the structures that can be covered are typically members of some super-tile random tiling ensemble [2, 22]. Super-tile random tilings consist of larger, inflated tiles (the supertiles), whose interior is ordered, but which are arranged randomly, forming a random tiling [25]. On a local scale, such structures look ordered; only at larger scales one can notice the disorder. Since real quasicrystals are not all perfectly quasiperiodic, clusters that select an entire super-tile random-tiling ensemble can be very relevant. It has even been proposed [26, 27] that one should relax overlap rules known to produce a perfectly ordered quasicrystal, in order to allow a larger class of structures, including ones containing some amount of disorder.


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After our principal tools are introduced in Sect. 3.2, most of the remainder of this chapter is a detailed discussion of the available results related to several two-dimensional quasiperiodic tilings, and the quasicrystal models built upon them. In particular, we shall discuss the decagonal Penrose tiling [5] and several other tilings related to it (Sect. 3.3), the octagonal Ammann–Beenker tiling [6, 28, 29] (Sect. 3.4), and the dodecagonal Socolar [6] and shield [30, 31] tilings (Sect. 3.5). Finally, we put these results into a more general context in Sect. 3.6, where we comment also on the case of the T¨ ubingen triangle tiling (TTT) [32]. Each of these models sheds some light on a different aspect of the general principles. Reviews of some parts of these results have appeared earlier [33, 34].

3.2 Important Concepts and Tools In this section we briefly review the principal concepts and tools needed to obtain the results presented in this chapter. The most important tool is certainly the concept of mutual local derivability [19], sometimes also called local equivalence. Mutual local derivability is a local equivalence relation between tilings, other discrete structures such as a Delone set, or the set of atom positions in a quasicrystal, including labels for the different chemical species. In order to be specific, we explain the concept for tilings. Consider two tilings, T1 and T2 , which are assumed to be embedded in the same space with fixed position, orientation, and scale. Tiling T1 is said to be locally derivable from tiling T2 if there exists a fixed δ < ∞ such that every patch PR,x (T1 ) of T1 , with radius R and center x, is uniquely determined by the corresponding patch PR+δ,x (T2 ). In this definition, the derivability radius δ must be independent of R and x. Loosely speaking, tiling T1 can be constructed from tiling T2 using local knowledge only. If T2 is also locally derivable from tiling T1 , possibly with a different derivability radius δ, the two tilings are said to be mutually locally derivable (MLD). Mutual local derivability clearly is a (local) equivalence relation. Another important equivalence concept for discrete structures such as tilings is that of local isomorphism (LI). Two tilings are said to be locally isomorphic if any finite patch of one tiling occurs also somewhere in the other (in the same orientation), and vice versa. All tilings locally isomorphic to a given one form the local isomorphism class (LI class) of that tiling. It is immediately clear from the definitions of local isomorphism and mutual local derivability that if two tilings are MLD, then this equivalence relation can be extended to a bijection between the respective LI classes of the tilings. For each tiling T1 in the first LI class, there exists a unique tiling T2 in the second LI class, such that the two tilings are MLD. Abusing the language a bit, we then say that the two LI classes are MLD. The concept of mutual local derivability can even be extended to more general classes of tilings. One can think of two ensembles of random tilings, such that for each tiling in one

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ensemble there is a unique partner tiling in the other ensemble which is MLD with it, and vice versa. We then say that the two ensembles are MLD. Note that in the case of mutual local derivability between such general ensembles of tilings, one should also require that the derivability radius δ is uniformly bounded over the ensembles. This extra requirement is not necessary for the LI class of a tiling, where δ is automatically constant over the LI class. In many examples, the concept of mutual local derivability can be used in this way to relate a new random-tiling ensemble, or a new random-covering ensemble, to other random-tiling ensembles already known from the literature. If two ensembles of tilings are MLD, many properties depending only on local information about the tilings can easily be transferred from one ensemble to the other. An important example of such a property is the existence of matching rules. A tiling T (or other discrete structure) is said to have matching rules of radius R if any other tiling with the property that all its patches of radius R occur also in T is locally isomorphic with T . In other words, the LI class of T is completely determined by the R-patches of T for some finite R. Note that such a finite atlas can never distinguish between different members of an LI class. Matching rules are therefore always matching rules for an LI class. It is clear that if two LI classes are MLD with derivability radius δ, and one has matching rules of radius R, then the other must also have matching rules, of radius not bigger than R + δ. Examples where such concepts have been applied to transfer matching rules from one tiling to another can be found in [35]. Unfortunately, it is not always possible to relate a tiling ensemble of interest to something else which is already known in all its details. This is so in particular for random-tiling ensembles. In such cases, a further important tool is that of Monte Carlo simulations. Such simulation methods are needed, in particular, for the study of cluster density maximization models, where it is very difficult to prove anything rigorously. Monte Carlo simulations can, for instance, be used to search for the tilings with the highest cluster density within some larger tiling ensemble. For this purpose, an energy function is defined on the space of tilings which gives the lowest energy to the tilings with the highest cluster density. One then simulates a tiling ensemble at temperature T , and by slow cooling (simulated annealing) one can reach the states with lowest energy. We cannot go into the details of these simulation techniques here; the reader is referred for general background to [36].

3.3 Penrose and Related Tilings 3.3.1 Perfect Decagon Coverings The aperiodic decagon introduced by Gummelt [11, 12] provides the first and most striking example where perfect quasiperiodic order can be obtained by a simple cluster-covering principle. The overlap rules for the decagons are


A4 A3

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B1

A2

A B

Fig. 3.1. Aperiodic decagon (top left), with the allowed overlap zones for A- and B-overlaps (top, middle, and right), and representative A- and B-overlaps (bottom)

encoded in their shading: if two decagons overlap, their shadings must agree in the entire overlap region (Fig. 3.1). A patch of decagons satisfying the overlap rules is shown in Fig. 3.2. These simple overlap rules are capable of enforcing a structure isomorphic to a perfect Penrose tiling. More precisely, the set of decagon coverings of the plane that obey the overlap rules is MLD with the set of perfect Penrose tilings. In other words, there is a one-toone correspondence beween decagon coverings and Penrose tilings, and this correspondence is local. We can only sketch the proof of this statement here. The proof requires a detailed analysis of those local decagon arrangements which can be extended to a complete covering. This analysis has been carried out in [12]. It is straightforward but somewhat tedious. The first important result of this analysis is that a local decagon arrangement can be continued to a full covering of the plane only if all decagon overlaps are of only two types, A and B

Fig. 3.2. A patch of an aperiodic decagon covering. The shading allows two kinds of decagon overlaps

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Fig. 3.3. Aperiodic decagon, superimposed with corresponding patch of a Penrose tiling

(Fig. 3.1). Moreover, these overlaps can occur only in certain overlap zones (Fig. 3.1). If other overlaps occur, even if they satisfy the condition of matching the shading, the decagon arrangement cannot be extended to a covering of the entire plane. Next, it has been shown that there is a one-to-one correspondence between local decagon arrangements in a covering and local tile arrangements in a Penrose tiling. This correspondence is indicated in Fig. 3.3 for a Penrose rhombus tiling (note that near the decagon boundary some edges of the Penrose tiling are not fully specified; these edges depend on the presence or absence of neighboring decagons). This correspondence has been established for all decagon arrangements whose decagons contain some common point. These local decagon arrangements are large enough that one can then make use of the (short-range) matching rules of the Penrose tiling: every tiling that looks locally, on the scale of these finite decagon arrangements, like a Penrose tiling is indeed a Penrose tiling. From the proof sketched above, it becomes clear that the decagon overlap rules are really just an efficient reformulation of the Penrose matching rules. Using the concepts of mutual local derivability, the matching rules, which are called overlap rules in the covering context, can be transferred from one system to the other, and vice versa. The proof given in [12] has later been simplified somewhat [18], but the basic idea is still the same. In the original formulation, the aperiodic decagon is an example of the cluster-covering principle. Jeong and Steinhardt [18] could show, however, that among all tilings with the two Penrose rhombuses, the Penrose tiling is the one with the highest density of the tile cluster corresponding to the aperiodic decagon. This holds true also, in particular, for tilings which are not covered by this cluster. Therefore, the cluster density maximization principle can also be used in this case, although it is mathematically much more difficult to obtain any rigorous results by doing so. In the description above, we have related the decagon coverings to the Penrose rhombus tilings. There are several other kinds of Penrose tilings [37], such as kite-and-dart Penrose tilings, Penrose tilings with Robinson triangles, and Penrose tilings with pentagons, rhombuses, and ship- and star-shaped tiles. All these tilings are MLD with each other, and so they are MLD with the decagon coverings. Instead of the Penrose rhombus tilings, any other kind of


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Penrose tilings could have been used to establish the relationship to decagon coverings [38]. In Sect. 3.3.2, we have found it advantageous to use Penrose pentagon tilings rather than rhombus tilings. Coverings by the aperiodic decagon have found many applications in the description of the structure of decagonal quasicrystals, and many authors have attempted to explain the stability and existence of these quasicrystals by a cluster-covering principle [9, 13, 14, 15, 16, 17]. The equivalence of the decagon overlap rules and those implied by the atomic structure of the clusters involved is not always completely evident, however. One reason is that it is not always clearly stated which overlaps of the clusters are considered admissible. 3.3.2 Random Decagon Coverings In Sect. 3.3.1 we have seen how a perfect Penrose tiling (and associated quasicrystals) can be obtained from a simple cluster-covering principle. However, many experimental decagonal quasicrystal structures are not perfectly quasiperiodic, and it is therefore interesting also to consider overlap rules which are less restrictive than those for the aperiodic decagon, and which do not enforce a perfectly ordered structure, but rather a (super-tile) randomtiling structure. The analysis of such relaxed overlap rules and their corresponding structures will be the subject of most of the rest of Sect. 3.3. Most of these results have been presented in [27]. Before we start the discussion of the relaxed overlap rules, we have to recall one further fact about perfect decagon coverings. In Sect. 3.3.1, we have established the relation between decagon coverings and Penrose rhombus tilings. For the transition to relaxed overlap rules, it is more natural to use Penrose pentagon tilings (PPTs), however. These consist of pentagons, rhombuses, and ship- and star-shaped tiles. It has been shown [38] that the decagon centers of a decagon covering obeying the “perfect” overlap rules form the vertex set of a PPT, and that every PPT can be obtained in this way from exactly one covering satisfying the perfect overlap rules. This is the mutual-local-derivability correspondence between perfect decagon coverings and PPTs. It is a one-to-one correspondence, and it is local. As the aperiodic decagon represents a cluster in the corresponding quasicrystal, we shall often also use the term “cluster” for the covering decagon. In order to allow partially disordered coverings, Gummelt and Bandt [26] have proposed that one should relax the overlap rules of the aperiodic decagon to some extent. To understand the type of relaxation, recall that if the perfect rules are obeyed, a decagon may have small A-overlaps with neighbor decagons in four possible directions, and bigger B-overlaps with neighbor decagons in two possible directions (Fig. 3.1). The coloring in the overlap region has an orientation, which must be respected. All possible overlaps are therefore oriented. As a relaxation of the perfect rules, Gummelt and Bandt have now proposed [26] that one should abandon this orientation constraint,

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1 0 0 1

111 000 000 111 000 111

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Fig. 3.4. Random Penrose pentagon tiling. Spiky tiles (shaded in gray) are surrounded by pentagons. This tiling is equivalent to an HBS tiling (indicated by gray lines), whose tile edges connect centers of neighboring pentagons

and retain only the condition which specifies the possible overlap zones, without any condition on the orientation inside them. These overlap rules will be referred to as the “fully relaxed” rules. There is a natural intermediate between the perfect rules and the fully relaxed rules. In these rules, which will be called just the relaxed rules, the orientation condition is abandoned only for the small A-overlaps, but is retained for the larger B-overlaps. These intermediate rules and the resulting structures will be the main topic in the following. Using an analysis of all admissible local decagon arrangements as for the aperiodic decagon, Gummelt and Bandt have shown [26] that every covering satisfying the fully relaxed rules has the property that its set of cluster centers forms the vertex set of a random PPT that has the additional property that all the spiky tiles (stars, ships, and rhombuses, shaded in gray in Fig. 3.4) are completely surrounded by pentagons (in the following, when we say “random PPT”, we always mean one satisfying this extra condition; more general ones do not play any role here). Such a random PPT is MLD with a random hexagon–boat–star (HBS) tiling, indicated by gray lines in Fig. 3.4. Since coverings satisfying the more restrictive relaxed rules also satisfy the fully relaxed rules, their set of cluster centers also forms the vertex set of a random PPT. Conversely, it is easy to see that every random PPT can arise both from relaxed and from fully relaxed coverings. The only difference between relaxed and fully relaxed coverings is the number of coverings associated with a given random PPT.


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To see this, we note that the orientation of a cluster is already determined by the presence of two B-neighbors or the presence of four A-neighbors. In Fig. 3.4, we see that A-neighbors are separated by an edge of a tile or by a long diagonal across a ship or a star, whereas B-neighbors are separated by a short diagonal of a rhombus, ship, or star. The only vertices whose cluster orientation is not fixed by the tiling are the obtuse corners of the rhombuses, where two cluster orientations are possible. For the fully relaxed rules, where no orientation conditions are to be observed, we therefore have altogether four choices per rhombus for the cluster orientations. For the relaxed rules, which have to obey the orientation condition for the B-overlaps, one can easily verify that the orientation condition for B-overlaps across ships and stars is always satisfied. In order that the same is true for B-overlaps across rhombuses, however, the orientations of the clusters on the obtuse rhombus corners cannot be chosen independently. The orientation condition is satisfied only for two of the four possible combinations. Summarizing, for a given random PPT we have four independent choices per rhombus for a fully relaxed cluster covering, and two choices per rhombus for a relaxed cluster covering. In the same way, one can quantify the relationship between the cluster coverings and certain variants of a random Penrose rhombus tiling. The random HBS tilings arise from random Penrose rhombus tilings which still satisfy the double-arrow condition [25, 39]. Such random Penrose rhombus tilings are also called 4-level (or 4-vertex) Penrose random tilings. The double-arrowed edges of the 4-level random Penrose tilings are simply wiped out to obtain the HBS random tilings, which are also known as 2-level Penrose tilings [25]. The relationship between 4-level and 2-level Penrose random tilings is not one-to-one: whereas the subdivision of boats and stars is unique, there are two choices for the subdivision of each hexagon into rhombuses, just as there are two possible cluster assignments on the obtuse rhombus corners in the PPT. Since rhombuses in the PPT and hexagons in the HBS tiling are in one-to-one correspondence, this implies that the multiplicity of relaxed cluster coverings and 4-level Penrose random tilings related to a given random PPT are the same. 3.3.3 Cluster Density Maximization In Sect. 3.3.2 we have considered cluster coverings, where our clusters have simply been decagons with certain overlap rules. Another variant of the ordering principle is cluster density maximization. The relaxed overlap rules allow a very natural realization in terms of a vertex cluster in a random PPT (we still require that spiky tiles are completely surrounded by pentagons). This vertex cluster is shown in Fig. 3.5, superimposed on the aperiodic decagon. The tile edges are drawn only as a guide to the eye; they are not part of the cluster, only the vertex set counts. It is easy to see that the orientation of the A-overlaps of the aperiodic decagon cannot be enforced by the vertex set of the cluster, whereas the orientation of the B-overlaps is enforced. The

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Fig. 3.5. Vertex cluster superimposed on aperiodic decagon. This cluster enforces the relaxed overlap rules

A-overlap consists of a rhombus, and the B-overlap consists of a hexagon with an extra vertex in an asymmetric position inside. With this cluster, we can now build a statistical model of cluster density maximization. We consider the set of all random PPTs and assign a weight to each tiling, which is simply the number of vertex clusters it contains. With Monte Carlo simulations, it is then possible to find the subensemble of those random PPTs which have maximal cluster density. For the simulations, we need a Monte Carlo dynamics which is ergodic in the ensemble of all random PPTs. We have found that the flip move shown in Fig. 3.6 has the required properties. By repeated flips, it is possible to turn any random PPT into any other. One must be careful, however, not to execute any flips which would introduce new kinds of tiles. This can be avoided if some local constraints on the flips are obeyed. With such a Monte Carlo model, the states of maximal cluster density have been determined by simulated annealing, using as the energy function the negative of the number of clusters, thus mimicking the total cohesion energy of the clusters. It turns out that these states of maximal cluster density 2 are precisely the super-tile random PPTs, whose √ tiles have an edge length τ times that of the small tiles, where τ = (1 + 5)/2 is the golden ratio. An example of such a super-tile tiling is shown in Fig. 3.7. In view of the results of Sect. 3.3.2, this is of course not too surprising. The cluster centers sit on the vertices of the super-tile tiling, covering all vertices of the small tiles. Since the vertex cluster is smaller than the aperiodic decagon, it does not cover the whole area, but only the vertices. There remain small pentagons uncovered, which sit at the centers of the super-tile pentagons. This does not affect the overlap constraints, however. Our results therefore imply that

Fig. 3.6. Flip move for Monte Carlo simulation


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Fig. 3.7. Structure with maximal cluster density. The cluster centers form the vertices of a super-tile random PPT

there is a one-to-one mutual-local-derivability correspondence between cluster coverings satisfying the relaxed rules, and structures with maximal density of the vertex cluster. Although these two ordering principles are very similar, they are conceptually slightly different and must be distinguished. 3.3.4 Entropy Density With the cluster model of Sect. 3.3.3, it is also possible to measure the entropy density of the ensemble of structures with maximal cluster density, and thus the entropy density of the relaxed cluster-covering ensemble. We have an energy model which assigns a cohesion energy to each cluster in the structure. In this model, the ground state, the state of maximal cluster density, consists of super-tile random PPTs, with an extra weight of two per rhombus, because for each rhombus there are two choices of cluster configuration with the same number of clusters. At infinite temperature, on the other hand, we have the full random PPT (with the small tiles). With entropic sampling techniques [40, 41], it is now possible to determine the entropy of the system as a function of energy, and in particular the difference in entropy between the ground state and the infinite-temperature state. An example of such an entropy function is shown in Fig. 3.8. The entropy of the ground state can be extracted in the following way. The entropies at zero and infinite temperature are both entropies of random PPTs, one with and one without extra degeneracy for each rhombus. Moreover, the two random tilings are on different scales. Taking both differences into account, we arrive at the following relation for the two entropy densities:

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Entropy S

S∞

S0 Energy E

Fig. 3.8. Entropy as a function of energy for a selected approximant

σ0 = τ −4 (σ∞ + ρrh kB ln 2) ,

(3.1)

where ρrh is the measured rhombus density in the high-temperature state. If we now write σ∞ = σ0 + ∆ σ, we obtain an equation for the ground-state entropy density σ0 , in which all other quantities are known. The groundstate entropy density has been determined in this way for several periodic approximants. By finite-size scaling, this can then be extrapolated to infinite system size. This is shown in Fig. 3.9, where one can see that the scaling works very nicely. At a scale where the super-tile edges (which separate A-overlap neighbors in the cluster model) have unit length, we obtain a value of σ0 /kB = 0.253 ± 0.001

(3.2)

for the entropy density. This value can be compared with the value Tang and Jarić [39] have obtained for the entropy density of the 4-level random Penrose tiling. In Sect. 3.3.2, we have seen that 4-level random Penrose tilings are in one-to-one correspondence with relaxed cluster coverings. If the different scales of the two tilings are corrected for, Tang and Jarić obtained a value of σ0 /kB = 0.255 ± 0.001 ,

(3.3)

which is compatible with ours. 3.3.5 Couplings Between Clusters The only difference between the perfect and the relaxed overlap rules is that the latter do not require oriented A-overlaps. Since not all relaxed coverings are perfect, there must be A-overlaps in a relaxed covering which do not obey the orientation condition of the perfect rules. A closer analysis shows [26] that there is actually only one kind of disoriented A-overlap which can occur in a relaxed covering. If we represent the orientation of a cluster on a vertex


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0.04

σ0 /k B − 0.2102

0.035 0.03 0.025 0.02

0.015 0

0.002

0.004

0.006

0.008

1/N

Fig. 3.9. Finite-size scaling of the entropy density

of a super-tile PPT by an arrow, we can represent the covering in a much more compact way, as shown in Fig. 3.10. The disoriented A-overlaps not permitted by the perfect rule are marked in Fig. 3.10. The corresponding tile edges have antiparallel arrows at their ends. This representation suggests that we should introduce a coupling of neighboring clusters in such a way that overlaps which are not permitted by the perfect rules are energetically penalized. We expect such a coupling to be weak, because these kinds of defects can be detected only on larger scales. However, at a low enough temperature it might still be able to order the super-tile random-tiling ground state of the relaxed cluster covering to a perfectly ordered structure. This suggests a scenario with two energy scales: the presence of each vertex cluster lowers the cohesion energy by a large amount, so that structures with maximal cluster density are strongly favored, even at relatively elevated temperatures. The equilibrium structures at these temperatures are therefore relaxed cluster coverings. Additionally, there is a small coupling between neighboring clusters, which at low temperatures can order the super-tile random tiling to a perfectly ordered tiling. We have verified the feasibility of this model by Monte Carlo simulations. Our model considers only the subensemble of states with maximal cluster density. In other words, we simulate at the level of the super-tile tiling. The cluster on each vertex is represented by an arrow, as in Fig. 3.10. This setup keeps the number of clusters constant, so that we cannot leave the states of maximal cluster density, which simplifies the simulation considerably. As the flip move, we can still use the one shown in Fig. 3.6, except that here we have to adjust the cluster orientations of the neighboring vertices of the vertex that is flipped to new values consistent with the new tile configuration. Additionally, we have to introduce a new type of flip, which only changes the

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!

!

!

! Fig. 3.10. Super-tile tiling with cluster orientations indicated by arrows

cluster orientations on the obtuse corners of a rhombus, and keeps the tiling fixed. With this model, we have verified that the coupling of the clusters can indeed order the model to a perfectly quasiperiodic structure. In other words, the ground state is a perfect quasicrystal, whereas the high-temperature state is the super-tile random tiling with the additional degeneracy of the cluster orientations on the obtuse corners of the rhombuses, corresponding to the relaxed cluster coverings. In this respect, “high temperature” means high compared with the cluster coupling, but still low compared with the energy required to break up clusters. Also in this model, it is possible to measure the entropy density of the relaxed covering ensemble. In this case, the ground state is ordered and has zero entropy (at least in the thermodynamic limit), and the high-temperature state is the one whose entropy we are interested in. We therefore need only to measure the difference between the entropies of the high-temperature state and the ground state, and extrapolate to the thermodynamic limit. It turns out, however, that finite-size scaling does not work as well for the present model as it does for the model with a random-tiling ground state. The results are therefore less precise, although they are consistent with the results reported in Sect. 3.3.4. We shall therefore not present any details here. 3.3.6 An Atomic Cluster Enforcing the Relaxed Overlap Rules We conclude Sect. 3.3 by showing that the relaxed overlap rules are very natural. We already have seen that they are equivalent to the natural overlap


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Fig. 3.11. Atomic cluster found by Henley and Roth [42] in a molecular-dynamics simulation. In units of the period in the z direction, white atoms are at z = 0, black atoms are at z = 1/2, and dotted atoms are at z = 1/4, 3/4

constraints of a simple vertex cluster. There is, however, also a natural atomic cluster that enforces essentially the same overlap constraints. This cluster was found by Roth and Henley [42] in a molecular-dynamics simulation of decagonal Frank–Kasper-type quasicrystals. It is shown in Fig. 3.11. The atoms drawn in black (which constitute one layer of the structure) form the vertices of a PPT. The only discrepancy with the overlap constraints of our vertex cluster is at the center of a star tile, where we would have expected a single atom at z = 0, not two atoms at z = 1/4, 3/4 as shown in the figure.

3.4 Octagonal Ammann–Beenker Tilings 3.4.1 The Alternation Condition In many respects, the octagonal Ammann–Beenker tiling [28, 29], shown in Fig. 3.12, is among the simplest of all quasiperiodic tilings. To some extent, this is true also for its cluster descriptions. We shall therefore use it as our second example to illustrate the general principles of cluster models. The matching rules that enforce a perfect octagonal tiling are rather complicated [6, 7, 29]. They are expressed in terms of a complicated, nonlocal decoration of the tiling. However, there is a simple, local subset of the matching rules, which can easily be enforced with a cluster density maximization principle or with a cluster-covering principle. This subset of the matching rules is the alternation condition [20]. It requires that along any lane of tiles, the two types of rhombuses must alternate, as illustrated in Fig. 3.13. The alternation condition can be enforced by a suitable arrowing of the tile edges. Opposite edges of squares have the same arrow direction, whereas opposite edges of

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Fig. 3.12. Octagonal Ammann–Beenker tiling, subdivided into super-tiles

rhombuses have opposite arrow directions. The alternation condition cannot enforce octagonal tilings, but it does enforce perfectly ordered, quasiperiodic tilings, which are at least 4-fold symmetric [21]. In fact, the tilings satisfying the alternation condition are all members of a one-parameter family of 4fold symmetric, quasiperiodic tilings. The unique member of this family with even 8-fold symmetry is the Ammann–Beenker tiling. The square–rhombus tilings satisfying the alternation condition can be constructed by dualization [28, 43, 44] of two superimposed square grids, which are rotated by 45◦ with respect to one another. These two square grids may have different scales, so that the resulting tiling is only 4-fold symmetric in the general case. If the two grids have the same scale, the octagonal Ammann–Beenker tiling is obtained. A cluster density maximization principle can now be implemented in the following way. In the first step, one selects suitable clusters whose presence fa-


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Fig. 3.13. The alternation condition requires that along a lane of tiles, the two kinds of rhombuses alternate. It can be enforced by an arrowing of the tiles

vors the alternation condition. Maximizing the density of these clusters with suitable weights will produce a tiling that satisfies the alternation condition. In the second step, the weights of these favored clusters are further tuned in such a way that the octagonal tiling has the highest weighted cluster density among all tilings that satisfy the alternation condition. This program has been carried out with the clusters shown in Fig. 3.14 [22], which clearly favor the alternation condition. They have been selected with the goal that two rhombuses which are neighbors along a lane of tiles are always covered by such a cluster (note that two such rhombuses are never separated by more than two squares). For the second step, we need to compute the densities of these clusters for an arbitrary square–rhombus tiling satisfying the alternation condition. We note that the clusters can all be assigned to a unique super-tile (Fig. 3.12), and that super-tiles of the same type always carry the same number of clusters. In particular, for each super-tile square we have 14 octagon clusters and 3 ship clusters, and for each super-tile rhombus we have 10 octagon clusters and 2 ship clusters. The cluster densities are therefore a simple function of the super-tile densities. The latter, on the other hand, are easily obtained from the construction of the tiling as the dual of two superimposed square grids. We simply have to compute the densities of grid line intersections of each type, suitably normalized by the total area of the corresponding tiles. If we denote by 1 + x the ratio of the grid line spacings of the two square grids, we obtain, in suitable units, 1 + (1 + x)2 , (2 + x)2 √ 2 2(1 + x) . ρship (x) = (2 + x)2

ρoct (x) =

(3.4)

(3.5)

For the octagonal tiling at x = 0, ρoct (x) has a maximum, but ρship (x) has a minimum. However, the weighted cluster density

Fig. 3.14. Octagon and ship clusters favoring the alternation condition

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ρ(x) = woct ρoct (x) + wship ρship (x) has a maximum at the octagonal point x = 0, provided 2woct √ > 2+1. wship

(3.6)

(3.7)

This latter condition on the cluster weights is sufficient for the octagonal tiling to have the highest weighted cluster density among all tilings satisfying the alternation condition. But what about other tilings? Indeed, there are tilings that have a higher octagon density than the octagonal tiling, but do not satisfy the alternation condition. In order that these do not have a higher weighted cluster density, the relative weight of the ship cluster must not be too small. Monte Carlo simulations have shown [22] that the most extreme case is the periodic tiling made of super-tile rhombuses only. In order to exclude also unwanted tilings which do not satisfy the alternation condition, the cluster weights must satisfy the double inequality √ √ 2 woct 2+1< < ( 2 + 1)5 . (3.8) wship Extensive Monte Carlo simulations have shown that whenever the cluster weights satisfy these inequalities, the octagonal Ammann–Beenker tiling has the highest weighted cluster density among all tilings [22]. At the upper bound, the periodic tiling with only super-tile rhombuses becomes competitive, whereas at the lower bound, the periodic tiling with only super-tile squares becomes competitive. We should emphasize that the interval of allowed cluster weight ratios is very large, which makes this model very robust. If only the density of octagons is maximized, but the tile stoichiometry is fixed, then the set of tilings with maximal octagon density includes all supertile random tilings with the given stoichiometry. In Fig. 3.12 one can see that super-tiles can be reshuffled without changing the number of octagons. We therefore obtain, in this case, a super-tile random tiling as the ground state with maximal octagon density [22]. Tilings in such a super-tile random-tiling ensemble look perfect on a local scale, but are disordered on larger scales. They can still be the basis of reasonable models for real quasicrystals. The cluster density maximization model discussed above has the disadvantage that it needs two clusters, whose weights must satisfy certain constraints. This can be avoided [23] by arrowing the octagon cluster (Fig. 3.15). A tiling completely covered by the arrowed octagon must necessarily satisfy the alternation condition, and among these, the Ammann–Beenker tiling has the highest octagon density. Under the assumption that tilings that are not completely covered by the arrowed octagon cannot have a higher arrowedoctagon density than the maximal density for tilings that are covered, maximizing the density of the arrowed octagon can therefore enforce the octagonal Ammann–Beenker tiling. Although there is no proof of the above assumption, it appears highly plausible. In any case, the arrowed Ammann–Beenker tiling is ideally suited to the maximal cluster-covering principle. It has the highest


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Fig. 3.15. Arrowed octagon cluster and inflated unarrowed octagon cluster. Both impose the same overlapping constraints

arrowed-octagon density among all tilings that are covered by the arrowed octagon. 3.4.2 An Atomic Model for Octagonal Mn–Si–Al In Sect. 3.4.1, we have seen that arrowed-octagon clusters can be used with a maximal cluster-covering principle to obtain the Ammann–Beenker tiling. If undecorated clusters are preferred for some reason, one can inflate the octagon once to arrive at a larger cluster, which has exactly the same asymmetries as the arrowed octagon (Fig. 3.15). This larger, undecorated cluster therefore imposes the same overlapping constraints, and can be used in place of the arrowed octagon. However, real quasicrystals are not tilings, but are at best decorations of a tiling. The tiling therefore has to be decorated with atoms, and it could be this atomic decoration which introduces the necessary asymmetry, and thus imposes the necessary overlapping constraints. This is indeed the case for the quasicrystal structure of octagonal Mn– Si–Al described by Jiang, Hovm¨ oller, and Zou [45], as has been discussed in detail in [46]. This quasicrystal has a layered structure . . . ABAB . . ., and can be regarded as a decoration of the Ammann–Beenker tiling. The decoration of the octagon motifs (which cover the whole structure) is shown in Fig. 3.16. Both the decoration of the edges in layer A and the decoration of the interiors of the squares in layers B and B show the same asymmetry as the arrowing. There are actually two possible decorations of an octagon, whose only difference is that the decorations of the layers B and B are exchanged. One kind of octagon is decorated with a stacking . . . ABAB . . ., and the other with a stacking . . . AB AB . . .. Since each octagon actually represents an infinite prism with an octagonal base, these two decorations correspond to prisms which are translated by half a lattice period in the z direction with respect to one another but are otherwise identical. We therefore have a covering by identical prisms, which, of course, can be chopped into identical, finite clusters.

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a)

b)

c)

d)

e)

f)

Fig. 3.16. Decoration of a small octagonal patch: (a) layer A, (b) layer B, (c) layer B , and (d) layers B and B together. Large dots denote Mn atoms, and small dots Si or Al atoms. In (d), only Mn atoms are shown; atoms from the B layer are shown as filled dots, and atoms from the B layer as open dots. In (e) and (f ), abstract representations of the layer stackings ABAB and AB AB, respectively, are given

It is most convenient to represent these prisms by an abstract decoration of the octagon. The different vertical positions of the prisms are encoded by a coloring, as shown in Figs. 3.16e,f. Tiles which differ in color but otherwise have the same decoration correspond to prisms shifted by half a lattice period in the z direction. It is interesting to note that since the octagonal prisms occur at two different positions in the z direction, the Bravais lattice of this octagonal quasicrystal must be an octagonal centered one [46]. This can also be seen in the colored and arrowed tiling of Fig. 3.17: if a tile is a translate of another tile by an odd number of tile edges, it has the opposite color. If an even number of tile edges separates the two tiles, they have like colors. In order to obtain a lattice translation, a (horizontal) translation by an odd number of tile edges must be combined with a translation in the z direction by half a lattice period, in order to make up for the color change. This results in an octagonal centered lattice.

3.5 Dodecagonal Socolar and Shield Tilings 3.5.1 The Socolar Tiling The case of the dodecagonal Socolar tiling [6] (Fig. 3.18) is very analogous to that of the octagonal Ammann–Beenker tiling. This tiling can be obtained


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Fig. 3.17. Colored and arrowed Ammann–Beenker tiling

by the projection method, or as the dual of two superimposed triangular grids [44]. Like the Ammann–Beenker tiling, it satisfies an alternation condition [20]: along any lane of tiles, the two kinds of rhombuses have to alternate (Fig. 3.19). This alternation is enforced by the arrowing of the tile edges. Besides the alternation condition, there is a further local constraint satisfied by the Socolar tiling. This becomes apparent if an inflated tiling is superimposed (Fig. 3.18). The edges of the inflated tiles have an asymmetric environment. There is a dodecagon cluster which is either to the right or to the left of the edge. We may say that square edges buckle inwards, rhombus edges buckle outwards, and the hexagon has opposite edges of different type. In order that an arrowed tiling by hexagons, squares, and rhombuses can be inflated, it must satisfy the local constraint that outward-buckling edges match only inward-buckling edges, and vice versa. This local constraint will henceforth be called the “buckling constraint”. Note that the edge types break the single mirror line of the arrowed hexagon, so that there are now left hexagons and right hexagons. Taking the buckling into account, arrowed hexagons are thus completely asymmetric. They then have exactly the same asymmetry as

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Fig. 3.18. The Socolar tiling, with superimposed super-tile tiling

inflated hexagons. A tiling satisfying both the alternation condition and the buckling constraint can be inflated infinitely often. Recall that in the case of the Ammann–Beenker tiling, the alternation condition enforces ordered tilings of at least 4-fold symmetry. A close analysis of the proof of Katz [21] shows that an analogous statement holds for the Socolar tiling [47]. One can show that every hexagon–square–rhombus tiling satisfying the alternation condition and the buckling constraint is quasiperiodically ordered and at least 6-fold symmetric. More precisely, such a tiling is the dual of two superimposed triangular grids, which are rotated by 90◦ with respect to each other. These triangular grids may have different scales, so that the symmetry of the tiling is only hexagonal in general. If the scales of the two grids are the same, the dodecagonal Socolar tiling is obtained. This implies that all tilings admitted by the alternation condition and the buckling constraint are quasiperiodic (or periodic) and at least hexagonally symmetric. The simplest such tiling is the one with only hexagons.

Fig. 3.19. The alternation condition for the Socolar tiling


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3.5.2 Cluster Covering and Cluster Densities The two tile clusters shown in Fig. 3.20 cover the entire Socolar tiling. This is best seen at the level of the superimposed super-tile tiling. Since super-tiles of a given shape all have the same decoration with small tiles, one just has to show that all super-tiles are covered. The dodecagon cluster alone covers most of the tiling. The only pieces that remain uncovered are a small square inside each large square, and a little hexagon and three small squares inside each large hexagon. There is one dodecagon cluster center near the middle of each super-tile edge (but not directly on the edge). Every dodecagon cluster can be assigned to a unique super-tile. There are two dodecagon clusters per super-tile rhombus, and three per super-tile hexagon. There are no dodecagon clusters assigned to any super-tile squares. The remaining uncovered pieces of the tiling are then covered by the butterfly cluster (Fig. 3.20). There is one butterfly cluster per super-tile square, and one per super-tile hexagon. Each of the two clusters always occurs with exactly the same decoration in the tiling. Since they are arrowed, every tiling that is covered by the clusters must necessarily satisfy the alternation condition, and, owing to the structure of the clusters, the same must be the case also for the buckling constraint. Using the results of Sect. 3.5.1, we now know that every tiling that is covered by the two clusters is a quasiperiodic super-tile tiling of at least hexagonal symmetry. It remains to prove that among these tilings, the dodecagonal one has the highest cluster density. Recall that each cluster is assigned to a unique supertile, and that each such super-tile of a given kind carries the same number of clusters. Moreover, the super-tile tiling is the dual of two superimposed triangular grids, so that we can compute the densities of the super-tiles, which are parameterized by the ratio 1 + x of the grid line spacings of the two grids. This parameterization is chosen such that the dodecagonal tiling corresponds to x = 0. From these super-tile densities, it is then easy to compute the cluster densities. In suitable units, we obtain √ √ (6 3 + 48)(1 + x) + 3 3x2 , (3.9) ρdod (x) = 1 + x + x2 /6 √ √ (2 3 + 6)(1 + x) + 3x2 . (3.10) ρbfl (x) = 1 + x + x2 /6 ρdod (x) has a maximum for x = 0, but ρbfl (x) has a minimum there. However, the weighted cluster density ρ(x) = wdod ρdod (x) + wbfl ρbfl (x)

(3.11)

has a maximum at the dodecagonal point √ x = 0, provided the weights satisfy the mild condition wbfl /wdod < 12 + 10 3. We have therefore shown that square–hexagon–rhombus tilings that are completely covered by the dodecagon cluster and the butterfly cluster satisfy the alternation condition and the buckling constraint. They are thus

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Fig. 3.20. The Socolar tiling, with superimposed super-tile tiling, and clusters indicated in gray

quasiperiodic and hexagonally symmetric. Among these tilings, the dodecagonal Socolar tiling has the highest weighted cluster density, provided a mild constraint on the cluster weights is satisfied. The dodecagonal Socolar tiling can therefore be obtained with a simple maximal cluster-covering principle. This does not rigorously rule out the possibility that there are tilings which are not completely covered, but have an even higher weighted cluster density. Such a scenario seems rather unlikely, however. These results are very similar to those for the Ammann–Beenker tiling. There is one key difference, however. For the dodecagonal Socolar tiling we need two covering clusters, whereas for the octagonal tiling (as well as for the Penrose tiling) one cluster was sufficient. By taking one larger cluster, it is possible to reduce the fraction of the uncovered area, even as far as one wants [48], but it does not seem possible to cover the whole tiling with one cluster of finite size. Whether such an “almost covering” can enforce an ordered tiling is unknown, however. 3.5.3 The Shield Tiling Just as there are several kinds of Penrose tiling which are all MLD with each other, there exist also for the Socolar tiling a whole family of other tilings which are MLD with it [49]. Since matching rules can be transferred between


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Fig. 3.21. The shield tiling, with associated Socolar tiling in gray

different tilings which are MLD [35], the same should also be possible with cluster coverings. An interesting tiling which is MLD with the Socolar tiling is the shield tiling [30, 31], which is shown in Fig. 3.21, together with the Socolar tiling. A close inspection of Fig. 3.21 makes it evident that these two tilings are related to one another, and that this relation is a local equivalence relation. The Socolar tiling has as vertices the triangle centers of the shield tiling, and also the shield tiling is easily constructed from the Socolar tiling. This local equivalence relation can be extended to a larger class of tilings. On the one hand, we have generalized Socolar tilings, and on the other hand, we have generalized shield tilings. For simplicity, we shall call these also just Socolar tilings and shield tilings, respectively. In the case of the Socolar tilings, this larger class consists of all those tilings which satisfy the local constraint on the buckling of edges (Sect. 3.5.1). This constraint is essential in order that an equivalent shield tiling can be derived. To each edge of the Socolar tiling there corresponds a vertex of the shield tiling, which is located on the side of the edge opposite to the buckling. Conversely, a shield tiling must satisfy the local condition that all squares and shields must be

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completely surrounded by triangles and double triangles, and all triangles and double triangles must be completely surrounded by squares and shields. Since the corresponding Socolar tiling has its vertices at the triangle centers of the shield tiling, one would otherwise obtain new tile types in the corresponding Socolar tiling. When the Socolar tiling is derived from such a shield tiling, the Socolar tiling then automatically satisfies the edge buckling constraint. The generalized Socolar and shield tilings are therefore MLD, with the same derivation rules as for the perfect tilings. An interesting subset of the generalized Socolar tilings consists of those which satisfy the alternation condition. For this subset there exists also a corresponding subset of generalized shield tilings, so that by local derivation we can define the alternation condition for the generalized shield tilings, also. Unfortunately, for the shield tilings it is not so easy to recognize directly whether the alternation condition is satisfied. We therefore have added marks to the vertices of the shield tiling in Fig. 3.21 whose purpose is to encode the arrow direction of the nearby bond of the corresponding Socolar tiling. We are now ready to define the covering clusters for the shield tiling. These clusters, shown in Fig. 3.22, are directly derived from the corresponding

Fig. 3.22. The shield tiling, with clusters indicated in gray


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clusters of the Socolar tiling. Their decoration with marks at the vertices has the purpose of ensuring that any tiling covered by them satisfies the alternation condition. In order for this to work, the following overlap rules have to be applied. Two overlapping clusters are compatible with each other if all tiles in the overlap agree and if all marks in the overlap agree, except perhaps those at the outermost triangle vertices of the clusters. Without this latter exception one would have to introduce further types of clusters. These overlap rules then ensure that any tiling completely covered by the clusters is a generalized shield tiling satisfying the alternation condition, and therefore is locally equivalent to a generalized Socolar tiling satisfying the alternation condition. Since the two are locally equivalent, the cluster densities are the same, and so we can easily transfer all the results that we have obtained for the Socolar tiling to the shield tiling. In other words, the dodecagonal shield tiling can be obtained with a maximal cluster-covering principle, where the same weighted cluster density as for the Socolar tiling has to be used.

3.6 Voronoi and Delone Clusters In recent papers [50, 51, 52, 53], Kramer has argued that the existence of covering clusters should be expected for theoretical reasons, at least for the canonical projection tilings. There are two kinds of such projection tiling [54]. The first kind has vertices at projected lattice positions, with an acceptance domain which is the projected Voronoi cell of the higher-dimensional lattice. The second kind has vertices at projected corners of the Voronoi cells, with acceptance domains which are the projected, dual Delone cells. According to this theory, for the first kind of canonical projection tiling we should expect covering clusters at projected corners of Voronoi cells, whose size is given by the corresponding projected Delone cell. Conversely, for the second kind of canonical projection tiling, covering clusters are expected at projected lattice points, with a size given by the projected Voronoi cell. In other words, covering clusters and vertices are, in a sense, dual to each other. There does not seem to be a proof that Voronoi or Delone clusters indeed cover the whole tiling without gaps in the general case, but for many particular examples this has been verified. Indeed, the examples discussed in the previous sections are exactly of this kind. The aperiodic decagon that covers the Penrose (rhombus) tilings is centered at projected lattice points, whereas the vertices of the Penrose tiling are at projected corners of the Voronoi cells. In the case of the octagonal Ammann–Beenker tiling, the vertices are at projected lattice points, whereas the octagon clusters are centered at projected corners of the Voronoi cells. In the case of the dodecagonal Socolar tiling, the dodecagon cluster is again of this kind. It is centered at projected lattice points, whereas the vertices of the Socolar tiling are projected corners of Voronoi cells. Only the butterfly

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Fig. 3.23. Pentagon clusters that cover the arrowed T¨ ubingen triangle tiling. Both clusters, as well as their mirror images, occur in 10 orientations

Fig. 3.24. Cluster covering of a periodic variant of the T¨ ubingen triangle tiling. A unit cell is outlined in gray

cluster does not really fit into this picture. In all these cases, the cluster size coincides with the size predicted by theory. Kramer has discussed in detail a further example, the T¨ ubingen triangle tiling (TTT) [50, 52]. Here, the clusters are located at projected corners of the Voronoi cells, whose sizes correspond to projected Delone cells. There are four translation classes of cluster centers, and thus also four different clusters, two small and two large pentagons. Disregarding orientation, there are just two clusters, large and small pentagons (Fig. 3.23), which indeed cover the TTT [52]. Unfortunately, obtaining the TTT by means of a cluster-covering principle or a maximal cluster-covering principle seems much more difficult. Like the Ammann–Beenker and Socolar tilings, the TTT has rather complicated matching rules, at least if one insists on short-range matching rules [8]. It is


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therefore rather difficult to encode these matching rules in terms of overlap rules for a small number of simple clusters. The matching rules can be formulated in terms of two kinds of decoration of the tiling, edge decorations and vertex decorations [8]. The edge decorations can easily be incorporated into the pentagon clusters (Fig. 3.23). The edge decoration alone is not enough, however, to enforce a quasiperiodic tiling. It only makes sure that at least the simplest periodic tilings can no longer be covered, but there are more complicated periodic approximants that still can be covered as well. One such approximant is shown in Fig. 3.24. There are wrong cluster arrangements around the centers of the decagons formed by ten large triangles. The vertex decoration of the matching rules would not allow these, but if such vertex decorations are introduced, each cluster splits into several subvariants, which differ in their decoration. All this suggests that there is no simple cluster model that could enforce a perfectly ordered TTT. This shows that the mere existence of simple covering clusters is not enough to enforce an ordered structure. There still is the possibility that applying a covering principle at fixed stoichiometry may select an interesting class of super-tile random tilings, as for the relaxed decagon overlap rules in the Penrose case, but this has not been worked out so far.

References 1. S. Burkov: Phys. Rev. Lett. 67, 614 (1991) 63 2. H.-C. Jeong, P. J. Steinhardt: Phys. Rev. Lett. 73, 1943 (1994) 63, 64, 66 3. C. L. Henley, “Matching rules for quasiperiodic tilings”. In: Quasicrystals: The State of the Art, ed. by D. P. DiVincenzo, P. J. Steinhardt (World Scientific, Singapore 1991) pp. 185–212 63 4. L. S. Levitov: Commun. Math. Phys. 119, 627 (1988) 63 5. N. G. de Bruijn: Ned. Akad. Wetensch. Proc. Ser. A 43, 39, 53 (1981) 64, 67 6. J. E. S. Socolar: Phys. Rev. B 39, 10519 (1989) 64, 65, 67, 79, 84 7. F. G¨ ahler: J. Non-Cryst. Solids 153&154, 160 (1993) 64, 65, 79 8. R. Klitzing, M. Baake, M. Schlottmann: Int. J. Mod. Phys. B 7, 1455 (1993) 64, 65, 92, 93 9. Y. Yan, S. J. Pennycook: Phys. Rev. Lett. 86, 1542 (2001) 64, 66, 71 10. F. G¨ ahler: Phys. Rev. Lett. 74, 334 (1995) 64 11. P. Gummelt: “Construction of Penrose tilings by a single aperiodic protoset”. In: Proceedings of the 5th International Conference on Quasicrystals, Avignon 1995, ed. by C. Janot, R. Mosseri (World Scientific, Singapore 1995) pp. 84–87 64, 68 12. P. Gummelt: Geometriae Dedicata 62, 1 (1996) 64, 68, 69, 70 13. P. J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A. P. Tsai: Nature 396, 55 (1998) 64, 71 14. R. Wittmann: Z. Kristallogr. 214 (1999) 501 64, 71 15. E. Abe, T. J. Sato, A. P. Tsai: Phys. Rev. Lett. 82, 5269 (1999) 64, 71 16. E. Cockayne, Mater. Sci. Eng. A 294–296, 224 (2000) 64, 71

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4 Voronoi and Delone Clusters in Dual Quasiperiodic Tilings Peter Kramer

4.1 Introduction The discovery and exploration of quasicrystalline materials, which began in 1984 has stimulated a wealth of new mathematical analysis. A new, aperiodic branch of crystallography was developed from classical crystallography to describe the structures of these materials. The atomic structure forms the background for the physics of quasicrystals, ranging from diffraction, scattering, and electronic states to transport phenomena, thermodynamics, and surface and material properties. The basis of the description of such structures is a new development of the classical theory of quasiperiodicity. For the spectacular noncrystallographic (fivefold, icosahedral, . . . ) point symmetries found in quasicrystals, it was possible to develop and apply a Fourier theory and from it develop a quasicrystallographic structural theory. Tiling theories [9] were developed to organize the atomic positions on tiles, whose matching rules provide the overall quasiperiodic property. The best-known example is the Penrose rhombus tiling. A systematic construction of quasiperiodic tiling theories by dual projection from a lattice Λ in Euclidean space E n has been developed by the present author and his collaborators since 1984. Most of the quasicrystallographic schemes used for quasicrystals can be derived from these dual schemes. The lattice gives rise to a full geometry, with n-polytopes, such as Voronoi and dual Delone cells along with their hierarchy of boundaries. A tiling subspace is selected by its point symmetry. All these geometric objects, when projected onto the tiling subspace, have a meaning in terms of tiles, their boundaries, and their matchings in the tiling. The qualitative and quantitative aspects of the tiling are encoded in geometric objects, projected from the lattice to the orthogonal complement of the tiling subspace, which are called the windows. Covering provides a recent alternative concept for describing atomic order in quasicrystals. Here the atomic positions are organized into a few clusters of fixed atomic occupation. The covering clusters encompass patches of tiles and thereby reveal new features of atomic correlations in quasicrystals. By overlap, these clusters build up the long-range quasiperiodic structure. Decagonal clusters were derived by Conway and Sloane [5] and Gummelt [10] in relation to the 2D Penrose–Robinson pattern. Steinhardt and coworkers [32, 33] used these decagonal clusters for decagonal quasicrystals and emphasized relations between clusters and quasi-unit cells. Duneau [6] constructed different cluster P. Kramer (Ed.): Convering of Discrete Quasiperiodic Sets, STMP 180, 97–165 (2002) c Springer-Verlag Berlin Heidelberg 2002

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coverings for the vertices of the octagonal and Penrose tilings and gave windows for them. The structures analyzed by these authors are all essentially 2D quasiperiodic, complemented by a 1D periodic structure. More recently Duneau [7] and Gratias et al. [8] have studied extended clusters in icosahedral quasicrystals. Their clusters are required to cover the atomic positions of specific models. Covering theory seems to offer a new view of quasiperiodic systems. For its generalization, one needs a general theory of quasiperiodic covering. What type of clusters are candidates for covering? What are their outer shape, internal structure, and symmetry? What are the objects to be covered, and how can a given covering be quantified? What are the mutual overlap rules of the clusters and with what frequency do overlaps appear? Can all the atomic positions be referred to these clusters, and what is the relation of these clusters to the notion of a unit cell in quasicrystals? With the new paradigm of covering on one hand and the theory of latticeprojected dual quasiperiodic tilings and their windows on the other, we examine these questions for dual tilings and look for corresponding coverings. Does the lattice Λ provide geometric objects whose projections can serve as candidates for clusters and for covering? We examine Delone and Voronoi polytopes of the lattice and from them project clusters into the tiling. We start with a summary of the geometry of lattices, quasiperiodic functions, and dual tiling theory. The notions of a fundamental domain and of covering are introduced. General results and constructive methods for Delone and Voronoi clusters are given. The analysis uses an explicit construction of windows for these clusters. The uniqueness and the symmetry breaking of the filling are shown to be general features. The dual 2D quasiperiodic tilings associated with the 4D root lattice A4 , the Penrose and the triangle tilings, illuminate all aspects and properties of these clusters. For the dual 3D icosahedral tilings we construct the Delone and Voronoi clusters. From the window theory, we prove a 98.7% covering of the vertices and an imperfect covering of the tiles. Finally, we propose a method for completing the incomplete covering of the tilings. Our previous work on covering can be found in [22, 23, 24, 25, 27]. We now describe briefly some notions which will be used in the following text. 4.1.1 Lattices in E n , Cells, Sections and Quasiperiodic Functions A lattice in E n arises as an orbit under the action of a discrete translation group Λ. Since no point is stable under Λ, the lattice can be identified with Λ. From Λ as a point set, we can pass to n-polytopes by using the Voronoi and the dual Delone construction. The set of points closer to q ∈ Λ than to any other lattice point form the Voronoi polytope V (q). The Voronoi complex is the set V (q), q ∈ Λ . Delone cells arise from the Voronoi cells by duality. Duality associates uniquely with any p-boundary X of a Voronoi cell a dual

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(n − p)-boundary X ∗ of a Delone cell, and vice versa. Both constructions yield cell complexes compatible with the action of Λ. Any complex-valued function f = f (x) on E n admits an action of Λ according to q ∈ Λ : f (x) → (Tq f )(x) := f (x − q). Λ-periodic functions are invariant under Λ. A linear subspace E m < E n of dimension m < n is an irrational section if it contains at most one point from any orbit on E n under Λ. The restriction of a Λ-periodic function to an irrational E m is a quasiperiodic function. As shown by H. Bohr in 1925 [3], any quasiperiodic function may be obtained by a construction of this type. Such a function can take values on an unbounded domain contained in E m . The lifting of quasiperiodic structures into periodic structures of higher dimension is the basis for the Fourier and structural theory of quasicrystals. A choice of an irrational subspace may arise from the maximal point group G or the holohedry of the lattice Λ as follows. Choose a subgroup H ∈ G such that the representation of H on E n is reducible. Split E n into linear subspaces irreducible under H. If the point group H is incompatible with any lattice in the irreducible subspaces then these subspaces must be irrational. All the irrational subspaces used for quasicrystals are related to point groups in this fashion. 4.1.2 Dual Quasiperiodic Canonical Tilings and Windows Consider a lattice in E n , an irrational subspace E m := E irreducible under the action of a point group H, and its orthogonal complement E n−m ⊥ E m , E n−m := E⊥ . Construct from the Voronoi and Delone complexes in E n the dual boundaries X, X ∗ of dimensions m and n − m. Project these boundaries onto E and E⊥ , respectively, and from their projections form ∗ the direct product polytopes X × X⊥ and X∗ × X⊥ . Then any section in E n parallel to E determines two dual quasiperiodic tilings, whose tiles are X and X∗ , respectively. We term these tilings (T , Λ) and (T ∗ , Λ) respectively. The windows for these dual tilings are perpendicular projections of a set of Delone polytopes Dj and of a set of Voronoi domains V , respectively. The vertices of the tiles project from vertices of Voronoi polytopes and from lattice points, respectively. The windows determine the projections of these sets of points from the lattice onto the tiling. Subwindows within these windows for points determine the hierarchy of projected boundaries up to the tiles of the tiling. 4.1.3 Fundamental Domains and Coverings for Quasiperiodic Tilings We consider a quasiperiodic tiling T whose tiles are all translated copies from a finite set Pi of prototiles. Given a linear space of complex-valued functions f with domain E , we call f compatible with the tiling T if its values

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are repeated on any translated copy of the tiling. The values of a compatible function f are specified on E once they are chosen on the set Pi , and we call this set a fundamental domain with respect to the tiling T . These compatible functions replace, in the absence of a lattice, the Λ-periodic functions on crystallographic unit cells. Like these, they can describe the physics of atomic positions or charge densities, whereas quantum states would belong to a more general class of functions. A function compatible with a tiling is still quasiperiodic, but in contrast to a general quasiperiodic function, takes values on a bounded domain of dimension dim(E ). This property is in line with the structure found in quasicrystals. For a tiling, any point of E belongs to one and, after appropriate treatment of boundaries, to only one tile. For a covering of a tiling we require clusters in the tiling, such that any object of the tiling (vertex, edge, . . . , tile) is covered by at least one such cluster. We admit multiple covering of an object. Of particular interest is the covering of vertex points and the covering of tiles in the tiling. The amount of overlap of the clusters is quantified by the thickness of the covering. 4.1.4 Voronoi and Delone Clusters: Theory and General Results In the dual tilings (T , Λ) and (T ∗ , Λ), respectively, we consider the parallel projections of Delone and Voronoi polytopes Di and V , respectively, and call these projections Delone and Voronoi clusters. The centers of these clusters are projections of vertices of Voronoi polytopes or of lattice points. We require that these projections be compatible with the tilings. We give general and constructive results related to the windows for these clusters. From these, we show that each one of them has a unique filling. The filling, in general, breaks the point symmetry of the outer shape of the cluster, and appears in all orientations allowed by the point group. The construction of windows allows us to test the fundamental domain and the covering properties of these clusters. It turns out that these depend on the details of the tilings. 4.1.5 Voronoi and Delone Clusters in 2D Quasiperiodic Tilings The well-known Penrose and T¨ ubingen triangle tiling can be interpreted as dual 2D tilings projected from the root lattice A4 . Voronoi clusters cover the Penrose tiling. In pairs, they form fundamental domains. Four types of Delone clusters correspond to four vertex classes of the Voronoi polytope. We construct in detail their subwindows within the overall window for the tiling. These subwindows cover both the vertices and the tiles of the T¨ ubingen triangle tiling and together form a fundamental domain. We determine the thickness of these coverings. We construct all 35 local configurations of the clusters and their relative frequencies.


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4.1.6 V- and D-clusters in Dual Canonical Icosahedral Tilings For dual icosahedral tilings belonging to the primitive P -module and to the F -module, we construct the Voronoi and Delone clusters according to the general theory and determine their windows. For tilings associated with the icosahedral F -module, we analyze in detail the Delone clusters projected from the three types of Delone cells. We construct their subwindows within the general window of the tiling. The Delone clusters cover 98.7% of the vertices but fail to cover all the tiles. Similar results are given for the tilings associated with the primitive P -module. We can augment the clusters with additional objects with corresponding windows and thereby achieve a covering of all the tiles and vertices. The thickness of this extended covering is determined. The text is organized as follows. In Sects. 4.2–4.6 we introduce general notions and results related to lattices, their holes, their dual cells and boundaries, dual tiling theory. We give general constructive theorems for Voronoi and Delone clusters within dual tilings. Up to their orientation, the clusters have a definite and unique filling. This filling breaks the point symmetry of the outer shape. It is compatible with the tiling and therefore organizes frames of reference which encompass several tiles and a variety of atomic positions. We examine the notion of a fundamental domain for point sets and for function spaces give a definition of it in the context of dual quasiperiodic tilings. We introduce the notion of a covering in the context of tilings. We obtain a distinction between a covering of vertices and a covering of tiles and give criteria for both. In the next sections, Sects. 4.7–4.11, we develop, on the basis of [24], the concept of Delone clusters in full detail for the triangle tiling with 5-fold symmetry. This tiling is dual to the Penrose rhombus tiling and, like that tiling, can be projected from the root lattice Λ = A4 ∈ E 4 (compare [1]). We examine the covering and fundamental-domain properties of the Delone clusters. We derive the linkage properties of the Delone clusters and obtain their types and relative frequencies. In Sects. 4.12–4.20 we implement Delone clusters and their windows in dual icosahedral tilings based on the primitive hypercubic lattices Λ = P and on the F -lattice, which is equivalent to the root lattice Λ = D6 . Here we elaborate results announced in [25] and derived in [26]. The lattice D6 appears very often in icosahedral quasicrystals. We start from D6 and show that the properties for P can be derived from it. We derive the unique fillings of these clusters in the tilings. We examine the fundamental domain and the covering properties and find specific relations depending on the lattice. The distinction between the covering of quasiperiodic points and of quasiperiodic tiles becomes manifest in the icosahedral tilings.

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4.2 Voronoi and Delone Polytopes and Dual Boundaries Consider, for even n, a lattice Λ whose basis spans E n . We pass from the lattices Λ ⊂ E n to dual polytopes and, by projection, form the building blocks for tilings and their windows. For general properties of these polytopes we refer to [5], and for their use in quasiperiodic tiling theory to [18]. The following properties apply to all lattices under consideration. The Voronoi polytope at a point q ∈ Λ, Λ ⊂ E n , is the set of points V (q) = x|q = q → |x − q| ≤ |x − q |. It is a convex polytope bounded by hyperplanes. A face or boundary X, X ∗ of dimension p will be termed a p-boundary, in a notation adopted from [31] p. 96. We denote the p-boundaries of dimension p, 0 ≤ p ≤ n,by X(q), where the argument q keeps track of the center of the bounded Voronoi domain. The 0-boundaries, the vertices of the Voronoi cells, are the holes ([5] p. 33) of the lattice. A p-boundary X(q) will in general also bound other Voronoi domains V (q ) with q = q. We define dual boundaries as follows (note that we have chosen to number definitions and propositions in a single sequence): Definition 1: Dual boundaries. For any fixed p-boundary X(q) ∈ V (q), let s(X) = q denote the set of all lattice points q (including q) which have X = X(q) as a proper boundary of dimension p of their Voronoi cell V (q ). (i) The intersection polytope Y := ∩q ∈s(X) V (q ) determines the boundary X =Y. (ii) The boundary X ∗ dual to X is the polytope defined as the convex hull X ∗ := conv(q ), q ∈ s(X). Boundaries and their duals have complementary dimensions (p, n−p), 0 ≤ p ≤ n. Given a boundary X(q), the vertices or 0-boundaries q ∈ X ∗ (q) of its dual X ∗ (q) determine, by Definition 1, the set s(X), so that we may write s(X) = q = s(X ∗ ). The duals to the 0-boundaries, or holes, h are the Delone cells Dh ([5] p. 35), of the lattice. These dual Delone cells are bounded by dual (n − p)-boundaries X ∗ . The following general inclusion properties arise [18] from duality: Proposition 2: Inclusion relations of dual boundaries. Consider dual pairs of boundaries X, X ∗ and Y, Y ∗ . (i) If X ⊂ Y then Y ∗ ⊂ X ∗ . (ii) If X ∗ ⊂ Y ∗ then Y ⊂ X. Under the action of Λ, both the holes h and the Delone cells Dh fall into a finite number of distinct orbits, which we denote by h = a, b, . . ..


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4.3 Dual Tilings and Their Windows A decomposition E n = E + E⊥ , dim(E ) = dim(E⊥ ) = n/2 = integer, irrational with respect to the lattice Λ ⊂ E n , arises in quasicrystals with a noncrystallographic point symmetry. For icosahedral point symmetry, compare Sect. 4.12. This decomposition allows one to project boundaries and their duals. For the (n/2)-boundaries of Voronoi and Delone cells, we define in E n the klotz polytopes [17, 18] as the direct products ∗ Xj (q) × Xj⊥ (q) , ∗ Xj (q) × Xj⊥ (q) .

(4.1)

where the index j labels different (n/2)-boundaries. The next two results were obtained in [18]: Proposition 3: Klotz polytopes. The klotz polytopes (4.1) in E n have the following properties: (i) They form a Λ-periodic tiling of E n . (ii) Any boundary of a klotz polytope is either parallel or perpendicular to the subspaces E and E⊥ . (iii) If the set of boundaries Xj (q) at a fixed lattice point q forms orbit representatives under Λ, then the corresponding set of representative klotz polytopes is a fundamental domain (Sect. 4.4) with respect to Λ. Proposition 4: Canonical tilings. The cuts E + c⊥ through the first or second klotz construction of (4.1) are two tilings (T , Λ), (T ∗ , Λ). The tiles ∗ are projections of the (n/2)-boundaries Xj and Xj from the Voronoi and Delone cells, respectively. It suffices to let c⊥ run over the projection of the Voronoi domain V⊥ (0) < E⊥ . Let us now turn to the window description of these tilings. A window w(X ∗ ) ∈ E⊥ is defined as a polytope in E n such that X ∗ ∈ T ∗ appears whenever (E + c⊥ ) ∩ w(X ∗ ) = 0. In this description, the windows must be attached to all lattice points. In what follows, we shall consider mainly the tilings (T ∗ , Λ). The technical advantage is that these tilings have a single window. For hypercubic lattices, the Voronoi and Delone complexes are essentially equivalent. Examples of dual pairs are the Penrose tiling (T , A4 ) and the triangle tiling (T ∗ , A4 ). Proposition 5: Tile windows. The windows for the tiles in the canonical ∗ ∗ , w(Xj ) = X⊥ , respectively. The window for the tilings are w(Xj ) = X⊥ projected lattice points q , which in (T ∗ , Λ) form the vertices of the tiles, is V⊥ . Proof: The first part follows from the properties of the klotz construc∗ appears whenever its dual Xj⊥ ∈ V⊥ intersects with tion (4.1). The tile Xj

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E + c⊥ . By use of the dual inclusion (Proposition 2), one finds that all pro∗ jected lattice points q are 0-boundaries of a tile Xj . The projected dual Voronoi cell V⊥ must therefore contain the tile window Xj⊥ , and this tile window in turn must contain the perpendicular projection q⊥ = q − q . An alternative description of the tilings uses windows at one fixed lattice point only, say q = 0. This can be seen as follows. Assume c⊥ ∈ V⊥ (0). The ∗ ) = 0 determines a union of tiles intersection of tile windows with c⊥ ∩ w(Xj ∗ ∗ ∪j Xj which share a vertex of (T , Λ) and close the solid angle around it. This is a vertex configuration. Each Delone edge or 1-boundary of a tile of (T ∗ , Λ) in this configuration is the projection q⊥ of a lattice vector q = q + q⊥ . We can move in the tiling along q = q − q⊥ from the initial vertex to the next one and refer it to the new V⊥ (q). This is equivalent to the replacement c⊥ → c⊥ − q⊥ while keeping the initial window V⊥ (0). So we may keep the initial window and its boundaries, transform the value of c⊥ by −q⊥ in the window V⊥ , and move by q = q − q⊥ from vertex to vertex in the tiling.

4.4 Fundamental Domains and Spaces of Functions To explore unit cell properties for quasicrystals, we need the concept of a fundamental domain as a particular point set under the action of translations. We distinguish a geometric action on point sets from an action by linear operators on elements of a linear space L of functions. The class of functions we have in mind should, in a crystal or a quasicrystal, describe observables, such as atomic positions or the electronic charge density, which exhibit a (quasi-)periodic symmetry. For these observables, the fundamental domain should provide a part of their domain of definition which is sufficient to know their values on the full domain. A different class of functions is provided by the electronic states. In a perfect crystal these states are Bloch states with a momentum-like Bloch label κ and transform according to an irreducible representation, characterized by the Bloch label, of the lattice translation group. This irreducible representation need not be the identity representation corresponding to κ = 0 with respect to the periodic symmetry, and so these states pick up a phase factor depending on κ on moving through the unit cell. The electronic states in a quasicrystal form a still more general class since, owing to the lack of periodicity, they cannot be characterized by an analog of a Bloch label. We return to the first class of observables and first follow the standard notions for crystals. For E n equipped with a lattice Λ, we recall the following well-known definition: Definition 6: Fundamental domain under translations. Consider the geometric action Λ × E n → E n given by q ∈ Λ, x ∈ E n : (q, x) → x = x + q. A fundamental domain F of E n under Λ is a subset which has exactly one point from each orbit under this action.


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Definition 7: Fundamental domain for periodic functions. Consider a complex-valued function f with a domain of definition E n and define, for q ∈ Λ, group operators by Tq : f (x) → (Tq f )(x) := f (x − q). Suppose that f is Λ-periodic. A fundamental domain for f is a subset F (Λ) of points such that any value taken by f on E n − F(Λ) can be obtained by the group action of Λ on f . Both notions yield the same candidates for fundamental domains, which may be used as cells for the lattice. The shape of the fundamental domain is not unique; the primitive cell and the Voronoi cell of a lattice are both examples of fundamental domains. The volume |F(Λ)| is unique. We need an extension of the notion of a fundamental domain to quasiperiodic systems, in the absence of periodic lattice symmetry. Consider an irrational linear subspace E ⊂ E n . The restriction of a Λ-periodic function f from E n to the domain E of dimension m determines a general quasiperiodic function on E compare [3, 2]. In general, the irrational subspace E will slice E n and hit a dense subset of the n-dimensional unit cell modulo Λ. It follows that a reasonable generalization of the fundamental domain for a general quasiperiodic function on the m-dimensional E is still the n−dimensional fundamental domain F = F (Λ). A different concept of fundamental domain can be obtained if we pass from the general class of quasiperiodic functions to specific subclasses associated with certain quasiperiodic tilings, projected again from a lattice Λ ∈ E n onto a subspace E of dimension dim(E ) < n. For these quasiperiodic tilings, we can require the quasiperiodic functions to be compatible with the tiling. We follow [17, 22] and extend the notions of Definitions 6 and 7 by Definition 8: Fundamental domain for quasiperiodic tilings. Let a quasiperiodic tiling (T , Λ) consist of a minimal finite set Pi of prototiles Pi ∈ E and their translates. A fundamental domain F (T , Λ) is a subset of points in E which contains one and only one translate of any point from any prototile. Definition 9: Fundamental domain for quasiperiodic functions on tilings. A quasiperiodic function f on E is called compatible with the tiling (T , Λ) if its values are repeated on all the translates in (T , Λ) of any prototile. A fundamental domain for such a function is a subset F (T , Λ) ⊂ E such that any value taken by f on E − F(T , Λ) arises by a translation between tiles in the tiling from an identical value on F (T , Λ) ⊂ E . To describe the atomic structure in a quasicrystal by a quasiperiodic tiling, it seems natural to assume the same atomic positions on tiles that have the same orientation but differ by a vector from the quasicrystal module. The notions Definitions 8 and 9 yield the same candidates for fundamental domains. In contrast to the situation for general quasiperiodic functions, these fundamental domains can be taken as bounded point sets in E of dimension dim(E ). One natural choice for them is the set of all points from all the prototiles. The volume of the fundamental domain for this and any

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other choice, can be determined in terms of the prototiles from |F(T , Λ)| = i |Pi | < ∞. The generalization from the periodic case should be kept in mind: The translation vectors of a prototile in a quasiperiodic tiling all belong to the module Λ . If two translates of a fixed prototile occur in the tiling, the sum of the two translation vectors need not in general be a translation vector in the tiling for this prototile. In standard periodic crystallography, translations combine with point group operations into elements of the space group. A set of orbit representatives under the full space group is then called the asymmetric unit. Usually the asymmetric unit can be taken as a part of the fundamental domain under translations. Since, in particular, the Voronoi domain is transformed into itself under the point group, one can often determine the asymmetric unit as a certain sector of the Voronoi domain. We emphasize that our Definitions 8 and 9 of fundamental domains, as they stand, do not yet include the action of point groups in quasicrystals, although it is possible to implement point symmetry and orientational order. The notion of a fundamental domain prepares the ground for examining the concept of a quasi-unit cell proposed by Jeong and Steinhardt [32]. A quasi-unit cell should be a connected geometric object or cluster which (i) allows one to assign all atomic positions to it, and (ii) allows one to build up the full quasicrystal by the quasiperiodic repetition of this object. In tiling theories, we would interpret the condition (i) by demanding that the geometric object should be a fundamental domain as defined in Definitions 8 and 9. For condition (ii) to hold, we would require that copies of the geometric object cover the tiling. We shall sharpen the notion of covering in Sect. 4.6. The detailed study of Voronoi and Delone clusters in the following Sects. 4.7–4.20 will allow us to address the notion of a quasi-unit cell for tilings with 5-fold and icosahedral symmetry.

4.5 Delone Clusters and Their Windows We explore the properties of Delone clusters here. Definition 10: Delone clusters. A Delone cluster Dh in the tiling (T ∗ , Λ) is (compare [22]), the parallel projection Dh of a Delone cell from the lattice Λ. It would be easy, in dual tiling theory, to give a prescription for projecting Delone polytopes. But, as we wish to relate these projections to the dual tilings, we require: Definition 11: Compatible projection. The projection of a Delone polytope Dh onto E is compatible with the tiling (T ∗ , Λ) if both the interior and the boundary of the projection are part of the tiling. We shall construct filled Delone clusters compatible with the tiling from a prescription for their windows w(Dh ).


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Proposition 12: Filling of a Delone cluster . Fix a hole point h⊥ in E⊥ and consider all the projected (n/2)-boundaries Xj⊥ (h), (i) each with a vertex h at this hole point. Determine a maximal intersection w of these projected boundaries which (ii) share at least one fixed interior point x⊥ = h⊥ , and define w = ∩j Xj⊥ (h) .

(4.2)

Then w = w(Dh ) is the window for a filled Delone cluster Dh . The Delone cluster is the union of all the projected tiles dual to the ones occuring in (4.2), ∗ Dh = ∪j Xj .

(4.3)

where j in (4.3) runs over the same set as in (4.2). This union is a filling of Dh with no gaps and no overlaps of dimension (n/2), and is compatible with the tiling. Proof: The hole point h, by assumption (i) is a 0-boundary contained in any of the (n/2)-boundary Xj (h) of the intersection (4.2), h ⊂ Xj (h). The dual to this 0-boundary is a Delone cell Dh . With Proposition 2 it follows that Xj∗ ⊂ Dh . This subset property extends to the projections. The tile windows for any pair Xj⊥ , Xl⊥ , j = l, by assumption (ii), intersect in at least one interior point x⊥ . Consider the corresponding klotz polytopes (4.1) indexed by j, l. Their projections onto E⊥ share an interior point. For their ∗ ∗ ∩ Xl = 0 except for points at projections onto E , we conclude that Xj boundaries of tiles, and so there is no overlap for j = l. Otherwise the full nD klotz polytopes would share interior points. We now describe the complex relation of the filled Delone cluster to the tiling. In the window description of Proposition 12, the intersecting tiles Xj⊥ (h) which form the window w(h) share the hole position h but belong to Voronoi cells at various lattice points q. These lattice points, for each coding tile, can be found as follows. Go within E⊥ , for each tile window in (4.2) to its ∗ ∗ . The vertices of Xj⊥ , by Definition 1, are produal according to Xj⊥ → Xj⊥ jections q⊥ of all the lattice points q whose Voronoi cells have Xj as a boundary. To find from (4.2) all vertex configurations which contribute to the filling (4.3), we must collect the full set of lattice points s(Dh ) = q |q ∈ Xj∗ , Xj⊥ ∈ w(Dh ). Starting from one such projection q⊥ ∈ V⊥ (q), an initial window w(h) must belong to V⊥ (q). We shift this initial window to all the positions w(h) + (q − q)⊥ , q ∈ s(Dh ). All these shifted windows are inside V⊥ (q) and determine parts of the full window for a fixed orientation. In the tiling the set of parallel projections q , q ∈ s(Dh ) becomes the complete vertex set from which the filling Dh can be seen in the tiling. The shifted windows in V⊥ (q) determine all the vertex configurations appearing in this filling. The vertex configurations inside the filling are complete, those at the boundary of the filling are incomplete.

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The filling Dh can occur in various orientations. By application of the local point group G at a fixed hole of type h in V⊥ , we find what we shall call a G-window wG (h) := G × w(h). This window codes all the orientations of the filling Dh , seen from a fixed vertex. By repeating this construction at every hole of type h in V⊥ one finds the total window for this type. All these constructions are exemplified in detail in Sects. 4.7–4.11 for Delone clusters in the quasiperiodic triangle tiling. An application of this tiling to atomic positions in quasicrystals and electronic stales is given in[21].

4.6 Covering by Delone Clusters Once the windows of the Delone clusters have been constructed, their covering properties can be explored by relating these windows to the windows of the tiling. We distinguish covering of vertices from covering of tiles. From dual tiling theory we give criteria for such coverings. Proposition 13: Window criterion for covering of vertex points. The Delone clusters form a covering of lattice points q in the tiling (T ∗ , Λ) ⊂ E or vertex i if and only if the collection i wG (D )(hi⊥ ) of all shifted G-windows covers V⊥ . Proof: If the criterion is fulfilled, any point in a tile window q⊥ ∈ X⊥ ⊂ w(X∗ ) is in the window of at least one Delone cluster. It follows that the corresponding point q = (q − q⊥) ∈ X∗ is covered by this Delone cluster. The converse works as well. Proposition 14: Window criterion for covering of tiles. A tile X∗ in the tiling is covered by at least one Delone cluster if and only if its window w(X∗ ) is covered by all the windows w(h) centered on the local hole vertices which belong to its tile window, h⊥ ∈ X⊥ . Proof: Any hole vertex h⊥ ∈ X⊥ is (the projection of) an intersection of the (n/2)-boundary Xj⊥ with a 0-boundary of a Voronoi cell. The dual ∗ inclusion relation, according to Proposition 2 in Sect. 4.2, is that Xj , as h a boundary, belongs to the Delone cell D . Covering properties characterize infinite quasiperiodic tilings in E . Once we have constructed the windows for the covering clusters, the criteria of Propositions 13 and 14 allow us to check the covering properties by studying the intersection of finite and finitely many windows in E⊥ . The distinction between the coverings according to Propositions 13 and 14 is relevant. Clearly the covering of tiles according to Proposition 14 implies the covering of lattice points Proposition 13. The converse is not true. It is easy to imagine a full covering of all the vertices in a tiling which does not cover all the points of the tiles. A full covering of all points x ∈ E in a tiling requires that every point of every tile be covered. In the 2D triangle tiling,


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to be analyzed in the next sections, the Delone clusters cover all tiles and therefore all vertices. We shall see that in icosahedral tilings this is not the case.

4.7 The Lattice A4 and the Triangle Tiling We refer to [1] for a detailed description of the root lattice A4 , its geometry and its projection. It proves useful to start from an orthonormal system of vectors in E 5 . A standard basis of the root lattice A4 in the subspace E 4 orthogonal to the vector e1 + e2 + e3 + e4 + e5 is ([11] p. 64) (b1 , . . . , b4 ) : b1 = e1 − e2 , b2 = e2 − e3 , b3 = e3 − e4 , b4 = e4 − e5 . (4.4) We prefer to use another system of vectors. In terms of the vectors (e1 , . . . , e5 ) ⊂ E 5 , we form five new vectors aj := ej −

5 5 1 el , j = 1, . . . 5, ai = 0 , 5 1 1

(4.5)

with one linear relation to express all relevant positions in the root lattice A4 ⊂ E 4 . The lattice points q ∈ A4 are then given by those integral linear combinations of the vectors (4.5) whose sums of coefficients are equal to 0 mod 5. The reasons for the use of the vectors (4.5) rather than a lattice basis will become clear after Definition 15. In the root lattice A4 , we construct the Voronoi cells V (q) centered on all lattice points q and the set of dual Delone cells Dh centered on all hole positions h (compare [5] p. 33), which form the vertices of V (q). The Voronoi cell V (q) is bounded by hyperplanes at equal distances between pairs of neighboring lattice points. Any face or lboundary X of dimension l, 0 ≤ l ≤ 4, from V (q) is uniquely determined by an intersection of hyperplanes between a minimal set s(X) := q (X) of lattice points. Its dual (4 − l)-boundary X ∗ is defined, from Definition 1, as the convex hull X ∗ := conv(q ), q ∈ s(X). This is the face or boundary of a dual Delone polytope. The holohedry of the root lattice A4 is given by its Weyl group isomorphic to S5 , [11] p. 66). It is convenient to express the elements of this group as permutations of the basis vectors (e1 , . . . , e5 ) of the hypercubic lattice in E 5 . This Weyl group has the Coxeter group I2 (5), ([12] p. 32), as a subgroup. I2 (5) has two generators I2 (5) = R1 , R2 | R12 = R22 = (R1 R2 )5 = e .

(4.6) 5

Within the Weyl group of A4 , expressed by the permutations in E , the generators of I2 (5) take the form R1 = (25)(34), R2 = (12)(35), R1 R2 = (54321) . 4

(4.7)

The representation in E of I2 (5) generated by the reflections (4.7) is reducible into two 2D inequivalent irreducible representations, each one with

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5-fold rotational symmetry. Moreover, the space inversion i in E4 reduces to corresponding rotations by π in E , E⊥ . With Z2 = e, i, we have the larger point subgroup I2 (5) × Z2 of A4 , which reduces to the same two irreducible subspaces. Note that the inversion is not an element of the Weyl group of A4 . From the crystallographic restriction in 2D which forbids 5-fold point symmetry in a 2D lattice, we can be sure that the two subspaces E , E⊥ corresponding to the two irreducible representations are irrational in the lattice A4 . We shall use these spaces as the spaces for the quasicrystal and for its windows, respectively. In the projection scheme for quasicrystals with 5-fold point symmetry, the irrational plane E serves as the position space and the irrational plane E⊥ as the window space. The vectors aj projected onto these two planes form the two 5-stars shown in Fig. 4.1. To characterize the (shallow and deep) holes in the lattice A4 (compare [5] pp. 108–110), we introduce, seen from the points q of the lattice, the hole positions h=

5

n j aj ,

(4.8)

1

and the modulo function r(h) =

5

nj mod 5 .

(4.9)

1

Definition 15: Holes of Λ = A4 . The shallow holes h of the root lattice A4 are denoted by replacing the letter h by a. They are the point classes (h, r(h)) = (a, 1), (a, 4). The deep holes are denoted by replacing the letter h by b. They are the point classes (h, r(h)) = (b, 3), (b, 2). The inversion i is a rotation by π in both E and E⊥ . Under this operation, the shallow and deep hole classes interchange their roles (see Proposition 16). The class q with r(q) = 0 describes the points of the lattice A4 .

4

3

5

5

2 1

2

3

4 1

Fig. 4.1. The vectors (a1 , . . . , a5 ) form two 5-stars in E (left) and E⊥ (right). They are used to describe the hole and lattice point positions of A4 and their projections onto these two spaces


111

Any hole position h is a vertex, or 0-boundary, of a Voronoi cell and so can be written as the intersection of bounding hyperplanes between a minimal set of lattice points s(h) := q . The Delone cell dual to h is then Dh = conv(q ), q ∈ s(h)(h). Both the simple action of the point group and the unified description of hole and lattice points are our reasons for using the vectors (a1 , . . . , a5 ) (4.5) and not a lattice basis. The notation aj for vectors with an index, should be strictly distinguished from the notation for shallow and deep holes, h = a, b. The Weyl group S5 and the point group I2 (5) × Z2 act with respect to lattice points of A4 . To explore the point symmetry at hole points, we consider the space group formed as the semidirect product A4 ×s (I2 (5) × Z2 ) of the translation group A4 and the point group (I2 (5)×Z2 ). We consider pairs (t, g) of Euclidean translations and point transformations with the standard action and (t, g) : x → gx + t and the standard multiplication rule (t1 , g1 )(t2 , g2 ) = (t1 + g1 t2 , g1 g2 ). Let h be a hole position and g ∈ I2 (5) × J. The element g preserves the point h if and only if its conjugate by the translation vector h from the lattice point q = 0 to the hole point is in the space group, i.e. (h, e)(0, g)(−h, e) = (h − gh, g) = (t, g) ∈ A4 ×s (I2 (5) × Z2 ) .

(4.10)

If this equation is fulfilled, the action (h, e)(0, g)(−h, e) : h → gh shows that we obtain a point symmetry at h. The crucial part of checking whether (4.10) is fulfilled is to see if the translation part is a translation vector from the lattice and fulfills h−gh ∈ A4 . If g is a permutation such as that corresponding to R1 , R2 in (4.7), the modulo function (4.9) is unchanged. In this case one easily finds r(h − gh) = 0, so that from (4.9), h − gh ∈ A4 for all g ∈ I2 (5). If g = i, the modulo function changes its value according to the scheme r(i(a, 1)) = r(a, 4), r(i(b, 3)) = r(b, 2). Then h − ih is not in A4 . We shall take advantage of the transformations i : (a, 1) → (a, 4), (b, 3) → (b, 2) when treating all four classes of holes. Proposition 16: Point symmetry at holes. The point symmetry group at any hole position is I2 (5) < (I2 (5) × Z2 ). We return to the triangle tiling (T ∗ , A4 ). The tiling dual to it is the Penrose tiling (T , A4 ), as treated in [1]. Definition 17: Vertex set of (T ∗ , A4 ). The triangle tiling has the vertex set {q |; q⊥ ∈ V⊥ } .

(4.11)

The window V⊥ for the vertex set is the perpendicular projection of a fixed ∗ ∗ , X2 of Delone cells proVoronoi cell. The tiles are dual 2-boundaries X1 jected onto E . The windows for the tiles are 2-boundaries X1⊥ , X2⊥ of the Voronoi cells projected onto E⊥ . The tiles are two of the golden triangles shown in Fig. 4.4, and the windows are two of the Penrose rhombus tiles shown in Fig. 4.2.

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4.8 Delone Clusters in the Triangle Tiling A Delone cell Dh in the lattice A4 is a 4D polytope with its center at a hole position h and is equipped with a hierarchy of dual boundaries. A Delone cluster is described in Definition 10 as a parallel projection Dh of a Delone cell from the lattice A4 . This cluster will be a polygon in E . Now we consider the filled Delone clusters of Proposition 12 for the tiling (T ∗ , A4 ). Definition 18: Filling of Delone clusters. A filling of (the polygon) Dh is a union of projected dual 2-boundaries Xj∗ which covers Dh exactly and forms a patch of the tiling without gaps or overlaps. Dual tiling theory, as outlined in detail in Sect. 4.5, provides the construction given in Proposition 12 for such a filling. It will be shown that this filling is unique, breaks the local symmetry inside the Delone cluster, and appears in the tiling with equal frequency in all orientations under I2 (5). Proposition 19: Windows and fillings of Delone clusters in (T ∗ , A4 ). In E⊥ , denote by X1⊥ (h), X2⊥ (h) some tiles in a standard orientation with a hole vertex of class h attached to a fixed hole position h⊥ . Determine, by the application of point group elements gl , gk ∈ I2 (5) with respect to the point h⊥ , following Proposition 12, a maximal intersection of tiles gl X1⊥ (h), gk X2⊥ (h) which share at least one interior point x⊥ = h⊥ . Construct in E , by dualization and application of the same point group elements gl , gk , the union of ∗ ∗ , gk X2 . This intersection and this union are the window the dual tiles gl X1 w(h) and the filling Dh , respectively, for a Delone cluster of fixed orientation: w(h) = ∩l,k (gl X1⊥ (h))(gk X2⊥ (h)) , ∗ ∗ (h))(gk X2 (h)) . Dh = ∪l,k (gl X1

(4.12)

We shall evaluate these expressions in the following subsections. 4.8.1 Standard Positions of Dual 2-Boundaries ∗ ∗ In E⊥ , X1⊥ , X1⊥ are a thick rhombus and an obtuse triangle, and X2⊥ , X2⊥ are a thin rhombus and an acute triangle. We refer the standard positions to the center q = 0 of a Voronoi cell. Each one of the boundaries X1 , X2 with a fixed orientation appears as three copies in a Voronoi window (Fig. 4.2). In the tiling (T ∗ , A4 ) ⊂ E , these three copies are the windows for the triangle tiles X1∗ , X2∗ seen from their three vertices. We have dropped the indices for parallel and perpendicular projections. This is allowed by the unique lifting property from both E , and E⊥ to E 4 . We express the boundaries in coordinates with respect to the center q of a Voronoi cell and indicate this by writing Xi (q), Xi∗ (q) (compare Fig. 4.2):

X1 (q) := P (+0 − −0) := (a1 − a3 − a4 )/2 + (λ2 a2 + λ5 a5 )/2 , X1∗ (q) := 0, a1 − a4 , a1 − a3 ,


113

(a, 1) : (a1 − a2 − a3 − a4 − a5 )/2 , a4 − a1 + X1 (q) = a4 − a1 + P (+0 − −0) = P (−0 − +0) , a4 − a1 + X1∗ (q) = a4 − a1 , 0, a4 − a3 , a3 − a1 + X1 (q) = a3 − a1 + P (+0 − −0) = P (−0 + −0) , a3 − a1 + X1∗ (q) = a3 − a1 , a3 − a4 , 0 , X2 (q) := P (+ − 00−) := (a1 − a2 − a5 )/2 + (λ3 a3 + λ4 a4 )/2 , X2∗ (q) := 0, a1 − a2 , a1 − a5 , (a, 1) : (a1 − a2 − a3 − a4 − a5 )/2 , a2 − a1 + X2 (q) = a2 − a1 + P (+ − 00−) = P (− + 00−) , a2 − a1 + X2∗ (q) = a2 − a1 , 0, a2 − a5 , a5 − a1 + X2 (q) = a5 − a1 + P (+ − 00−) = P (− − 00+) , a5 − a1 + X2∗ (q) = a5 − a1 , a5 − a2 , 0 , |λj | ≤ 1 .

(4.13)

The notation is taken from [1]; the triangles Xj∗ are denoted by their vertex set. All other 2-boundaries are obtained from (4.13) by the action of the Coxeter group I2 (5) and of the inversion i.

Fig. 4.2. Positions for rhombus boundaries of fixed orientation in the decagonal Voronoi window V⊥ with center q⊥ (black square). The two standard positions X1⊥ , X2⊥ have reflection symmetry under (25)(34). The hole vertex (h, r(h)) = (a, 1) is marked on each rhombus by a black circle. The hole vertices h = b are marked by white circles; the full class identification is given in Fig. 4.9

(a,j)

4.8.2 Delone Clusters D

and Their Windows

Each rhombus tile X1 , X2 has a single vertex of hole type h = a. In the standard positions of (4.13) we list the corresponding coordinates. For the Delone cluster it is convenient to rewrite the boundaries in terms of coordinates with respect to this unique hole position. The vector for X1 , X2 in the standard position from q = 0 to this hole position is always t = a1 . We denote the boundaries referred to a hole position (a, 1) as Xi (a, 1), Xi∗ (a, 1) and find

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X1 (a, 1) = −a1 + P (+0 − −0) , X1∗ (a, 1) = −a1 + 0, a1 − a4 , a1 − a3 , (a, 1) : 0 , X2 (a, 1) = −a1 + P (+ − 00−) , X2∗ (a, 1) = −a1 + 0, a1 − a2 , a1 − a5 , (a, 1) : 0 .

(4.14)

The Window for a Fixed Orientation and Hole Class (a, 1). After choosing a particular orientation, we arrive at the following expressions for the intersections and unions of tiles entering (4.12). The combination of these tiles yields expressions for the window w(a, 1) = (a,1) (a,1) w(D ) and the filling D according to (4.12), in the form w(a, 1) = (−a1 + P (+0 − −0)) ∩ (−a2 + P (− + −00)) ∩(−a5 + P (−00 − +)) , (a,1) D

= (−a1 + 0, a1 − a4 , a1 − a3 ) ∪ (−a2 + 0, a2 − a3 , a2 − a1 ) ∪(−a5 + 0, a5 − a1 , a5 − a4 ) .

(4.15)

It is understood that all expressions for windows refer to E⊥ and all expressions for fillings refer to E . The window w(a, 1) is a cone at the hole (a, 1) with an opening angle of 2π/5, taken from a decagon scaled by τ −2 with respect to V⊥ . This window is shown in Fig. 4.3. Table 4.1. Translation t and rotation g of tiles for the windows and filling of hole class (a, 1) t

a1

g

e

gX1 (a, 1)

−a1 + P (+0 − −0)

gX1∗ (a, 1)

−a1 + 0, a1 − a4 , a1 − a3

t

a1

a1

g

(12345)

(15432)

gX2 (a, 1)

−a2 + P (− + −00)

−a5 + P (−00 − +)

gX2∗ (a, 1)

−a2 + 0, a2 − a3 , a2 − a1

−a5 + 0, a5 − a1 , a5 − a4

We now wish to characterize the filling D within the tiling (T ∗ , A4 ). In terms of windows, we must relate the window w(a, 1) shown in Fig. 4.3 to the window V⊥ for the tiling. In the present case it is possible to represent the window w(a, 1) as an intersection of rhombus tiles which belong to a single (a,1)


115

Fig. 4.3. The window w(a, 1) (4.15) for the Delone (a,1) is the shaded intersection cone of 3 rotated cluster D rhombus tiles at a hole position (h, r(h)) = (a, 1) (black circle). The filling is given in Fig. 4.4

Voronoi window V⊥ . This is shown in Fig. 4.4 on the left, along with the (a,1) on the right. filling D All Windows of Hole Class (a, 1) for a Fixed Orientation. The contributing 2-boundaries Xl which intersect in the window w(a, 1) (4.15) refer to a variety of different Voronoi cells V (q). To obtain the set of all lattice points q which participate in the union of tiles, recall that the dual boundaries Xl∗ are convex hulls for subsets of lattice points from this set. It follows that we can find all the centers q of these Voronoi cells by collecting all the different vertices of the dual tiles gX1∗ , gX2∗ Table 4.1. By inspection we find, seen from the hole position (a, 1), the Voronoi centers q − a = −a1 , −a2 , −a3 , −a4 , −a5 .

(4.16)

The inverses of these vectors are, by application of (4.8) and (4.9), five particular hole positions of type (h, r(h)) = (a, 1) belonging to V (q). (a,1) Now we look for the general occurrence of the filling D with a fixed orientation in the tiling. When checking the tiling vertex by vertex, one can

Fig. 4.4. Left: Window cone w(a, 1), shaded, as intersection of rotated rhombus tiles X1 , X2 at a hole position (h, r(h)) = (a, 1), (black circle), in the decagon V⊥ centered on a lattice point q⊥ , (black square). Right: Filled pentagonal Delone (a,1) ∗ ∗ as union of dual rotated triangles X1 , X2 cluster D

116

Peter Kramer

identify a filling whenever one arrives at one of its vertices. Similarly, as we found three different rhombus windows within V⊥ for a single triangle tile (Fig. 4.2), corresponding to its three vertices, we expect to find 5 windows (a,1) with fixed orientation (Fig. 4.4) corresponding to the for the filling D number of its vertices, 5. The 5 windows can be constructed from w(a) by rewriting this window as w(q) = w(a, 1)−(q−a), seen from its set of 5 Voronoi centers q given in (4.16). We obtain w(a, 1) + a1 = (P (+0 − −0)) ∩ (a1 − a2 + P (− + −00)) ∩(a1 − a5 + P (−00 − +)) = (P (+0 − −0)) ∩ (P (+ − −00)) ∩ (P (+00 − −)) , w(a, 1) + a2 = (a2 − a1 + P (+0 − −0)) ∩ (P (− + −00)) ∩(a2 − a5 + P (−00 − +)) , w(a, 1) + a3 = (a3 − a1 + P (+0 − −0)) ∩ (a3 − a2 + P (− + −00)) ∩(a3 − a5 + P (−00 − +)) = (P (−0 + −0)) ∩ (P (− − +00) ∩ (a3 − a5 + P (−00 − +)) , w(a, 1) + a4 = (a4 − a1 + P (+0 − −0)) ∩ (a4 − a2 + P (− + −00)) ∩(a4 − a5 + P (−00 − +)) = (P (−0 − +0)) ∩ (a4 − a2 + P (− + −00)) ∩ (P (−00 + −)) , w(a, 1) + a5 = (a5 − a1 + P (+0 − −0)) ∩ (a5 − a2 + P (− + −00)) ∩(P (−00 − +)) .

(4.17)

In these expressions we have, by application of (4.13) and its rotated versions in a second step, eliminated all the translations which provide another boundary of the chosen Voronoi cell. In particular, the window w(a, 1) + a1 is an intersection of unshifted boundaries. This window is shown in Fig. 4.4 (a,1) in Fig. 4.4 (right) seen from (left) and corresponds to the filling of D the top vertex. In all other cases the window in the Voronoi domain is an intersection cone that involves one or two translated boundaries. We show the position of all 5 window cones obtained from (4.17) on the left of Fig. 4.5. There is a one-to-one correspondence between the hole position of a window cone and a vertex in the filling. Total Window for All Orientations and Hole Class (a, 1). Each cone (a,1) in (4.17) is the window of a filled Delone cluster D of the same fixed orientation but seen from a different vertex. Any new orientation obtained by 5-fold rotation yields, in the Voronoi window, another set of 5 window cones. The reflection (25)(34) transforms both the initial windows and the filling in Fig. 4.5 into themselves. The total window under all these operations can be described as follows: It consists of 5 decagons centered on the 5 hole positions and scaled linearly, in comparison with the Voronoi window, as τ −2 V⊥ . This total window is shown on the right of Fig. 4.5.


117

Fig. 4.5. Left: the 5 cones w(a, 1) at hole positions (h, r(h)) = (a, 1) in V⊥ are the (a,1) seen from its 5 vertices. Right: 5-fold rotations and windows for the filling D superposition of the cones on the left-hand side generate the total window for all holes of class (a, 1). The total window consists of 5 scaled decagons τ −2 V⊥ centered on the 5 hole positions of class (h, r(h)) = (a, 1)

Windows for the Hole Class (a, 4). Finally, we apply the inversion i. The 5 hole positions (h, r(h)) = (a, 1) go into 5 hole positions of the class (h, r(h)) = (a, 4). There are 5 new decagonal windows and 5 new orientations of the filled Delone cluster Da . These inverted windows and fillings are not shown in the figures. We summarize the results on the Delone clusters Da as follows: (a,j)

Proposition 20: Filling of Delone clusters D(a,j) . The Delone clusters D have a unique filling. The filling has a mirror symmetry and appears in 10 orientations, 5 for class (h, r(h)) = (a, 1) and 5 for class (h, r(h)) = (a, 4). The total windows for all orientations are scaled decagons τ −2 V⊥ , centered on all hole positions h = a of and intersected with V⊥ . (b,j)

4.8.3 Delone Clusters D

and their Windows

We again start from the window side. The tiles X1 , X2 each have three vertex holes of type b. Therefore we obtain a variety of vectors t from the standard positions (4.13). Again we denote by Xl (b, j) the coding tiles, in coordinates relative to the fixed hole position, and denote by Xl∗ (b, j) their duals. In Sect. 4.7, it was shown that the point group elements g ∈ I2 (5), applied with respect to hole positions, are symmetries of the lattice. In Table 4.2 we apply the inversion i, which does not belong to I2 (5), with respect to hole positions. Instead of introducing a second set of standard 2-boundaries, we have extended Proposition 12. The interpretation is that i acts geometrically on the standard rhombus tiles and their duals but, as was explained before Proposition 16, interchanges the subclasses of holes (a1, a4) and (b3, b2) at the vertices of the rhombus tiles compared to their labels given in Fig. 4.9. In the

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Peter Kramer

intersection (Fig. 4.6), this combined action ensures that all the intersecting rhombus tiles share the hole vertex of class (b, 3). The Window for a Fixed Orientation and Hole Class (b, 3). After choosing a fixed orientation, we find that the window w(b, 3) is obtained as the intersection of the 7 rhombus tiles (Table 4.2). It is a cone of opening angle 2π/10 and part of a scaled decagon τ −2 V⊥ (Fig. 4.6). Table 4.2. Translation t and rotation g of tiles for the windows and filling of hole class (b, 3) t

(−a3 − a4 )

(a1 + a2 )

g

(12345)

i(14253)

gX(b, 3) a4 + a5 + P (0 + 0 − −)

a4 + a5 + P (+ + 0 − 0)

∗

gX (b, 3) a4 + a5 + 0, a2 − a4 , a2 − a5

a4 + a5 + 0, −a4 + a2 , −a4 + a1

t

(a1 + a5 )

(a1 + a5 )

g

i(15432)

i(13524) a2 + a3 + P (+0 − 0+)

gX(b, 3) a4 + a5 + P (0 + +0−)

gX ∗ (b, 3) a4 + a5 + 0, −a5 + a2 , −a5 + a3 a2 + a3 + 0, −a3 + a1 , −a3 + a5 t

(−a2 − a5 )

(−a2 − a5 )

g

(13524)

e

gX(b, 3) a2 + a4 + P (0 − + − 0)

a2 + a5 + P (+ − 00−)

∗

gX (b, 3) a2 + a4 + 0, a3 − a4 , a3 − a2 t

(a1 + a4 )

g

i(14253)

a2 + a5 + 0, a1 − a2 , a1 − a5

gX(b, 3) a2 + a4 + P (00 + −+) gX ∗ (b, 3) a2 + a4 + 0, −a4 + a5 , −a4 + a3

(b,3)

Again we ask about the occurrence of the filling D in the tiling and look into the relation of its window shown in Fig. 4.6 to the Voronoi window. It is not possible to represent the window w(b, 3) as an intersection of rhombus boundaries of a single Voronoi cell. Seen from the point of view of the tiling, (b,3) the reason is that in D (Fig. 4.7), there is no vertex q shared by all the tiles Xj∗ of the filling, in contrast to D . To represent the window w(b, 3) within V⊥ , we must admit rotated and translated rhombus tiles in a single decagon. One such representation is given in (4.18) and shown in Fig. 4.7. The window on the left-hand side yields the filling on the right-hand side, seen from its lower internal vertex configuration. (a,j)


119

Fig. 4.6. The window w(b, 3) is the shaded intersection cone of 7 rhombus tiles attached to a hole (h, r(h)) = (b, 3), (white circle). The 7 tiles are marked, in addition, (b,3) is by their holes h = a (black circles). The filling D shown in Fig. 4.7

The window and filling for q − b = −a2 − a5 are given by w(b, 3) − a2 − a5 = (−a2 + a4 + P (+ + 0 − 0)) ∩ (−a2 + a4 + P (+ + 0 − 0)) ∩(−a2 + a4 + P (0 + +0−)) ∩ (a3 − a5 + P (+0 − 0+)) ∩(a4 − a5 + P (0 − + − 0)) ∩ P (+ − 00−) ∩(a4 − a5 + P (+ − 00−)) , (b,3) D

= (−a2 + a4 + 0, a2 − a4 , a2 − a5 ) ∪(−a2 + a4 + 0, −a4 + a2 , −a4 + a1 ) ∪(−a2 + a4 + 0, −a5 + a2 , −a5 + a3 ) ∪(a3 − a5 + 0, −a3 + a1 , −a3 + a5 ) ∪(a4 − a5 + 0, a3 − a4 , a3 − a2 ) ∪ 0, a1 − a2 , a1 − a5 ∪(a4 − a5 + 0, −a4 + a5 , −a4 + a3 ) .

(4.18)

All Windows for a Fixed Orientation and Hole Class (b, 3). We evaluate, as seen from the hole vertex, all the Voronoi centres which appear in Table 4.2 as vertices of any X1∗ (b), X2∗ (b), to obtain q − b = a 1 + a 2 , a1 + a 5 , a2 + a 3 , a2 + a 4 , a2 + a5 , a3 + a 4 .

(4.19)

The 7 positions b−q are holes of class (h, r(h)) = (b, 3) but do not exhaust the representatives of this class within one Voronoi cell. By the same reasoning as before, the 7 vectors b − q produce 7 window cones w(b) + b − q on V⊥ . (b,j) In the tiling, these cones correspond to the 7 vertices of the filling D (Fig. 4.7). These windows are shown on the left of Fig. 4.8. Again there is a one-to-one relation between the hole position of a window cone and a vertex of the filling. The filling has two internal vertices. These vertices correspond to window cones at hole positions on vertices of the decagon.

120

Peter Kramer

Fig. 4.7. Left: window cone w(b, 3), shaded, (4.18), as the intersection of rotated and translated rhombus tiles X1⊥ , X2⊥ at a hole position (b, 3) (white circle), in (b,3) as the union of dual the decagon V⊥ . Right: filled pentagonal Delone cluster D ∗ ∗ rotated and translated triangles X1 , X2

Fig. 4.8. Left: the 7 cones w(b, 3) at hole positions (b, 3), in V⊥ are the windows (b,3) seen from its 7 vertices. Right: 5-fold rotations, the reflection for the filling D (25)(34) in the vertical line, and superposition of the cones on the left-hand side generate the total window for all holes of class (b, 3) (white circles). The total window consists of 10 scaled decagons τ −2 V⊥ , centered on 10 hole positions (b, 3) and intersecting with V⊥

Total Window for all Orientations and Hole Class (b, 3). Applying all 5-fold rotations to the 7 windows, we obtain altogether 7 × 5 = 35 window cones, located now at all the 10 holes of type (b, 3) of the Voronoi cell. Next we include the reflection (25)(34), which still keeps the same class of holes. (b,3) are changed Both the initial 7 windows w(b, 3) and the initial filling D into different forms by reflection. Under 5-fold rotation, the reflected window cones fit precisely in between the first 35 window cones and fill up the scaled decagons. The total window can now be described as follows. First consider


121

scaled decagons τ −2 V⊥ at all 10 representative hole positions. Drop from these decagons all the sectors which fall outside of V⊥ . The total number of cones is 5(10+4) = 70. This total window is shown in Fig. 4.8. By comparison with the previous section, we observe that the window cones w(b, 3) at all hole positions on the edges of the decagon V⊥ correspond to internal vertices (b,3) of the Delone filling D . Windows for Hole Class (b, 2). Next we apply the inversion i. This transforms the 10 holes from class (b, 3) to class (b, 2) and, together with the reflection (25)(34), gives a new total window. To the windows of this kind (b,2) there correspond 10 more rotated fillings D . These inverted windows are not shown in the figures. We summarize the results on the Delone clusters Db as follows: (b,j)

Proposition 21: Filling of Delone clusters D(b,j) . The Delone clusters D have a unique filling. This filling has no point symmetry with respect to its center and appears in 20 orientations, 10 for class (b, 3) and 10 for class (b, 2). The windows, for all orientations, are scaled decagons τ −2 V⊥ , centered on all hole positions h = b of and intersected with V⊥ .

4.9 Delone Covering of the Triangle Tiling Given a vertex of the tiling (T ∗ , A4 ), does it always belong to at least one Delone cluster? Are all the tiles of the tiling covered by Delone clusters, and what is the covering fraction? We analyze these points in this section. 4.9.1 Covering of Vertices and Tiles We analyze the covering of a vertex q ⊂ (T ∗ , A4 ) in terms of the windows. By applying the criterion of Proposition 13 to the triangle tiling, we find that a vertex is covered if and only if q⊥ belongs to the window of at least one Delone cluster. To check what fraction of vertices in the tiling is covered by a Delone cluster one must superpose the total windows for all four hole classes. Note that these windows include the occurrence of fillings seen from any one of their vertices. It is easy to see from Fig. 4.5 and 4.8 and their versions rotated by 2π/10 that the four total windows together cover all the points of the decagon V⊥ . As a result, we obtain Proposition 22: Delone covering of all vertices of (T ∗ , A4 ). Any vertex of the tiling (T ∗ , A4 ) is covered by at least one Delone cluster. Consider next the Delone covering for complete tiles of the tiling. We apply the window criterion Proposition 14 to the triangle tiling. To this end,

122

Peter Kramer

(b, 3) (a, 1) (b, 2)

(b, 2)

(b, 2)

(b, 2) (b, 3)

(a, 1) Fig. 4.9. The two rhombus tile windows X1⊥ , X2⊥ are completely covered by cones from small decagons (heavy edge lines) centered on the hole positions marked (a, j), (b, l) (black and white circles) on their four vertices

we consider the windows X1⊥ , X2⊥ , construct all the holes (a, j), (b, j) at their vertices, attach the decagonal windows w(D(a,j) ), w(D(b,j) ) to these hole positions, and consider the intersections of these decagon windows with the tile windows, see Fig. 4.9. As can be seen from Fig. 4.9, the decagon windows cover the tiles completely. So the criterion Proposition 14 is fulfilled from the point of view of window for all the tiles of the triangle tiling. There follows Proposition 23: Delone covering of all tiles of (T ∗ , A4 ). Any tile of the tiling (T ∗ , A4 ) is completely covered by at least one Delone cluster. A patch of the triangle tiling, along with the Delone clusters, is given in Fig. 4.10. 4.9.2 Thickness of the Covering All vertices and all tiles of the tiling has been found to be covered by Delone clusters. To quantify the efficiency of this covering, we use, in analogy to the packing fraction, the thickness: Definition 24: Thickness. Consider a large patch of the tiling and its covering Delone clusters. We define the thickness Θcov of the covering as the limit, for infinite patch size, of the ratio between the area Fcov of all Delone clusters on the patch and the area F occupied by the patch. A true tiling would yield a covering fraction Θcov = 1. For a covering by Delone clusters, we expect a value larger than 1 owing to overlap. Consider ∗ and a large patch of the tiling of area F built from acute triangles X1 ∗ obtuse triangles X2 . Let n(1), n(2) denote the numbers of these tiles, for a fixed orientation, in the patch. The windows of the two tiles are the thick and the thin Penrose rhombus. The relative frequencies of the two tiles in the


123

Fig. 4.10. A patch of the triangle tiling (T ∗ , A4 ) and its covering by small and (a,j) (b,j) (edges shown by heavy lines). The large pentagonal Delone clusters D , D four shaded Delone clusters together form a fundamental domain for the tiling

infinite tiling limit are proportional to the area occupied by their windows. Using arrows in front of asymptotic values valid for large patches, we have n(1)/n(2) → τ .

(4.20)

The area F of the patch can be written in terms of the tiles as ∗ ∗ F := 10(n(1)|X1 | + n(2)|X2 |) ∗ |. → 10(τ 2 + 1)n(2)|X2

(4.21)

The factor 10 accounts for the possible orientations. In the second line we ∗ ∗ used (4.20) and |X1 |/|X2 | = τ . Let n(q) denote the number of vertices in

124

Peter Kramer

the patch. To relate this number to the number of tiles, we attach to each vertex of a tile a fraction equal to (2π) (internal angle at the vertex). Exept for the boundaries of the patch these fractions add up to the number n(q) of vertices, and so they must yield the correct asymptotics. Inside any triangle tile the internal angles sum to π and so contribute a fraction 1/2 to the number of vertices. Therefore we can write, using (4.20) n(q) → (10/2)(n(1) + n(2)) → (10/2)τ 2 n(2) .

(4.22)

With these results, we obtain from (4.21) and (4.22) ∗ |, F → 2τ −2 (τ 2 + 1)n(q)|X2

(4.23)

valid for n(q) 1. We now turn to the Delone clusters, and wish to relate the area covered by them to the number n(q) of vertices in the patch. The areas of the Delone clusters are ∗ ∗ ∗ |Da | = |X1 | + 2|X2 | = (τ + 2)|X2 |, ∗ ∗ ∗ |Db | = 4|X1 | + 3|X2 | = (4τ + 3)|X2 |.

(4.24)

Let n(a, 1), n(a, 4), n(b, 3), n(b, 2) denote the numbers of oriented Delone clusters in the patch. These numbers have asymptotic values given by the ratio of the windows of the Delone clusters to |V⊥ |, multiplied by the number of vertices: 5n(a, 1), 5n(a, 4) → τ −4 n(q) , 10n(b, 3), 10n(b, 2) → τ −4 n(q) .

(4.25)

The numbers 5 and 10 count the possible orientations of the two Delone clusters. The area covered by all Delone clusters on the patch is, from (4.24) and (4.25), Fcov := 5(n(a, 1) + n(a2))|Da | + 10(n(b, 3) + n(b, 2))|Db | ∗ → 2τ −4 n(q)(|Da | + |Db |) = 10τ −2 n(q)|X2 |,

(4.26)

again valid for n(q) 1. From (4.26) and (4.23), we can now compute the covering fraction defined in Definition 24 as the limit Θcov =

lim (Fcov /F )

n(q)→∞

= (10τ −2 )/(2τ −2 (τ 2 + 1)) = 5/(τ + 2) = 1.38 .

(4.27)

So, on average, there is an excess of 38% in the covering of the triangle tiling by Delone clusters.

4.10 Fundamental Domains in the Triangle Tiling We now enquire if the Delone clusters can be related to a fundamental domain. In Definitions 8 and 9, the notion of a fundamental domain F (T , Λ)


125

is described for functions compatible with a class of quasiperiodic tilings, built from a minimal set of prototiles and their translates. The tiling (T , A4 ) ∗ ∗ belongs to this class. We may choose the two triangles X1 , X1 , each in 10 possible orientations as prototiles. A possible fundamental domain F (T , A4 ) is then given by all points from these 20 prototiles. As shown in Fig. 4.10, it is possible to choose filled oriented Delone clusters for four different hole classes which encompass all these 20 prototiles. Proposition 25: Fundamental domain for (T ∗ , A4 ). Four filled Delone clusters of the four hole classes (a, 1), (a, 4), (b, 3), (b, 2) can be oriented so that they form a fundamental domain F (T ∗ , A4 ) for quasiperiodic functions compatible with the tiling (T ∗ , A4 ).

4.11 Linkage of Delone Clusters in (T , A4 ) How are the Delone clusters linked in the covering of the tiling? Consider the linkage of Delone clusters by a shared vertex q ∈ ∪h Dh of the tiling. From the window point of view, this linkage is characterized by the condition q⊥ ∈ ∩h w(h) ⊂ V⊥ . By constructing all possible intersections of the windows w(h) ⊂ V⊥ , we can find all linkages of Delone clusters through a vertex. Any window w(h) is a sector of a small decagon around the hole position h⊥ . Inside V⊥ ⊂ E⊥ , we must find all possible intersections of these small decagons. It suffices to analyze a large sector of of V⊥ opening angle 2π/10. Such a sector is shown in Fig. 4.11. The small decagon windows around 8 hole positions in V⊥ contribute to the large sector. There are 21 intersection polygons which form the windows for linked Delone clusters. The reflection i(15)(24) in the symmetry axis of the sector interchanges subclasses of holes. The 7 intersection polygons 3, 5, 11, 18–21 are invariant under this reflection; the other 14 have images in the sector under this reflection. For each hole that contributes to an intersection polygon, we determine its centre position and the specific sector of its decagon. The specific sector is obtained from a standard sector of Fig. 4.3 or 4.6 by a point group element g. We pass to the tiling in E , mark the center position of the hole, and apply the group element g to the standard position in Fig. 4.4 or 4.7 of the filled Delone cluster. As a result we obtain the 21 sets of Delone clusters linked by a vertex shown in Figs. 4.12–4.20. The central part of any linkage is a vertex configuration of the tiling (T ∗ , A4 ) as classified in [1]. Observe that for any vertex configuration, the covering enforces a continuation into only a few of the linkages of Delone clusters. The reflection i(15)(24), applied now in E , leaves 7 linked clusters invariant and produces 14 new images (not shown), with interchanged subclasses of holes. These linkages can appear in all orientations under I2 (5). For each linkage one can compute the area of the window or intersection polygon and divide it by the area of the large sector.

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Peter Kramer

Table 4.3. Centers of Delone clusters in the linkages j = 1, . . . , 21, and their orientations. Rows 2–9 list the centres and the orientations, expressed by the action of s1 =: (25)(34), g5 =: (12345), i on the standard positions in Fig. 4.4 and 4.7. Row 10 gives the relative frequency ν(j), and row 11 (vert) the central vertex configuration in the enumeration of [1] p. 2243 h

1

2

3

4

5

6

7

8

9

10

11

(a, 1) a1

g54

g54

−

g54

g54

g54

g54

g54

g54

g52

g52

(a, 4) −a5

−

−

−

−

i

i

g52 i −

g52 i

g52 i g52 i −

(b, 3) −a2 − a5 g52 s1 e

g52 s1 g52 s1 g52 s1 g52 s1 e

e

e

(b, 3) −a2 − a3 −

−

−

−

−

−

−

−

−

g 5 s1 g 5 s1

(b, 3) −a4 − a5 −

−

−

−

−

−

−

−

−

−

−

g52 i

g52 i

−

−

−

−

−

−

−

−

s1 i s1 i

s1 i g53 i

−

−

−

−

(b, 2) a1 + a4

−

−

g52 i

(b, 2) a3 + a4

−

−

−

−

−

(b, 2) a1 + a2

−

−

−

−

− −7

2τ

−7

τ

−9

τ

−10

2τ

−9

τ

−8

τ

−8

ν(j)

2τ

vert

1

2

1

1

1

1

2

12

13

14

15

16

17

18

(a, 1) a1

e

e

e

e

e

e

(a, 4) −a5

g52 i

g52 i

g54 i

g54 i

−

−

−

−

g5

g53 s1

h

(b, 3) −a2 − a5 − (b, 3) −a2 − a3 g5

g5

(b, 3) −a4 − a5 g54 s1 g54 s1 g52 (b, 2) a1 + a4 (b, 2) a3 + a4

−

−

−

s1 i

g53 i

g53 s1 i

−

τ

− 2τ −7 3

19

20

21

e

e

−

−

−

g54 i

g54 i

−

−

−

−

−

−

−

−

g53 s1

g53 s1

g5

g53 s1

g53

g53

g52

g52

g52

g52

g52

g52

g52 s1

−

−

−

−

−

−

−

g 5 s1 i g 5 s1 i

g53 s1 i

g5 i

g 5 s1 i g 5 s1 i

g52 s1 i

g52 s1 i

g52 s1 i

g52 s1 i g52 i

−

ν(j)

τ −7 τ −8 τ −9 τ −10 τ −9 τ −10 τ −10 τ −11 (τ + 2) τ −9 τ −6

vert

4

5

− 6

−

2

−6

3

g5 i

2

τ

−7

(b, 2) a1 + a2

4

−

τ

−

−8

−

7

7

5

6

8

9


127

(b, 2) (b, 3)

21 20 17 16 19 (b, 2) 15 18 14 13 12 (a, 1)

(b, 3)

(a, 4)

11 10

8

9 7

2 1

6 5 43

(b, 2)

(b, 3) Fig. 4.11. A sector of the decagon window V⊥ of opening angle 2π/10. The small decagons (heavy lines) around 8 hole positions marked (a, 1), (a, 4), (b, 3), (b, 2) are the windows of 8 Delone clusters, with sectors (thin lines) corresponding to possible orientations. The small decagons and their sectors intersect in the 21 numbered polygons, up to a reflection i(15)(24) in the symmetry axis of the large sector. Each polygon is the window for a set of Delone clusters linked by a shared vertex. The patches of linked Delone clusters can be constructed from the small decagon sectors which participate in the polygonal window. The patches are shown in Figs. 4.12–4.20

128

Peter Kramer

1

2

3

4

5

Fig. 4.13.

6

8

Fig. 4.12.

7

9

Fig. 4.14.

Fig. 4.15.


10

11

Fig. 4.16.

12

13

14

15

16

Fig. 4.17.

17 Fig. 4.18.

129

130

Peter Kramer

18

19 Fig. 4.19.

20

21 Fig. 4.20.

Figs. 4.11.– 4.20. The Delone clusters form 21 linkages through a vertex; these are shown in Figs. 4.12–4.20. Their windows are the 21 numbered polygons of Fig. 4.11. The linking vertex is marked by a black square; the centers of the Delone clusters are marked for holes of type a and b by black and white circles, respectively. 7 linkages are invariant under the reflection i(15)(24), 14 more (not shown) result from the action of this reflection on the remaining linkages

The resulting number ν(j) yields the relative frequency of occurrence of the linkage j in the tiling. All the relevant quantities are listed in Table 4.3. Proposition 26: Linkage of Delone clusters in (T ∗ , A4 ). The Delone clusters in the tiling (T ∗ , A4 ) appear, up to orientation, in 35 different linkages at a vertex. They extend the 9 vertex configurations of the tiling. The linkages are shown in Fig. 4.12–4.20, and their relative frequencies are given in Table 4.3. The information about the linkages given here is exhaustive. Other information, for example, the linkages of pairs of Delone clusters, could easily be derived from it.

4.12 6D Lattices and the Icosahedral Coxeter Group The first quasicrystals, discovered in 1984 by Shechtman et al. [30], had a Bragg-type diffraction pattern of overall icosahedral point symmetry. From


131

standard crystallography, this could not be a crystal, since, by the crystallographic lemma, icosahedral point symmetry is not compatible with the longrange order provided by a periodic lattice. The very existence of an ordered structure with a diffraction pattern of icosahedral point symmetry implied that scientists needed to search for alternatives to periodic lattices to describe the long-range order of certain solid materials. In functional analysis, there already existed generalizations from periodic functions to quasiperiodic and almost periodic functions, due in particular to the work of H. Bohr [3] in 1925. Bohr had shown that a quasiperiodic function on E m may be described and interpreted in terms of an irrational cut through a lattice in a space of dimension m > n. Penrose [29], in 1974 generalized the notion of a lattice cell and constructed a 2D paradigm of a quasiperiodic tiling with 5-fold point symmetry from two rhombus tiles. De Bruijn [4], in 1981, obtained the Penrose rhombus tiling by a projection from a hypercubic lattice in E 5 . De Bruijn’s approach showed that one could relate the idea of a forbidden point symmetry to an irrational cut through a high-dimensional lattice. This relation was established for the icosahedral point symmetry by Kramer and Neri [16] in 1984 along the following lines. The hypercubic lattice in E 6 admits the icosahedral point group as a subgroup of its hyperoctahedral holohedry. The corresponding 6D representation of the icosahedral group is reducible in two 3D irreducible representations, one of them being the standard 3D irreducible representation of this group which describes the point symmetry of the icosahedron. The subspace, say E , for this irreducible representation must be irrational, for if it were rational it would carry a periodic structure, in conflict with the crystallographic lemma. Moreover, it was shown in [16] by an analysis similar to de Bruijn’s that there is a tiling of E with two rhombohedral tiles and overall icosahedral point symmetry. This tiling will be denoted as (T ∗ , P ) and analyzed in what follows. The 3D orthogonal subspace E⊥ , complementary in E 6 to E , plays an important part in the analysis since it provides the windows for this tiling. There are two more lattices that can be obtained from the primitive hypercubic lattice; they are obtained by face and body centering, and are denoted by F and I. They admit the same holohedry, and hence the same subgroup embedding of the icosahedral group. Of these, the F -lattice has been found in many icosahedral quasicrystals. This lattice is also known as the root lattice D6 [5]. We now turn to a detailed description of the lattices Λ = P, D6 and then examine their tilings, Delone clusters, and covering properties. Application of those lattices to atomic positions in icosahedral quasicrystals are given for example in [13, 14]. 4.12.1 Lattices D6 , P and Their Holes We introduce here the lattices and point groups used for the construction of icosahedral tilings. By Λ, we denote both a lattice and its translation group.

132

Peter Kramer

We denote the primitive hypercubic lattice Λ = P in E 6 spanned by six orthonormal unit vectors (e1 , . . . , e6 ) as 6 nj ej , n ∈ Z , el · ek = δlk . (4.28) P = j

The holohedry of this lattice is the hyperoctahedral group Ω(6), |Ω(6)| = 26 6!, generated by all permutations and reflections of the basis vectors ei , i = 1, . . . , 6. We shall express all elements of this group as signed permutations in cycle notation. We shall use the basis (4.28) to express general lattice or hole points in the lattices by their components, as 6

mj ej → (m1 m2 m3 m4 m5 m6 ) .

(4.29)

j

The standard root lattice Λ = D6 [5] may be taken as a centering F of P . We keep a factor 2 in order to have simple relations to the lattice P . A basis of D6 ∼ F is (b1 , . . . , b6 ) = (e1 , . . . , e6 )Z 2F ,   100000 0 1 0 0 0 0   0 0 1 0 0 0 2F  . Z =  0 0 0 1 0 0 0 0 0 0 1 0 111112

(4.30)

(4.31)

The primitive lattice Λ = P may be viewed as the gluing of two lattices Λ = D6 , P = D6 + ((000001) + D6 ) .

(4.32)

The two copies of D6 form the even and odd lattice points of Λ = P . We shall adopt this point of view in what follows and shall derive the properties for the lattice P from those of the lattice D6 . The projections Λ of the lattice basis vectors of P or D6 onto E provide two of the three irreducible icosahedral modules [19] in the form Λ . The module bases are the parallel projections of the lattice bases (4.28) and (4.30) from E 6 onto E . The third icosahedral module corresponds to the centering I. Since it is reciprocal to the F -lattice, it appears in the diffraction analysis of quasicrystals whose structure is characterized by the F -lattice. There is a unique lifting of the modules into corresponding vectors in E 6 . We shall exploit this unique lifting by writing many algebraic vector expressions without a symbol for the projection. The corresponding expressions in E , E⊥ are uniquely defined. We turn now to the description of holes in the lattice D6 . For this purpose we consider, in E 6 , four classes q, a, c, b of points with coordinates


133

1 (m1 , m2 , m3 , m4 , m5 , m6 ), mj integer , (4.33) 2 with respect to the primitive basis (4.28), where the numbers mj in the classes obey 1 mj = even , q : ∀ mj = even, 2 j 1 a : ∀ mj = odd, mj = odd , 2 j k=

c : ∀ mj = odd, b : ∀ mj = even,

1 mj = even , 2 j

1 mj = odd . 2 j

(4.34)

We note the following properties. (i) The points q are the even points of the primitive hypercubic lattice P , and hence the points of the lattice D6 . (ii) Addition of any vector q to a point (a, c, b) from a fixed class yields another point from the same class. So these points form translation classes or orbits under the translation group D6 . (iii) An analysis of the holes as vertices of the Voronoi polytope V (q) of D6 [5, 19, 20], shows the following: Proposition 27: Holes in Λ = D6 . The classes a, c, b described algebraically in (4.34) exhaust the three types of holes in the root lattice Λ = D6 . Representatives on V (0) for the three classes of holes are given in Table 4.5. Note that the classes of holes (a, b, c) refer to Λ = D6 and have no relation to the classes of holes for Λ = A4 . In the primitive lattice Λ = P , the class b of points become lattice points, and the classes a, c of points together yield a single translation class of holes. 4.12.2 Point Groups and Icosahedral Symmetry We turn now to the point group symmetry of the lattice Λ = D6 . The Weyl group of D6 is a Coxeter group which has the icosahedral Coxeter group H3 , |H3 | = 120, as a subgroup. We give the Coxeter diagrams for both groups in Fig. 4.21. The Coxeter relations for the generators of H3 read R12 = R22 = R32 = (R1 R2 )5 = (R2 R3 )3 = (R3 R1 )2 = e .

(4.35)

In the lattices D6 , P these generators can be expressed as signed permutations acting on the basis vectors (4.28), R1 = (23)(46),

R2 = (36)(45),

R3 = (15)(23) .

(4.36)

We denote the 6D representation of H3 generated by the generators (4.36) as D(H3 ). This representation is reducible to two inequivalent 3D irreducible

134

Peter Kramer D6

5

H3

Fig. 4.21. Coxeter diagrams for D6 and its subgroup H3

representations D (H3 ), D⊥ (H3 ) in orthogonal subspaces E , E⊥ . An explicit reduction D(H3 ) = M −1 (D (H3 ) ⊕ D⊥ (H3 ))M

(4.37)

is provided [19, 20] by the matrix   011τ 0τ 1 τ τ 0 1 0    τ 0 0 1 τ 1   . M = 1/2(τ + 2)   0 τ τ 1 0 1  τ 1 1 0 τ 0  100τ 1τ

(4.38)

D (H3 ) is the standard defining irreducible representation of H3 . The reduction (4.37) may be rewritten as M D(H3 ) = (D (H3 ) ⊕ D⊥ (H3 ))M

(4.39)

with the following interpretation. The columns of the orthogonal matrix M (4.38) are explicit expressions for the six basis vectors (e1 , . . . , e6 ) of the hypercubic lattice P . The representation D(H3 ) on the left-hand side acts with signed permutation matrices from the right. These permutation matrices are elements of the general linear group Gl(6, Z). On the right-hand side, the representation of H3 appears in two diagonal 3 × 3 orthogonal irreducible blocks and acts on the components of the column vectors of M . This leads to the following interpretation of the matrix M : the parallel and perpendicular projections ej , ej⊥ of the six unit vectors (4.28) are given by the six sets of column vectors with entries from the upper and lower three rows, respectively of (4.38). To the three generators (4.36) of the Coxeter group H3 there correspond, in E , three Weyl vectors perpendicular to reflection planes, R1 → (e6 − e4 ) ,

R2 → (e6 − e3 ) ,

R3 → (−e2 − e3 ) .

(4.40)

The Weyl reflections generate the 5-, 3- and 2-fold rotations of the icosahedral group according to (4.35). For the rotation axes of the icosahedral


135

group in E , E⊥ we introduce three sets of vectors along the 5-, 2-, and 3-fold axes (Table 4.4). They are enumerated for each axis and indexed by the order of the rotation. If these vectors are projected from E6 onto E , E⊥ by use of (4.38), they point along the rotation axes in these spaces and have lengths as indicated. An intuitive picture of the axis directions and of the generator g5 = (R1 R2 ) = (23456) may be given as follows. In the irreducible representation space E of D (H3 ), the vectors i2 , i3 , i4 , i5 , i6 form a counterclockwise consecutive forward quintuple with respect to the 5-fold rotation g5 with axis i1 . In the irreducible representation space E⊥ of D⊥ (H3 ), the vectors i2⊥ , i5⊥ , i3⊥ , i2⊥ , i4⊥ form a counterclockwise backward quintuple with respect to i1⊥ . The representation D⊥ is obtained from the permutations (4.36), applied to the vectors i5⊥ . In this representation, g5 = (23456) acts as the second power of a 5-fold rotation. The stereographic projections of all other axes are given in Figs. 12.2 and 12.3 of [19]. Table 4.4. Algebraic expressions for the vectors parallel to the rotation axes and for their lengths in E , E⊥ i5 : |i5 | = |i5⊥ | = ➄ i5 = ei , i = 1, . . . , 5 j2 : |j2 | = ➁, |j2⊥ | = τ ➁ 12 = e1 − e5 ,

22 = e1 − e4 ,

32 = e1 − e2 ,

42 = e1 − e3 ,

52 = e1 − e6

62 = e2 + e4 ,

72 = e3 + e5 ,

82 = e4 + e6 ,

92 = e5 + e2 ,

102 = e6 + e3

112 = e4 − e3 ,

122 = e5 − e4 ,

132 = e5 − e6 ,

l3 (±) : |l3 (+)| = |l3⊥ (−)| = ➂ 13 (±) = (±(e1 + e2 + e3 ) − e4 + e5 − e6 )/2 23 (±) = (±(e1 + e3 + e4 ) + e6 − e5 − e2 )/2 33 (±) = (±(e1 + e4 + e5 ) + e2 − e6 − e3 )/2 43 (±) = (±(e1 + e6 + e5 ) + e3 − e4 − e2 )/2 53 (±) = (±(e1 + e6 + e2 ) + e4 − e3 − e5 )/2 63 (±) = (±(e2 + e3 − e5 ) + e6 + e4 − e1 )/2 73 (±) = (±(e3 + e4 − e6 ) + e2 − e1 + e5 )/2 83 (±) = (±(e4 + e5 − e2 ) + e3 − e1 + e6 )/2 93 (±) = (±(e6 + e5 − e3 ) + e2 + e4 − e1 )/2 103 (±) = (±(e2 + e6 − e4 ) + e3 + e5 − e1 )/2

142 = e6 − e2 ,

152 = e2 − e3

136

Peter Kramer

We characterize all three types of axes by vectors, although there is no a priori direction for the 2-fold axes. With respect to 3-fold axis vectors, the six vectors along 5-fold axes form one narrow and one wide forward triple. The signs + and − for E , E⊥ , respectively, provide vectors along 3 fold axes of length ➂. The symbols ➄ , ➁ , ➂ denote standards of length [20] along 5-, 2-, and 3-fold axes:

1 2 3 , ➁= , ➂= . (4.41) ➄= 2 τ +2 2τ + 2 The triples of 2-fold vectors (152 , 12 , 82 ) form an orthogonal system both in both E and E⊥ . The right-handed triples (152 , 12 , 82 ) and (152 , 12 , −82 )⊥ in E and E⊥ , after normalization provide the orthonormal systems for the entries of the column vectors in (4.38). This can be verified by forming these linear combinations along the 2-fold axes given in Table 4.4 with the 3D projections taken from (4.38). The three Coxeter reflection planes corresponding to (4.40) intersect pairwise in closest 5-, 3- and 2-fold axes and bound an infinite Coxeter cone. The points within this infinite Coxeter cone form a fundamental domain under the action of the Coxeter group on E . Under D (H3 ), all 120 Coxeter cones are mapped into one another. Under pure rotations, we can distinguish 60 right and 60 left Coxeter cones. By a spherical Coxeter cone, we mean the intersection of an infinite Coxeter cone with the unit sphere in E . In later sections we shall require intersections of the infinite Coxeter cone with polyhedra centered on the intersection of the reflection planes. In D⊥ the corresponding Weyl reflections, given by the signed permutations (4.36), act on the perpendicular projections of the vectors (4.28). Since the representation D⊥ (H3 ) in E⊥ is inequivalent to D , the notion of a Coxeter cone in E⊥ as a fundamental domain with respect to the action of the Coxeter group must be redefined in E⊥ ; compare Sect. 4.15. Representatives of the three translation classes a, c, b of holes in the lattice Λ = D6 are given in Table 4.5. They all belong to the Voronoi polytope V (q) at q = 0. Under the icosahedral Coxeter group, applied to the representative lattice point q = 0, the translation classes of holes on V (0) fall into various icosahedral orbits, with representatives given in Table 4.5. The point group H3 acts in E 6 exclusively with respect to lattice points q. For the later analysis of Delone polytopes we shall need the point symmetry with respect to any hole point a, c, b. We could employ the full Weyl group of D6 for this analysis but prefer to use only the icosahedral group H3 . To obtain the point symmetry at hole positions, we must use the space group of the lattice, whose elements are Euclidean pairs (t, g) of lattice translations t ∈ Λ = D6 and point group elements g ∈ H3 . This space group is the semidirect product D6 ×s H3 , where the translation group D6 is the normal subgroup. The usual Euclidean multiplication rule is (t1 , g1 )(t2 , g2 ) = (t1 + g1 t2 , g1 g2 ) .

(4.42)


137

Table 4.5. Representative hole positions on the projected Voronoi polytopes V , V⊥ . The first column (D6 class) gives the class name from (4.34) under the translations of D6 on V (0). The second column (H3 rep. h) gives representatives of this class for orbits under the point group H3 applied with respect to q = 0. Columns 3 and 4 (distance,⊥ ) give the distances of points on these orbits from the origin in E and E⊥ . The last column gives the length of the orbit under H3 D6 class H3 rep. h c a

Distance Distance⊥ |Orbit|

(1/2)(111111) τ −1 ➄

τ➄

12

(1/2)(111111) τ ➂

➂

20

(1/2)(111111) ➂

τ➂

20

(1/2)(111111) τ ➄ b

(000001)

➄

τ

−1

➄

➄

12 12

Denote a hole position by h. A general point transformation g which preserves this hole position can be expressed by a conjugation in the Euclidean group, (h, e)(0, g)(−h, e) = (h − gh, g) .

(4.43)

We must now check if, for a given hole position, the space group D6 ×s H3 provides the operation appearing on the right-hand side of this equation. This will be the case if and only if the translation vector t in (h − gh, g) = (t, g) is in D6 . We choose h from the icosahedral orbits of Table 4.5 and g as a generator of the Coxeter group H3 . Any one of these generators preserves the icosahedral point group orbit of h given in Table 4.5. But each one of these orbits, from (4.34), belongs to one single translation class under D6 . Therefore, for any g ∈ H3 and any hole position h, there exists a t ∈ D6 such that h − gh = t ∈ D6 . There follows Proposition 28: Point symmetry at holes for Λ = D6 . With respect to any hole position h = a, c, b, the space group symmetry of the lattice D6 is the full icosahedral point group H3 . 4.12.3 Scaling Symmetry in Icosahedral Lattices The lattice D6 has, beyond its translational and point group symmetry, a specific scaling symmetry. For the scaling symmetry [19], we use the projectors P = M −1 I M, P⊥ = M −1 I⊥ M to the subspaces E , E⊥ , with the matrix M taken from (4.38). Here I , I⊥ are 6 × 6 matrices with diagonal 3 × 3 unit √ or zero blocks, I = I ⊕ 0, I⊥ = 0 ⊕ I. We define, where τ := (1 + 5)/2, the matrix

138

Peter Kramer



11 1 1  1 1 S(τ ) = τ P − (1/τ )P⊥ = (1/2)  1 1  1 1 11

11 11 11 11 11 11

 11 1 1  1 1 . 1 1  1 1 11

(4.44)

The scaling symmetries of the lattices can now be expressed in the basis of P as (S(τ ))3 = S(τ 3 ) : P → P , S(τ ) : D6 → D6 ,

(4.45)

where the matrix S is to be applied from the right to the basis vectors (4.28). The scaling matrix has the properties S(τ ν ) = (S(τ ))ν , S(τ 2 ) = S(τ ) + I6 .

(4.46)

and so, by comparison with the quadratic equation x2 = x + 1, x = τ, −τ −1 ,

(4.47)

−1

has the eigenvalues τ, −τ . The scaling transformation S(τ ) (4.44) commutes with the action of the icosahedral group. With respect to the two irreducible subspaces E , E⊥ , it acts as (τ I ⊕ −τ −1 I). That the scaling with τ is not a symmetry of Λ = P can be seen from the half-integer entries in (4.44). Only the third power of S(τ ) becomes an element of Gl(6, Z). In fact, from (4.46) one finds (S(τ ))3 = S(τ 3 ) = 2S(τ ) + I6 ,

(4.48)

which has integer entries and is an element of Gl(n, Z). The lattice D6 is transformed into itself by the scaling S(τ ). This can be seen as follows. If the scaling transformation (4.44) is transformed with the matrix in (4.30) to the lattice basis B 2F , (4.49) B := M Z = B⊥ where M is given in (4.38), then the transform S (τ ) = (Z 2F )−1 S(τ )Z 2F ,

(4.50)

becomes an element S ∈ Gl(6, Z) and hence a symmetry of Λ = D6 . The action of scaling on the lattice can now be written in terms of the basis matrix B as B B (τ I3 ⊕ (−τ −1 I3 )) (4.51) = S (τ ) . B⊥ B⊥ This means that all the projections of lattice vectors are scaled in E by τ , and in E⊥ by −τ −1 .


139

We emphasize one important consequence of the scaling. Consider the operator O(n1 , n2 ) := n1 I6 + n2 S (τ )

(4.52)

acting on an arbitrary lattice vector q ∈ D6 . We can state that its image q = (n1 I6 + n2 S (τ ))q

(4.53)

is another lattice vector q ∈ D6 . This follows since S (τ ) is a lattice symmetry, and multiplication by integers transforms lattice vectors into lattice vectors. Note that the operator in (4.52) is, in general, not invertible with respect to D6 . Exceptions exist for the Fibonacci numbers (n1 , n2 ) = (n, m), n + mτ = τ ν , when the operator in (4.52) becomes S (τ ν ) ∈ Gl(n, Z). Now consider (4.53) written out with respect to the lattice basis and its projections according to (4.51), which yields q = (n1 + n2 τ )q , q⊥

= (n1 − n2 τ

−1

(4.54)

)q⊥ .

So if two parallel projections of lattice points q, q are on a line, with a factor (n1 + n2 τ ), their perpendicular projections are again on a line with the factor (n1 − n2 τ −1 ). From the point of view of (icosahedral) modules, the parallel projection of the operator in (4.52) represents an element of a ring R which extends the icosahedral module into an R-module. Equation (4.54) then relates pairs of elements of two icosahedral R-modules which are on a single line in the corresponding spaces E , E⊥ . The scaling transformation acts differently in E and E⊥ . It is convenient to use expressions which avoid the number τ for the relevant vectors. For this we use the quintuples and triples of vectors discussed in relation to Table 4.4. We can derive the following rules from (4.44), valid both in E , E⊥ for the scaling or inflation of the vectors k3 = l3 , k3⊥ = τ −1 l3⊥ of length ➂ and i5 , i5⊥ of length ➄: k3,⊥ = ((narrow forward triple) − (wide forward triple))/2 ,

(4.55)

τ k3,⊥ = ((narrow forward triple) + (wide forward triple))/2 , τ i5,⊥ = τ ei,⊥ = (forward quintuple + ei,⊥ )/2 , τ −1 i5,⊥ = (forward quintuple − ei⊥ )/2 . The parallel and perpendicular projections of the basis vectors in the matrix M of (4.38) may be linked algebraically as follows. Define 6×6 orthogonal matrices in terms of 3 × 3 orthogonal blocks I, R as follows:   010 0 −R Q= ,R = 1 0 0  . (4.56) R0 0 0 −1

140

Peter Kramer

These matrices obey R2 = I3 , Q2 = −I6 . It is easy to show that Q M = M (33)(44)(66)(16)(23)(45) .

(4.57)

This equation shows that upon left multiplication with Q, the matrix M can be expressed by right multiplication with a point group element g ∈ Ω(6) of the hyperoctahedral group. It follows that the operation Q is a lattice symmetry. Consider now the scaling transformation (4.44) acting in block diagonal form on M . We easily find (QM )−1 (τ I3 ⊕ (−τ −1 I3 ))QM = M −1 (−τ −1 I3 ⊕ τ I3 )M .

(4.58)

For the scaling transformation, this implies (16)(23)(45)(33)(44)(66)S(τ )(33)(44)(66)(16)(23)(45) = S(−τ −1 ) . (4.59) The scalings in the parallel and perpendicular spaces are interchanged. Next consider the action of the icosahedral group. A similar computation shows that (QM )−1 (D ⊕ D⊥ )(QM ) = M −1 (RD⊥ R−1 ⊕ RD R−1 )M ;

(4.60)

the two irreducible representations of H3 are interchanged up to a conjugation with R. When the scaling transformation acts on the holes of the lattice D6 , it has the following effect: the three translation classes of holes are cyclically interchanged according to S(τ ) : a → c → b → a .

(4.61)

4.13 The Icosahedral Tiling (T , D6 ) The tiling (T ∗ , D6 ) [20, 19] belongs to the face-centered hypercubic lattice with the basis (4.30). There are three Delone polytopes Da , Db , Dc , centered on three types of holes h = a, b, c. Their representatives are given in Table 4.5. The projection Λ = (D6 ) is the icosahedral 2F -module. Its module basis is the six vectors (b1 , . . . , b6 ) in (4.30). In E these vectors point along 2-fold axes. The tiles are six tetrahedra (A∗ , B ∗ , C ∗ , D∗ , F ∗ , G∗ ) which will be described below. Their vertices are parallel projections of lattice points, and their edges point along 2-fold icosahedral axes. The tetrahedra (F ∗ , G∗ ) may be described as the convex hulls of four even vertices of the two standard rhombohedra of the tiling (T ∗ , P ). Conversely they may easily be blown up into these rhombohedra. We shall come back to this mutual relation in Sect. 4.14. The Voronoi cell V for Λ = D6 differs from the hypercube associated with Λ = P , but its icosahedral projection V⊥ is again a Kepler triacontahedron. It is the window for the vertices of the tiling (T ∗ , D6 ). The dual tile windows (A, B, C, D, F, G)⊥ are four pyramids and two rhombohedra. Each pyramid has a standard rhombus base like a face of a triacontahedron.


141

The vertices of any rhombus base contain vertex pairs of perpendicularly projected holes a and c in opposite positions. The fifth vertex of any pyramid is a perpendicularly projected hole point b. The two tiles (F, G)⊥ are an obtuse and an oblate rhombohedron, with the same types of rhombus faces and the same distribution of vertices as for the bases of the pyramids. The tiles and windows are given in detail in Tables 4.8–4.10. The single window V⊥ and the tiles of (T ∗ , D6 ) are shown in Fig. 4.22. In Fig. 4.23 we show the three Delone windows and the tiles of the dual tiling (T , D6 ). These tiles are the parallel projections of the 3-boundaries whose perpendicular projections yield the windows of the tiling (T ∗ , D6 ).

Fig. 4.22. Top: the triacontahedral vertex window V⊥ ∈ E⊥ of the tiling (T ∗ , D6 ). The holes a, c are marked by black and white circles. Bottom: the six tetrahedral tiles of the tiling in the order (F ∗ , B ∗ , D∗ ; G∗ , A∗ , C ∗ ) ∈ E . The symbol τ denotes edges of length τ ➁

142

Peter Kramer

a c b Fig. 4.23. Top: the three Delone windows D⊥ , D⊥ , D⊥ ∈ E⊥ of the tiling (T , D6 ). Bottom: the six tiles of the tiling in the order (G, A, C; F, B, D) ∈ E are four pyramids on a rhombus base and two rhombohedra (G, F ) already known from the primitive tiling. The holes a, c are marked by black and white circles, and the holes b by double circles

We turn now to an algebraic description of the dual 3-boundaries which project into tiles and tile windows. For polytopes, we use the following short hand symbols: By a circle ◦, we denote the join of two polytopes. If U, V denote the set of points of the two polytopes, their join is defined as the set of all points U ◦ V = z|z = µx + (1 − µ)y, x ∈ U, y ∈ V, 0 ≤ µ ≤ 1 .

(4.62)

By vectors within angle brackets , we denote the convex hull of the corresponding points. For hypercubes of dimension j, we write [19]


143

j 6 1 P (0 . . . 0<j+1 . . . 1 then so too would their images q 1 , . . . , q k (again since L is in general position). The points of L on l are equally spaced, so their images under πW are equally spaced along a line and only finitely many of them fall into the bounded window Ω. Hence the line on which q 1 , . . . , q k lie contains only finitely many points of Q. Similar reasoning shows that no proper affine subspace of V contains infinitely many points of Q. Nevertheless, the quasilattice associated with the Penrose tiling with decagonal symmetry, shown in Fig. 6.1, possesses infinite lines of points. The most prominent are parallel to the Ammann lines of the Penrose tiling (in the figure, horizontal, diagonal, and in two steeper directions) but in fact any line joining two points of the quasilattice passes through infinitely many other points. Likewise, in 3 dimensions, the well known icosahedral quasilattices, as well as containing infinite lines of points, have plane sections identical to the Penrose quasilattices and to other 2-dimensional quasilattices. Again, every line joining two points passes through infinitely many and every plane containing 3 noncollinear points has a positive density of points. Plane quasicrystal surfaces are also seen experimentally in physical quasicrystals. 6.3.3 An Example How does this happen? The answer is that although the lattices from which these quasilattices are derived comply with the V - and W -conditions, in other ways they are far from being “in general position” relative to V and W . As an example of this behavior, we look at the Z[τ ]-module Z[τ ]3 in E3 . This module does not have full icosahedral symmetry, but it has finite index in the standard icosahedral modules and hence the vector spaces over Q spanned by these modules are identical, which is all that concerns us for √ the moment. Here τ is the golden ratio (1 + 5)/2 (a root of x2 − x − 1 = 0) and Z[τ ] = {a + bτ | a, b ∈ Z}. For α = a + bτ ∈ Z[τ ], we denote √ its algebraic conjugate by α = a + bσ, where (following [2]) σ := τ = (1 − 5)/2 = −1/τ

194

Peter A. B. Pleasants

Fig. 6.1. Quasilattice associated with the Penrose “cartwheel” tiling. The points shown lie on the long edges of kites and darts, dividing each long edge in the ratio τ : 1. Dense lines of points can be seen horizontally, approximately parallel to the diagonals of the figure, and at two steeper angles (these five directions having angles π/5 between them)

is the other root of the defining equation of τ . We now define a mapping from Z[τ ]3 to E6 by (α, β, γ) → (α, β, γ, α , β , γ ) .

(6.10)

The image of Z[τ ]3 under this mapping is a lattice L of density 5−3/2 in E6 with basis (1, 0, 0, 1, 0, 0), (τ, 0, 0, σ, 0, 0) ,

(6.11)

together with four other vectors derived similarly from the second and third components of Z[τ ]3 . If we take V = {(x1 , x2 , x3 , 0, 0, 0) | x1 , x2 , x3 ∈ R} and W = {(0, 0, 0, x4 , x5 , x6 ) | x4 , x5 , x6 ∈ R} then the action of πV on L is the inverse of the mapping (6.10) and we can obtain quasilattices Q ⊂ Z[τ ]3 by choosing windows Ω ⊂ W . Since α = β = γ = 0 implies α = β = γ = 0, L satisfies the W -condition. For rather less obvious reasons it also satisfies the V -condition ([8] Lemma 3.3(iv)). The points of Z[τ ]3 on the x1 axis are mapped by (6.10) to the 2dimensional subspace S2 of E6 spanned by the two vectors (6.11) of L. Clearly πV (S2 ) is the x1 axis in V , but also V ∩ S2 = {(r(τ − σ), 0, 0, 0, 0, 0) | r ∈ R} ,

6 Lines and Planes in 2- and 3-Dimensional Quasicrystals

195

again the x1 axis in V . Similarly, πW (S2 ) and W ∩ S2 both coincide with the x4 axis in W . The subspace S2 generated by lattice vectors is thus situated in a special way relative to V and W and is not at all in general position. Because of this special behavior of S2 , it can be used to produce a 1dimensional quasilattice on the x1 axis in V by the cut-and-project method: the x1 and x4 axes are orthogonal subspaces of S2 , and L ∩ S2 is a 2dimensional lattice. If we choose for the window the interval in which the x4 axis meets Ω then this quasilattice consists precisely of the points of Q on the x1 axis. In the same way, the use of planes parallel to S2 through points of L shows that every line parallel to the x1 axis through a point of Q meets Q in a 1-dimensional quasilattice that differs from the quasilattice on the x1 axis only in that it has a different window, owing to the intersection of Ω with the parallel plane being different from its intersection with S2 . These lines parallel to the x1 axis meet the (x2 , x3 ) plane (the quotient of V by the x1 axis) in a set of points which is a 2-dimensional quasilattice, being given by the quotient lattice L/(L ∩ S2 ) in the 4-dimensional space E6 /S2 and the window which is the orthogonal projection of Ω on the (x5 , x6 ) plane. As the simplest example of a plane section of Q, we take the set of points of Z[τ ]3 in the (x1 , x2 ) plane. These map by (6.10) to the points of the lattice of density 1/5 with basis {(1, 0, 0, 1, 0, 0), (τ, 0, 0, σ, 0, 0), (0, 1, 0, 0, 1, 0), (0, τ, 0, 0, σ, 0)} that spans a 4-dimensional subspace S4 of E6 . Again, because πV (S4 ) = V ∩ S4 = (x1 , x2 ) plane in V and πW (S4 ) = W ∩ S4 = (x4 , x5 ) plane in W , this lattice enables us to exhibit the points of Q in the (x1 , x2 ) plane as a 2-dimensional quasilattice and the points of Q on planes parallel to the (x1 , x2 ) plane as quasilattices derived from the same lattice but with different windows. Also, the points where these planes meet the x3 axis form a 1dimensional quasilattice derived from the quotient of L by this lattice. So every plane parallel to the (x1 , x2 ) plane through a point of Q meets Q in a 2-dimensional quasilattice and the spacings between these planes form a 1dimensional quasilattice. As another example, consider the line l = R(1, τ 2 , 0). The mapping (6.10) takes the points of Z[τ ]3 on l to the lattice in E6 with basis {(1, τ 2 , 0, 1, σ 2 , 0), (τ, τ 3 , 0, σ, σ 3 , 0)} , √ with density 1/3 5. Again the subspace spanned by this lattice both projects onto l and meets V in l, so can be used to exhibit the points of Q on l as a 1-dimensional quasilattice. The points of Q on parallel lines form related 1-dimensional quasilattices and the lines themselves are arranged as a 2dimensional quasilattice.

196


In general, the subspace of V spanned by any 1- or 2-dimensional Z[τ ]submodule of Z[τ ]3 meets Q in a quasilattice, as do all parallel subspaces that meet Q, and the arrangement of these subspaces forms a quasilattice of the complementary dimension.

6.4 Subquasilattices and Quotients We now describe more generally what is required for a quasilattice to contain quasilattices of lower dimension. Readers primarily interested in how to locate dense lines and planes in specific quasicrystals and how to calculate their densities may prefer to pass over this section. Let (V ⊕ W, L) be a cut-and-project scheme and suppose there is a subspace T of V ⊕ W , generated by lattice vectors, such that πV (T ) = V ∩ T . Any such T also has πW (T ) = W ∩ T , since, for any x ∈ T , πW (x) = x − πV (x) is in T . There is a unique maximal lattice-vector-generated subspace S of V ⊕ W with πV (S) = V ∩ T

(6.12)

(since πV (S1 ) = πV (S2 ) = V ∩ T implies that πV (S1 + S2 ) = V ∩ T ) and for this S we clearly have πV (S) = V ∩ S =: VS

(6.13)

say, and therefore πW (S) = W ∩ S =: WS .

(6.14)

Since V ∩ W = {0} and (V ∩ S) + (W ∩ S) ⊆ S ⊆ πV (S) + πW (S) , (6.13) and (6.14) imply that S = VS ⊕ WS , and since S is generated by lattice vectors, LS := L ∩ S is a lattice in S. Clearly MS := πV (LS ) ⊆ M ∩ S, and the maximality of S shows that these two modules are the same. Indeed, if x ∈ M ∩ S then x = πV (l) for some l ∈ L and πV (S + l) ⊆ V ∩ S, whence l ∈ S by the maximality of S subject to (6.12). We call (VS ⊕ WS , LS )

(6.15)

a subscheme of (V ⊕ W, L). It satisfies the W -condition because (V ⊕ W, L) does. If it also satisfies the V -condition it is a cut-and-project scheme in its own right. For the remainder of this section we assume that this is the case.


197

Given a window Ω ⊂ W that picks out a quasilattice Q ⊂ M , the window Ω ∩ S gives the subquasilattice QS := Q ∩ S ⊂ VS , and (6.9) applied to QS gives densm (QS ) = densk LS volk−m (Ω ∩ S) ,

(6.16)

where m = dim VS and k = dim S (so that dim WS = k − m), and the suffixes indicate the dimensions of the densities and volume. For each l ∈ L, MS + πV (l) is a coset of MS in M (lying in the affine subspace πV (S + l) of V ) and the points of Q in this coset are those obtained from the scheme (6.15) by using the window (Ω − πW (l)) ∩ S in WS and translating the resulting points in VS by πV (l). Consequently, the density of Q ∩ πV (S + l) is given by replacing Ω ∩ S with (Ω − πW (l)) ∩ S in the right-hand side of (6.16). Different vectors l ∈ L from the same coset of LS give rise to the same coset of MS , which is uniquely determined by the points v = (VS + πV (l)) ∩ VS⊥ and w = (WS + πW (l)) ∩ WS⊥ , where VS⊥ and WS⊥ are the subspaces of V and W orthogonal to VS and WS . The points v + w, for different cosets, form a lattice in VS⊥ ⊕ WS⊥ which is the projection of L on VS⊥ ⊕ WS⊥ parallel to S and so can be identified with the quotient L/LS . Then the intersections with VS⊥ of the affine subspaces containing the cosets of MS in M form a module derived from the quotient scheme (VS⊥ ⊕ WS⊥ , L/LS ) .

(6.17)

If there were N − 1 − dim S vectors of L/LS that generated a subspace U containing VS⊥ then U + S would be a subspace containing V generated by N − 1 vectors of L, violating the V -condition for (V ⊕ W, L). Hence there are no such vectors and the quotient scheme (6.17) satisfies the V -condition. The cosets that have nonempty intersection with Q are those for which W ∩(S +l) = WS +πW (l) meets Ω,2 so the window for the scheme (6.17) that produces the quotient quasilattice QS := Q/VS consisting of the orthogonal projection of Q on VS⊥ is Ω/WS , the orthogonal projection of Ω on WS⊥ . If the scheme (6.17) satisfies the W -condition, then (6.9) applied to QS gives densn−m (QS ) = (densN L/densk LS ) volN −n−k+m (Ω/WS )(sin φ/ sin φS ) = dens LS vol(Ω/WS )(sin φ/ sin φS ) ,

(6.18)

where LS is the quotient lattice L/LS . Again the sine term is 1 when V and W are orthogonal. An extreme example of a subquasilattice is given by taking S = 0 ⊂ V ⊕ W . Then πV (S) = V ∩ S = {0} and, in view of the W -condition, S is the maximal lattice-vector-generated subspace with πV (S) = {0}. For this S, the cosets of MS = {0} are the individual points of M , QS = {0}, and the quotient quasilattice QS is Q itself. At the other extreme, S = V ⊕ W 2

There are typically exceptions to this (confined to a finite number of hyperplanes in VS⊥ ) due to cosets that meet Ω only at its boundary.

198


has πV (S) = V ∩ S = V and is clearly maximal. For this S, QS = Q and QS = {0}. When V and W are in general position relative to L, these are the only subquasilattices and quotient quasilattices of Q. Finally, we note that whenever S is a lattice-vector-generated subspace that gives a subscheme, the -map can be extended from points of M to affine subspaces of V parallel to VS that meet M , by putting : VS + πV (l) → WS + πW (l) . For different S, this -map respects inclusions: if S1 and S2 are two such subspaces and l1 , l2 two lattice vectors with VS1 + πV (l1 ) ⊂ VS2 + πV (l2 ), then WS1 + πW (l1 ) ⊂ WS2 + πW (l2 ). (We can, in fact, choose l1 = l2 here.) In particular, taking S = V ⊕ W gives V = W for every cut-and-project scheme (V ⊕ W, L). From (6.9), (6.16), and (6.18) we see that, when (6.15) satisfies the V condition, (6.17) satisfies the W -condition, and Q has density 1, we have dens(QS ) dens(QS ) =

vol(Ω ∩ S) vol(Ω/WS ) . vol Ω

(6.19)

6.5 Quasilattices from Quadratic Fields Most of the following notation (and all of the following facts) about√algebraic number theory can be found in [1]. A real √ quadratic field K = Q( d) is the natural set of all numbers α of the form r + s d, where d is a squarefree √ number ≥ 2 and r, s are in Q. The conjugate of α is α := r − s d, the norm of α is norm α := α α = r2 − ds2 , the trace √ of α is tr α := α+ α = 2r and the different of α is dif(α) := α − α = 2s d. The ring of integers O = OK of K is the unique maximal subring of K that is finitely generated as an abelian group. It consists of all numbers of the form a + ωb with a, b ∈ Z, where √ ( d + 1)/2 when d ≡ 1 (mod 4) ; ω= √ d otherwise . More generally, a finitely generated subgroup M of the additive group of K is called a module and is full if QM = K. A full module M has a Z-basis {ω1 , ω2 } with M = Zω1 + Zω2 , and the discriminant of M is disc M := det(tr(ωi ωj )) = (ω1 ω2 − ω1 ω2 )2 = − det(dif(ωi ωj )) , a rational number that is independent of the choice of basis (see [1] Sect. 2.2). The ring of integers O is a full module and its discriminant (often referred to as the discriminant of K) is d when d ≡ 1 (mod 4) and 4d otherwise. A module M with OM = M is called an O-module or a fractional ideal of K. (It is an integral ideal if M ⊂ O.) Two O-modules M1 and M2 are equivalent if M2 = αM1 , for some α ∈ K. The number of equivalence classes of O-modules in K is called the class number of K. The class number is 1 if


199

and only if every integer in O has an expression as a product of irreducible integers that is unique up to multiplication of the factors by units. Here a unit means a number ε in K such that ε and 1/ε are both in O, and an integer α in O is irreducible if in every factorization of it as a product of two factors in O one of the factors is a unit. Every real quadratic field has a fundamental unit ζ (unique up to sign) such that the set of units in K is {±ζ r | r ∈ Z}. The quadratic √ fields we shall √ use in the examples considered in this chapter are Q(τ ), Q( 2), and Q( 3) which all have class number 1 (see [1], first three entries of Table 1) as also do their associated cyclotomic fields Q(ξ5 ), Q(ξ8 ) and Q(ξ12 ), where ξm denotes a primitive mth root of 1 [11]. (These are the only cyclotomic fields of degree 4.) The units of these cyclotomic fields are of the form wr ζ s , where w is a root of unity and ζ is a fundamental unit. In the case of Q(ξ12 ), however, the fundamental unit is unavoidably complex √ and not the same as the fundamental unit of its maximal real subfield, Q( 3). 6.5.1 Modules over K For a quadratic number field K and an n-dimensional space En , we define a module in En over K to be a finitely generated subgroup M of the additive group of En such that v · w ∈ K for all v, w ∈ M . The module is full if it contains 2n vectors linearly independent over Q (the maximum possible number). The modules associated with the commonly studied quasicrystals that have a quadratic inflation multiplier are full modules in this sense. A full module M over K has a Z-basis of 2n vectors {b1 , . . . , b2n } and its discriminant is disc M := det(tr(bi · bj )) .

(6.20)

This generalizes the notions of the discriminant of a lattice and the discriminant of a module in K and is independent of the choice of basis. A -map for a full module M in En over K is an injective additive group homomorphism : M → En that either conjugates or negative conjugates dot products, that is, v · w = ±(v · w)

for all v, w ∈ M .

Any given -map has the same sign on the right for all v and w, so there are two kinds of -map: positive -maps (where the sign is +) and negative -maps (where the sign is −). A -map is a kind of “conjugate isometry”. It is semilinear, in the sense that if v ∈ M and if α ∈ K is such that αv ∈ M , then (αv) = α v , and it takes submodules of M to submodules of the image of M of the same dimension and the same absolute value of discriminant. A -map can be extended to KM by semilinearity. It is nowhere continuous however. Since isometries preserve inner products, composing a map with an isometry on En gives another -map, but the -map is unique up to isometry: if 1 and 2 are two -maps then there is an isometry T on En such that v 1 = T v 2 for all v ∈ M .

200


Given a module M in En over K and a -map on M , the set L := {(m, m ) | m ∈ M }

(6.21)

in E2n is a lattice and, by (6.1) and (6.20), disc L = ± disc M ,

(6.22)

with the minus sign just when the -map and n is odd. √ √ is negative It is worth noting here that, since d g = − d, if M is a module over K with a -map then the expanded set d1/4 M is also a module over K and the map defined by d1/4 α → d1/4 α is a -map of the opposite sign. The lattice derived from this new module and -map is just the lattice derived from M expanded by a factor d1/4 . Now (En ⊕ En , L)

(6.23)

is a cut-and-project scheme with module M and a -map agreeing with the -map on M . (The W -condition is satisfied because 0 = 0 and the V condition is satisfied by [8] Lemma 3.3(iv) again.) A window Ω in the second En factor gives a quasilattice Q ⊂ M and, since the En factors are orthogonal, vol Ω . dens Q = dens L vol Ω = √ disc M

(6.24)

The density is thus the product of a module factor, 1/ | discM |, and a window factor, vol Ω. Now let U be a linear subspace of En spanned by a subset of the vectors of the full module M over K. Then M ∩ U is a module over K that is full within U , and the above -map construction gives a cut-and-project scheme that is a subscheme of (En ⊕ En , L). In particular, if Q ∩ A has nonzero density for any affine subspace A of En parallel to U then M ∩ U is a full d-dimensional submodule of M , where d = dim U . So all affine subspaces that meet Q in a relatively dense set arise from a full submodule of M . An argument like that of [8] Lemma 3.2 shows that the subschemes obtained in this way are the only subschemes of (En ⊕ En , L). We note that the containing space En ⊕ En and the lattice L are no longer needed for this construction. Given the module M and the -map, the quasilattice can be defined as Q = {m ∈ M | m ∈ Ω} and its density, as given by the expression on the right of (6.24), can be calculated without reference to L. With this construction the sine terms in (6.16) and (6.18) are 1, so by (6.16), (6.18), and (6.22), the module factor of a quasilattice Q ∩ U is 1 | disc(M ∩ U )|

(6.25)


and the module factor for the corresponding quotient quasilattice is | disc(M ∩ U )| . | discM |

201

(6.26)

6.5.2 Dual Modules For a full module M in En over K, we define its dual module M ∗ by M ∗ := {m∗ | m · m∗ ∈ K and tr(m · m∗ ) ∈ Z for all m ∈ M } .

(6.27)

(In the special case when n = 1 and M is a fractional ideal in K, M ∗ is the fractional ideal inverse to dM , where d, called the different of M is the smallest fractional ideal containing {dif(α) | α ∈ O}.) A Z-basis for M ∗ is the dual basis {b∗1 , . . . , b∗2n } that is uniquely defined by tr(bi · b∗j ) = δij for i, j = 1, . . . , 2n. Using the dual basis, it can be shown that disc M ∗ = 1/ disc M . When M has a -map M has a naturally related -map of the same sign, and when -map is positive it follows from (6.2), (6.21), and (6.27) that the lattice derived from M ∗ using (6.21) is the dual L∗ of the lattice L derived from M . The dual module can be useful for finding discriminants of submodules, in the manner of (6.5) for lattices. Let M be a module over K that is also an O-module and let h be a hyperplane whose intersection with M is full in h. Suppose that h∩M has an O-basis {b1 , . . . , bn−1 } that can be extended to an O-basis {b1 , . . . , bn−1 , bn } of M . Then {b1 , ωb1 , . . . , bn , ωbn } is a Z-basis of M and the vectors in the dual of this Z-basis come in pairs b∗ , −b∗ /ω . The last pair of dual basis vectors has the form b∗n , −b∗n /ω , with b∗n orthogonal to h, and is a Z-basis for l ∩ M ∗ , where l is the line through 0 orthogonal to h. From (6.8), with r = n − 2 and the inner product tr(x · y), we now have disc(h ∩ M ) = disc(l ∩ M ∗ ) disc M ,

(6.28)

provided h ∩ M has an O-basis that can be extended to an O-basis of M . Finally, we show that this italicized condition holds when K has class number 1 and M is an O-module. To see this, we shall show that (for r = 1, . . . , n − 1) any O-basis {b1 , . . . , br } of an r-dimensional section hr ∩ M of M can be extended to an O-basis of an (r + 1)-dimensional section hr+1 ∩ M . Choose an mr+1 ∈ M that is not in hr ∩M , let hr+1 be the subspace spanned by hr and mr+1 , and define a linear functional A(x) by the (r + 1) × (r + 1) determinant (bi · bj ) (bi · x) A(x) := det mr+1 · x (mr+1 · bj ) (so that A(x) = 0 is a form of the equation of hr , in the sense that if x is in hr+1 then x is in hr if and only if A(x) = 0). The set {A(m) | m ∈ hr+1 ∩M }

202


is an O-module in K, so if K has class number 1 it has the form A(br+1 )O for some br+1 ∈ hr+1 ∩ M . Then {b1 , . . . , br , br+1 } is an O-basis of hr+1 ∩ M , since for any m ∈ hr+1 ∩ M we have A(m) = αA(br+1 ) for some α ∈ O, so A(m − αbr+1 ) = 0 and m − αbr+1 is in hr ∩ M . Now applying the process n − 1 times to h ∩ M (starting from the empty basis) shows that h ∩ M has an O-basis, and applying the process once more shows that it can be extended to an O-basis of M . Consequently (6.28) is valid whenever M is an O-module and K has class number 1. It is worth emphasizing for future reference the principle we have established here: If O is the ring of integers of K, M is a module over K in En that is also an O-module, and K has class number 1, then for every subspace S of En the module M ∩ S has an O-basis and for every subspace T ⊃ S any O-basis of M ∩ S can be extended to an O-basis of M ∩ T .

6.6 2-Dimensional Examples In this section we identify some of the densest lines in well known 10-fold, 8-fold, and 12-fold quasilattices and present tables listing their directions and densities. We also show how to find the densities of all lines and how to count the number of lines within a given range of densities. 6.6.1 10-Fold Quasilattices The module M in E2 with Z-basis {(cos(2mπ/5), sin(2mπ/5)) | m = 0, 1, 2, 3} is a full module over K := Q(τ ), with discriminant 53 /24 , invariant under the dihedral group D10 , and the mapping given by (cos(2mπ/5), sin(2mπ/5)) := (cos(4mπ/5), sin(4mπ/5)) is a positive -map as defined in the previous section. By (6.24), the density of the quasilattice Q arising from a window Ω is 4/53/2 times the area of the window. Viewed as a subset of the complex plane, M is precisely the ring of integers Z[ξ5 ] of the fifth cyclotomic field Q(ξ5 ), where ξm denotes a primitive mth root of 1. This ring has discriminant 53 in Q(ξ5 ), and the discriminant of M over K could be derived from this by noting that the trace from K to Q of mi · mj is half the trace from Q(ξ5 ) to Q of the corresponding product of complex numbers. The -map on M corresponds to the automorphism ξ5 → ξ52 of Q(ξ5 ) (which extends the automorphism τ → σ of K). From this representation or from the equivalent representation M = OL (where O := Z[τ ] is the ring of integers of K and L is the lattice with basis


203

{(1, 0), (cos(2π/5), sin(2π/5))}) it can be seen that M is an O-module. Since K has class number 1, this implies that any nontrivial intersection of M with a line through 0 has the form Om for some m ∈ M . Using the basis {m, τ m} for Om, we have tr(m · m) tr(τ m · m) disc(Om) = det tr(m · τ m) tr(τ m · τ m) = (µ + µ )(τ 2 µ + σ 2 µ ) − (τ µ + σµ )2 = 5µ µ = 5 norm(m · m) , where µ := m · m. Now norm(m · m) = Norm α , where α is the complex number corresponding to m and “Norm” indicates the norm from Q(ξ5 ) to Q (as distinct from “norm”, which is the norm from K to Q). Since all such α’s are in Z[ξ5 ], the minimum discriminant of 1dimensional submodules is 5 and is achieved by α’s with Norm α = 1, that is, the units of Q(ξ5 ). These have the form α = ±ξ5r τ s , where 0 ≤ r ≤ 4 and s ∈ Z. There are thus precisely 5 different 1-dimensional submodules of discriminant 5 corresponding to the 5th roots of unity. The integers of Q(ξ5 ) with the next smallest norm are 1 − ξ5 and its unit multiples, which have norm 5. These give 5 1-dimensional submodules of discriminant 25, which bisect the angles between adjacent submodules of discriminant 5. The next smallest norm is given by σ − ξ5 , τ − ξ5 , their complex conjugates, and their unit multiples, which have norm 11, giving 20 1-dimensional submodules of discriminant 55. These 30 submodules account for all lines that can be seen radiating from the center in Fig. 6.1 except for the 4 lines joining the center to the points closest to the corners, which correspond to submodules of discriminant 205 arising from integers of Q(ξ5 ) of norm 41. We see that 1-dimensional submodules occur in sets of 5 (corresponding to multiplication by roots of unity). These sets are orbits under the action of the group of rotations C5 and (since in Q(ξ5 ) it takes roots of unity to roots of unity) the -map takes orbits to orbits. Table 6.2 gives, for the orbits with the three smallest discriminants, the smallest angles, θ and θ , that any submodule of the orbit and of the image of the orbit under the -map makes with the x axis. (So π/10 < θ, θ ≤ π/10.) The plus/minus pairs (corresponding to complex conjugation in C) combine to form a single orbit of 10 submodules under the action of the dihedral group D5 , with the exception of the orbits of submodules of discriminant 5 and 25 which have mirror symmetry in the x axis so that the C5 orbits are themselves D5 orbits. We see, for example, that there are two D5 orbits of 1-dimensional submodules of discriminant 55 which, depending on the window, may give rise to quasilattices of slightly different densities. More generally, the field Q(ξ5 ) has class number 1, implying that the factorization of integers in Z into irreducibles of Z[ξ5 ] is unique up to multiplication by units. Since the norm of an irreducible in Z[ξ5 ] that divides

204


Table 6.2. Positions of submodules with the three smallest discriminants in the 10-fold 2-dimensional module. θ is the smallest angle that the x axis makes with any 1-dimensional submodule of a set of 5 with given discriminant, and θ is the corresponding angle for the image of the set under the -map Discriminant

θ

θ

5

0

0

25

π/10 √ ±sec−1 (2 4 − τ /3) √ ±(π/5 − sec−1 (2 4 − σ/3))

π/10 √ ∓(π/5 − sec−1 (2 4 − σ/3)) √ ∓sec−1 (2 4 − τ /3)

55 55

a prime p in Z is a power of p and the norm function is multiplicative, we can find all integers α ∈ Z[ξ5 ] with Norm α = a by finding the irreducible factors in Z[ξ5 ] of the prime factors p of a. We have seen that the irreducible factors of 5 are unit multiples√(or associates) of 1 − ξ5 and have norm 5 and we note that (1 − ξ5 )2 = −ξ5 5/τ is a root of unity times an integer of O. Primes p ≡ 1 (mod 5) (for example, 11) have 4 nonassociated, nonreal, irreducible factors each of norm p, in 2 complex conjugate pairs, and primes p ≡ −1 (mod 5) have 2 irreducible factors of norm p2 , both in O and therefore real. All other primes p in Z are themselves irreducible in Z[ξ5 ] with norm p4 . So if α ∈ Z[ξ5 ] is the generator of a 1-dimensional full submodule of M over K then Norm α has the form 5r ps11 . . . pskk , where r = 0 or 1 and each pi is a prime ≡ 1 (mod 5). Hence discriminants of 1-dimensional full submodules of M over K have the form d = 5r ps11 . . . pskk : r = 1 or 2, pi ≡ 1 (mod 5) and the number of submodules with discriminant d is g(d) = 5 × 22k × s1 × · · · × sk , since there are 4si ways of choosing si (not necessarily distinct) irreducibles of norm pi with no complex conjugate pair (whose product would be a nonunit integer of O). There is a strong link between 1-dimensional submodules of M and the coincidence site modules of M , studied in [9]. A coincidence site module for M is a submodule of finite index in M that is the intersection of M with a congruent copy of itself. For the current M , d is the discriminant of a 1-dimensional submodule if and only if it is 5 or 25 times the index of a coincidence site module and g(d) is 5 times the number of coincidence site modules of M with that index. In the terminology of [9], the primes p ≡ 1 (mod 5) are the complex splitting primes for Q(ξ5 ); that is, they are primes whose factorizations into irreducibles of Z[ξ5 ] consist of pairs of nonassociated complex conjugates. In [9] we expressed the Dirichlet series generating function


205

∞ f (d) d=1

ds

for the number f (d) of coincidence site modules of index d in M in terms of the ζ-functions of Q(τ ) and Q(ξ5 ), where the ζ-function, ζF (s), of an algebraic number field F is the Dirichlet series generating function for the number of ideals of given index in the ring of integers of F . (The index of an ideal in the ring of integers is standardly called the absolute norm of the ideal.) A similar analysis gives the Dirichlet series generating function for the number of 1-dimensional submodules of discriminant d as ∞ g(d) ζF (s) , (6.29) = 51−s ds ζK (2s) d=1

where F := Q(ξ5 ). The principal use of generating functions of sequences is to find information about the asymptotic distribution of the sequence. In the present case, for example, (6.29) can be used to show that the average number of 1-dimensional submodules with a given random integer as discriminant is 6 log τ . π2 This number can, equivalently, be viewed as the density among the positive integers of the discriminants of 1-dimensional submodules of M , counted with multiplicities. A fuller description of the derivation of such averages can be found in [9]. We can now use (6.16) and (6.18) to calculate, for quasilattices Q derived from M by various choices of window, the maximum density of subquasilattices along lines in a given direction and the density of the corresponding quotient quasilattice consisting of the set of quasilattice lines in that direction. The windows we shall choose are the unit circle, the regular decagon with the tenth roots of unity as vertices (which leads to the quasilattice of Fig. 6.1, associated with the Penrose tiling), and the regular decagon which results from rotating this decagon through π/10 and expanding it by a factor 2 cos(π/10) (which leads to a quasilattice associated with the T¨ ubingen tiling). In each of these cases the window factor for the densest line in the direction l is the length of the diameter of the window parallel to l , and the window factor for the quotient quasilattice is the length of the projection of the window orthogonal to l (for the decagonal windows, equal to the longest projection orthogonal to l of any diagonal). Finally, for comparison with the densities of subquasilattices and quotient quasilattices of other quasilattices, it is convenient to normalize (as we did for crystals in Table 6.1) by scaling M so that it has density 1. To do this we multiply by a normalizing factor equal to the reciprocal of the square root of the density of M , namely 53/4 (disc M )1/4 √ = √ . area Ω 2 area Ω

(6.30)

206


Table 6.3. The areas of regular 2m-gons, their length of cross section on the line y = x tan θ , and their length of projection on the orthogonal line y = −x cot θ . Here c := cos(π/2m). Both 2m-gons have their centers at (0, 0) and their vertices in the module generated by (1, 0) and (cos(π/m), sin(π/m)) (which, regarded as a subset of C , consists of the integers of Q(ξ2m )). The first has (1, 0) as a vertex and the second has (2c2 , 0) as the midpoint of an edge. Also included are the corresponding data for the unit circle Unit circle

First 2m-gon

Second 2m-gon

Area

π

m sin(π/m)

4mc2 sin(π/m)

Cross section

2

2c sec (π/2m − |θ |)

4c2 sec|θ |

Projection (m odd)

2

2 cos (π/2m − |θ |)

4c cos |θ |

Projection (m even)

2

2 cos |θ |

4c cos (π/2m − |θ |)

Table 6.3 lists the areas of the windows, the lengths of their intersections with the line through 0 making an angle θ with the x axis, and the lengths of their projections in the direction of this line (given in terms of θ ), not just for decagonal windows but, more generally, for windows that are regular polygons. As examples, we find from (6.25), (6.26), Tables 6.2 and 6.3, and (6.30) that for the normalized Penrose quasilattice the lines with the greatest density of points are the x axis and its orbit under C5 , that the density of points on these lines, given by module factor × window factor × normalizing factor, is 51/4 1 √ ×2× = 21/2 5−3/8 τ 1/4 ≈ 0.8723 , 5 2 sin(π/5)

(6.31)

and that the density of the intersections with the y axis of quasilattice lines parallel to the x axis is π 51/4 4 × 2 cos × = 23/2 5−5/8 τ 3/4 ≈ 1.4840 . 5 10 2 sin(π/5) For the densest lines in the normalized T¨ ubingen quasicrystal (again the orbit of the x axis), the corresponding calculations are 1 51/4 π √ × 2 cos × = 2−1/2 5−1/8 τ 3/4 ≈ 0.8296 10 2 sin(π/5) 5 and 51/4 4 = 25/2 5−7/8 τ 1/4 ≈ 1.5604 . ×2× 5 2 sin(π/5)

(6.32)


207

We note that these normalized densities are independent of the size of the window and are valid for all other quasilattices obtained from a window of the same shape and orientation but of a different size. So in calculating these densities for the T¨ ubingen quasilattice we could just as well have worked with an unexpanded, rotated decagon as the window. Because the window for the Penrose quasilattice has a pair of opposite sides parallel to the x axis, there is a lower bound for the densities of points on lines parallel to the x axis (or to the other lines of its orbit) equal to 1/2τ times the maximum density (6.31). (Lines in directions such as these, where there is a positive lower bound on densities of points, were called Ammann bars in [8].) Because the window for the T¨ ubingen quasilattice has a pair of opposite sides parallel to the y axis, a proportion 1/2τ of quasilattice lines parallel to the x axis achieves the maximum density (6.32). For the next densest lines (corresponding to discriminant 25) in these quasilattices, which are rotated through an angle of π/10 relative to the lines corresponding to discriminant 5, this situation is reversed and a proportion 1/2τ of such lines in the Penrose quasilattice achieve the maximum density 2−1/2 5−5/8 τ 3/4 ≈ 0.3710, whereas in the T¨ ubingen quasilattice these lines are Ammann bars with a density bounded below by 2−1/2 5−7/8 τ −3/4 ≈ 0.1205. Apart from these cases, in all other directions in the Penrose and T¨ ubingen quasilattices there are lines of arbitrarily small density and only the line through the origin achieves the maximum density for that direction. What is the 4-dimensional lattice L = {(m, m ) | m ∈ M } in this construction? We have

(m, m ) 2 = m 2 + m 2 = tr( m 2 ) , √ there are 10 minimum vectors of length 2, corresponding to the roots of unity in F ,√and in fact it is readily verified that the lattice is A∗4 (or, more precisely, A∗4 / 2). The familiar cut-and-project constructions of the quasilattice Q start from the 4-dimensional lattice A4 (or perhaps from the 5-dimensional lattice Z5 with the quotient lattice A4 occurring implicitly). How does this relate to the present construction? The lattice used in a cut-and-project construction is by no means uniquely determined by Q, and in fact an arbitrary linear transformation applied to the internal space W changes the lattice without changing either√M or Q. In the present case, if we√shrink√V by √ a factor 1/ τ then M/ τ is still a module over K and m/ τ → τ m (for m in M ) is a negative√-map for it. The squared length of the lattice point corresponding to m/ τ is dif( m 2 /τ ). This modified lattice has 20 minimum vectors of length 51/4 , corresponding to τ ±1/2 times √ the roots of unity, and is A4 (or 2−1/2 51/4 A4 ). Shrinking M by a factor 1/ τ has had the √ effect of expanding M by a factor τ , and up to scale these two operations are equivalent to keeping V fixed and expanding W by a factor τ . Shrinking W by a factor 1/τ would also make the lattice A4 (up to scale) without changing M .

208


6.6.2 8-Fold Quasilattices Similarly, the module M in E2 with Z-basis {(cos(mπ/4), sin(mπ/4)) | m = 0, 1, 2, 3} √ is a full module over K := Q( 2), invariant under D8 , with discriminant 24 and a positive -map given by (cos(mπ/4), sin(mπ/4)) := (cos(5mπ/4), sin(5mπ/4)) . As a subset of C, M is the ring of integers of Q(ξ8 ), and the √ -map√is then the automorphism ξ8 → ξ85 , extending the automorphism 2 → − 2 of K but fixing i. It can also be described as √ √ M = {(α + βi)/ 2 | α, β ∈ O, α + β ≡ 0 (mod 2)} , √ where O := Z[ 2] (the ring √ √ of integers of K) and α + β being divisible by 2 means that (α + β)/ 2 belongs to O. As in the 10-fold case, M is an O-module and K has class number 1, so any nontrivial intersection of M with a line through 0 has the form Om for some m ∈ M . A similar calculation as before shows that disc(Om) = 8 norm(m · m) = 8 Norm α , where α is the complex number corresponding to m and “Norm” is the norm from Q(ξ8 ) to Q. Since all such α’s are integers of Q(ξ8 ), the minimum discriminant of 1-dimensional submodules √ is 8 and is achieved by the units of Q(ξ8 ), which have the form ±ξ8r (1 + 2)s with 0 ≤ r ≤ 3 and s ∈ Z. Hence there are 4 different 1-dimensional submodules of discriminant 8, corresponding to the x and y axes and the bisectors of the angles between them. The integers of next smallest norm are 1 + ξ8 and its unit multiples, which have norm 2, giving 4 1-dimensional submodules of discriminant 16, which bisect the angles between adjacent submodules of discriminant 8. Apart from (1 + ξ8 )2 and (1 + ξ8 )3 (which have norms 4 and 8 and are unit multiples of a real number and a √ real number times 1 + ξ8 , respectively), the next smallest norm is given by 2 ± i and their unit multiples, which have norm 9 and give 8 1-dimensional submodules of discriminant 72. Information about the positions of these 1-dimensional submodules with the three smallest discriminants is given in Table 6.4, which corresponds to Table 6.2 for the 10-fold module. Each entry represents an orbit of 4 submodules under the action of the cyclic group C8 generated by rotation through π/4, and pairs of C8 orbits combine to form orbits of 8 submodules under the action of the dihedral group D8 , with the exception of the submodules of discriminants 8 and 16, whose C8 orbits are also D8 orbits. For example, there is only one D8 orbit of modules of discriminant 72, which contains 8 submodules. The cyclotomic field Q(ξ8 ) has class number 1, and the complex splitting primes for this field are the primes p congruent to 1 or ±3 (mod 8). Primes


209

Table 6.4. Positions of submodules with the three smallest discriminants in the 8-fold 2-dimensional module. θ is the smallest angle that the x axis makes with any 1-dimensional submodule of a set of 4 with given discriminant and θ is the corresponding angle for the image of the set under the -map Discriminant

θ

θ

8

0

0

16

π/8 ±(π/8 − cot−1

72

√

π/8 2)

∓(π/8 − cot−1

√

2)

p ≡ 1 (mod 8) have 4 irreducible factors in Z[ξ8 ] of norm p in two complex conjugate pairs, and primes p ≡ ±3 (mod 8) have 2 complex conjugate irrefourth power of ducible factors of norm p2 . The prime 2 is (up to a unit) the √ the irreducible integer 1 + ξ8 with norm 2, whose square is i z. As a result, the discriminants of 1-dimensional full submodules of M over K have the form d = 2r ps11 . . . pskk q12t1 . . . ql2tl : r = 3 or 4, pi ≡ 1 (mod 8), qi ≡ ±3 (mod 8) and the number of submodules with discriminant d is g(d) = 22k+l+2 × s1 × · · · × sk . We have the Dirichlet series generating function ∞ g(d) d=1

ds

= 22−3s

ζF (s) , ζK (2s)

where F := Q(ξ8 ), and the average number of 1-dimensional submodules with a given discriminant is √ 3 log(1 + 2) √ . 2π 2 The densities of 1-dimensional subquasilattices and their quotients can now be calculated √ from (6.25), (6.26), and Table 6.3. The normalizing factor in this case is 2/ area Ω. As an example, for the normalized Ammann– Beenker quasilattice (whose window is the first octagon in Table 6.3), the lines with the greatest density of points are the x axis and its orbit under C8 , where the density is 1 1 √ ×2× = 2−1/4 ≈ 0.8409 , 8 sin(π/4) and the density of the intersections with the y axis of quasilattice lines parallel to the x axis is

210


√ 1 8 √ ×2× = 23/4 ≈ 1.6818 . 16 sin(π/4) Since the window for the Ammann–Beenker quasilattice has pairs of opposite sides parallel√to and orthogonal to the submodules of discriminant 16, a proportion 1/(1+ 2) of√the quasilattice lines in these directions achieve the maximum density, ((1 +√ 2)/8)1/2 , for these directions, and there is a positive lower bound (8(1 + 2))−1/2 on the densities of lines in these directions. The lines in the orbit of the x axis show a similar behavior in the quasilattice derived from the second, rotated, octagonal window. Apart from these cases, in all other directions in these quasilattices there are lines of arbitrarily small density and only the line through the origin achieves the maximum density for that direction. 6.6.3 12-Fold Quasilattices The module M in E2 with Z-basis {(cos(mπ/6), sin(mπ/6)) | m = 0, 1, 2, 3} √ is a full module over K := Q( 3), invariant under D12 , with discriminant 32 and a positive -map given by (cos(mπ/6), sin(mπ/6)) := (cos(5mπ/6), sin(5mπ/6)) . It can also be described as

√ M = {(α + βi)/2 | α, β ∈ O, α + β ≡ 0 (mod 1 + 3)} , √ with O := Z[ 3], the ring of integers of K. In this description, the -map corresponds to conjugation in K acting on α and β. As a subset of C, M is 5 is the automorphism ξ12 → ξ12 , the ring of integers of Q(ξ12 ) and √ the -map √ extending the automorphism 3 → − 3 of K but fixing i. Again M is an O-module and K has class number 1, so any nontrivial intersection of M with a line through 0 has the form Om for some m ∈ M . Also, disc(Om) = 12 norm(m · m) = 12 Norm α ,

where α is the complex number corresponding to m and “Norm” is the norm from Q(ξ12 ) to Q. Hence the minimum discriminant of 1-dimensional submodules is 12 and √ is achieved by the units of Q(ξ12 ), which have the form r ±ξ12 (1 + ξ12 )s (2 + 3)t with 0 ≤ r ≤ 5, s = 0 or 1, and t ∈ Z. So there are 12 different 1-dimensional submodules of discriminant 12 (given by the 12 possible sets of values of r and s), corresponding to the x axis and iterated rotations of it through π/12 about the origin. √ In fact 1 + ξ12 is a fundamental unit of Q(ξ12 ) with (1 + ξ12 )2 = ξ12 (2 + 3), so this field differs from the previous two √we have looked at in not √ having a real fundamental unit. The integers 1 ± 3 have norm 4 and ± 3 have norm 9, but these are real so give no new submodules. The integers of next smallest norm are 2 ± ξ12 ,


211

Table 6.5. Positions of submodules with the three smallest discriminants in the 12-fold 2-dimensional module. θ is the smallest angle that the x axis makes with any 1-dimensional submodule of a set of 6 with given discriminant, and θ is the corresponding angle for the image of the set under the -map Discriminant

θ

θ

12

0

0

12 156

π/12 ± cot−1 (4 + ± cot

−1

(4 −

√ √

π/12 3)

∓(π/6 − cot−1 (4 − ∓(π/6 − cot

−1

√

3))

156

3) √ ±(π/12 − cot (4 + 3)) √ ±(π/12 − cot−1 (4 − 3))

3)) √ ±(π/12 − cot (4 − 3)) √ ±(π/12 − cot−1 (4 + 3))

300

±(π/6 − cot−1 1/2)

±(π/6 − cot−1 1/2)

300

±(π/12 − cot−1 1/2)

±(π/12 − cot−1 1/2)

156 156

−1

(4 +

√

−1

their complex conjugates, and their unit multiples, which have norm 13 √and give 48 1-dimensional submodules of discriminant 156. Apart from (1 ± 3)2 , which has norm 16 and is again real, the next smallest norm is given by 2 ± i and their unit multiples, which have norm 25 and give 24 1-dimensional submodules of discriminant 300. Information about the positions of the 1-dimensional submodules with the three smallest discriminants is collected in Table 6.5. Each entry represents an orbit of 6 submodules under the action of the cyclic group C12 generated by rotation through π/6, and pairs of C12 orbits combine to form orbits of 12 submodules under the action of the dihedral group D12 , with the exception of the submodules of discriminant 12 whose C12 orbits are also D12 orbits. The cyclotomic field Q(ξ12 ) has class number 1 and its complex splitting primes are the primes p congruent to 1 or ±5 (mod 12). Primes p ≡ 1 (mod 12) have 4 irreducible factors in Z[ξ12 ] of norm p in two complex conjugate pairs and primes p ≡ ±5 (mod 12) have 2 complex conjugate irreducible factors of norm p2 . All irreducible factors of 2 and 3 have the form of a real number times unit. Consequently, the discriminants of 1-dimensional full submodules of M over K have the form d = 12pr11 . . . prkk q12s1 . . . ql2sl : pi ≡ 1 (mod 12), qi ≡ ±5 (mod 12) and the number of submodules with discriminant d is g(d) = 3 × 22k+l+2 × r1 × · · · × rk . We have the Dirichlet series generating function ∞ g(d) ζF (s) , = 121−s s d ζK (2s) d=1

212


where F := Q(ξ12 ), and the average number of 1-dimensional submodules with a given discriminant is √ √ 3 log(2 + 3) . π2 As an example of calculating densities of 1-dimensional subquasilattices and their quotients, we again choose the submodule along the x axis and the regular dodecagon with a vertex on the x axis as the window. Then the density of points on the x axis is 1 1 √ × 2 = √ ≈ 0.5774 , 2 3 3 and the density of the intersections with the y axis of quasilattice lines parallel to the x axis is 4 2 √ × 2 = √ ≈ 2.3094 . 3 3 No normalizing factor is needed in this case, since the area of the first dodecagonal window happens to be the square root of the discriminant of the module, so the 2-dimensional quasilattice already has density 1. Since the first dodecagonal window has pairs of opposite sides parallel to and orthogonal to√the second set of submodules with discriminant 12, a proportion the quasilattice lines in these directions achieve the maximum 1/(2 + 3) of √ density, ((2 + √3)/12)1/2 , for these directions, and there is a positive lower bound (12(2 + 3))−1/2 on the densities of lines in these directions; and in the quasilattice derived from the second dodecagonal window, the lines in the first set of submodules of discriminant 12 show a similar behavior. Apart from these cases, in all other directions in these quasilattices there are lines of arbitrarily small density and only the line through the origin achieves the maximum density for that direction. Although 1-dimensional modules of equal discriminant occur in sets of 24 (except for those of discriminant 12), in the quasilattices with dodecagonal windows (which do not have D24 symmetry) the corresponding lines fall into two orbits of 12 under the action of D12 , the densities along lines in different orbits being slightly different. This behavior is caused by the nonreal fundamental unit in F . In the quasilattice with a circular window the lines of both orbits have the same density, but nevertheless the 1-dimensional quasilattices on lines of different orbits are not congruent. √ Similarly 10-fold case, M/ 2 + 3 is also a module over K, with to the √ √ -map m/ 2 + 3 → 2 + 3 m (positive in√this case), and the squared length of the vector associated with m/ 2 + 3 in its corresponding 4√ dimensional lattice is tr( m 2 /(2 + 3)). This lattice, obtained (up to scale) from the √ lattice derived from M by expanding the internal space W by √ a factor 2+ 3, is D4 = D4∗ , and its 24 minimum vectors correspond to (2+ 3)±1/2 times the roots of unity in F .


213

6.7 3-Dimensional Examples Throughout this section K is the quadratic algebraic number field Q(τ ) and O := Z[τ ] is its ring of integers. The O-module O3 , considered at the end of Sect. 6.3, is clearly invariant under the symmetry group of the cube but not under the icosahedral group, because a typical matrix of an element of order 5 is   τ −1 −σ 1 1 −σ −τ  , (6.33) 2 −σ τ 1 which has noninteger entries. However, it has a finite index in an O-module that does have icosahedral symmetry, namely

1 A (α, β, γ) | α, β, γ ∈ O, (α, β, γ) ≡ 0 or (1, σ, τ ) (mod 2) . I0 := 2 Here, and in what follows, the superfix A indicates, as in [2] Chap. 8, that, as well as the vector itself, all vectors obtained from it by even permutations of the the coordinates are allowed. Note that 2 is a prime in O and that O has 4 residue classes mod 2, which can be represented by 0, 1, σ, and τ . An alternative way of describing the vectors (α, β, γ) that occur in I0 is that they satisfy α + σβ + τ γ ≡ α + β + γ ≡ 0 (mod 2) .

(6.34)

6.7.1 Icosians The module I0 is a full module over K and, since O has 4 residue classes mod 2, it has index 16 in ( 12 O)3 and hence has discriminant 53 /24 . Our notation is chosen to exploit the fact that I0 can be regarded as a subset of the division ring of Hamiltonian quaternions, H := {a + bi + cj + dk | a, b, c, d ∈ R} , a 4-dimensional vector space over R with a noncommutative multiplication determined by i2 = j 2 = k2 = ijk = −1 . ¯ := a − bi − cj − dk, The conjugate of α = a + bi + cj + dk ∈ H is defined by α ¯ = a2 + b2 + c2 + d2 , and the reduced the reduced norm of α by n(α) := αα ¯ = 2a. (In [2], n(α) is called the quaternionic trace of α by t(α) := α + α norm of α.) Now H0 := {α ∈ H | t(α) = 0} = {bi + cj + dk | b, c, d ∈ R} is the set of pure imaginary quaternions and has dimension 3 as a vector space over R. The multiplicative conjugation action on H0 , β → αβα−1

214


(β ∈ H0 ), of any nonzero α ∈ H is a positive isometry of H0 , and this gives a homomorphism from the multiplicative group of nonzero elements of H onto SO(3, R) with kernel R. The icosian ring I (whose members are called icosians) is the additive subgroup of H generated by the quaternions 1 1 1 (±2, 0, 0, 0)A, (±1, ±1, ±1, ±1)A, (0, ±1, ±σ, ±τ )A , 2 2 2 where we identify H with E4 via the basis {1, i, j, k} and the superfix A indicates that all even permutations of the coordinates are allowed. It is a ring because these generators (which have have reduced norm 1) form a multiplicative group, the icosian group, of order 120. It is also an O-module. In its conjugation action on H0 , the icosian group gives each rotation symmetry of the icosahedron twice (±α giving the same rotation). For example, the rotation (6.33) corresponds to conjugation by τ + i − σk. Now I0 = I ∩ H0 and is generated as an additive group by

1 I1 = (±1, 0, 0)A , (±1, ±σ, ±τ )A , 2 where we identify H0 with E3 via the basis {i, j, k}. Algebraically, I1 consists of all pure imaginary icosians with reduced norm 1 and geometrically it consists of all unit vectors parallel to the axes of 2-fold icosahedral symmetry. As well as I0 itself, H0 contains two other nonsimilar modules over K with icosahedral symmetry, in which I0 has finite index. One of these is

1 (1,σ,τ ) (α, β, γ) | α, β, γ ∈ O, α + σβ + τ γ ≡ 0 (mod 2) , (6.35) := I0 2 where the requirement α + β + γ ≡ 0 (mod 2) in (6.34) is omitted. Since O (1,σ,τ ) has 4 residue classes mod 2, I0 has index 4 in I0 . Intermediate between (1,σ,τ ) (1) (σ) (τ ) are three modules I0 , I0 , I0 over K defined by replacing I0 and I0 the requirement α + β + γ ≡ 0 (mod 2) in (6.34) by α + β + γ ≡ 0 or υ (mod 2) , (1,σ,τ )

where υ = 1, σ, or τ , respectively. While it is easily verified that I0 and I0 (1) (σ) (τ ) are O-modules, I0 , I0 , and I0 are not invariant under multiplication by (τ ) (σ) (1) τ and are Z[2τ ]-modules only. In fact I0 , I0 , and I0 are similar to each other and the similarity factor τ permutes them cyclically. The situation is illustrated in Fig. 6.2, where the vertical lines are inclusions with index 2 and (1) the horizontal arrows expansions by a factor τ . Up to similarity, I0 , I0 , and (1,σ,τ ) are the only finitely generated O-modules with icosahedral symmetry; I0 see, for example, [10].


215

Algebraic conjugation in K or, equivalently, an odd permutation of the coordinates takes the icosian ring I to a ring I and QI contains images of (1,σ,τ ) , again obtained either by algebraic conjugation the modules I0 , . . . , I0 or by an odd permutation of the last three coordinates, so as to keep H0 invariant. Geometrically, interchanging the y and z axes, for example, corresponds to reflection in the plane y = z, so the icosahedral modules are taken to their images by this reflection and by any isometry in the same coset of the icosahedral group (for example, a rotation of order 4 about the z axis). Algebraic conjugation in K also provides a positive -map on each of these modules. (1,σ,τ )

I0

✁ ❊❅ ✁ ❊ ❅ ❅ ❊ ✁ ❅ ❊ ✁✁ ❅ (1) (τ ) ✛ ❊ I0 I0 ❊ ✸✡ ✑ ◗ ❊ ✑ ❇ ◗◗ s I(σ) ✑ ✡ 0 ❇ ✡ ❇ ✡ ✄ ❇ ✄ ✡ ❇ ✄ ✡ ❇ ✄ ✡ ❇ ✄✡ I0

Fig. 6.2. Icosahedral modules. A schematic diagram of the three similarity types of icosahedral module in E 3 . The arrows represent magnification by a factor τ (so that the modules in the central 3-cycle are similar to each other) and lines without arrows represent inclusion with index 2 (the upper module containing the lower one). The three vertical levels correspond to three different similarity classes

6.7.2 1-Dimensional Submodules Since I0 is an O-module and K has class number 1 the intersection of I0 with any line through 0, being a 1-dimensional O-module, has the form Oα for some α ∈ I0 . It can be verified that disc(Oα) = 5 norm(n(α)), where the norm is the norm in K and we have used the fact that the inner product α · α is n(α). If we write, for α ∈ QI0 , Norm α := normK/Q (n(α)) then to find lines of high density in quasilattices derived from I0 we need to find elements α of I0 with Norm α small. We have already given the name I1 to the set of members of I0 with reduced norm 1 and we now define some other sets of icosians with small norm. Put

√ A 1 A 1 2 2 A I4 := (±1, ±1, 0) , (±σ, ±τ, ± 5) , (±τ , ±σ , ±1) , 2 2 I5 := {(±1, ±τ, 0)A} ,

216


I5

√ A 1 √ A A 1 2 2 := (±1, ±σ, 0) , (±τ , ±τ, ± 5) , (±σ, ±σ , ± 5) , 2 2

I9 := {(±1, ±1, ±1), (±τ, ±σ, 0)A} ,

√ 1 1 I9 := (±σ, ±τ, 0)A , (±3, ±σ, ±τ )A , (±τ 2 , ±σ 2 , ± 5)A . 2 2 Each of these sets is permuted transitively by the icosahedral group. The set I4 consists of all elements of I0 with reduced norm 2, and the corresponding rays through 0 meet the surface of a regular icosahedron in the points that divide the 60 medians of the triangular faces in the ratio τ 2 : 1 and are nearer the vertex ends of the medians. (The rays meet the surface of a regular dodecahedron at the midpoints of the 60 face diagonals.) Up to multiplication by√powers of τ , I5 and I5 together give all elements of I0 with reduced norm 5 times a power of τ . The points of I5 lie on the axes of 5-fold icosahedral symmetry and the rays through I5 meet the surface of a regular icosahedron at the midpoints of the 60 medians of the faces. (These are the rays obtained by applying to the 5-fold icosahedral axes those symmetries of the 5 cubes inscribed in the dodecahedron that are not icosahedral symmetries.) Similarly, I9 and I9 together comprise all elements of I0 with reduced norm 3. The points of I9 lie on the axes of 3-fold icosahedral symmetry and the rays through I9 meet the 30 edges of an icosahedron in pairs of points that divide the edges in the ratio 1 : 2τ : 1. Again these rays are obtained by applying to the 3-fold icosahedral axes symmetries of the 5 cubes that are not icosahedral symmetries. (1,σ,τ ) , defined by (6.35). In addition to I0 it contains We now turn to I0 quaternions of the form α/2 for α in the following 12 extra residue classes of O3 mod 2: (1, 1, 1), (σ, σ, σ), (τ, τ, τ ), (τ, σ, 0)A , (1, τ, 0)A , (σ, 1, 0)A .

(6.36)

The reduced norm of any such additional quaternion is of the form α/4 for some α ∈ O. Since none of the quaternions in I1 ∩ O3 is congruent mod 2 (1,σ,τ ) contains no quaternions of reduced to any of the vectors in (6.36), I0 norm 1/4. The quaternions in I5 and I9 are congruent mod 2 to vectors in √ (1,σ,τ ) (6.36), however, and give 12 members of I0 of reduced norm τ 5/4 and 20 members of reduced norm 3/4. None of the quaternions of O3 in I4 , I5 , or I9 is congruent mod 2 to any of the vectors in (6.36). (1) (1,σ,τ ) Finally, I0 is obtained from I0 by removing quaternions (α, β, γ)/2 with α + β + γ ≡ σ or τ (mod 2). The effect of this on 1-dimensional sub(1,σ,τ ) modules of I0 is that those of the form αO with α ∈ I0 are left unchanged but those of the form 12 (α, β, γ)O, where without loss of generality (1) α + β + γ ≡ 1 (mod 2), are reduced to 12 (α, β, γ)Z[2τ ] in I0 , which has index 2 in 12 (α, β, γ)O and hence a discriminant of 4 times the discriminant of 12 (α, β, γ)O.


217

Table 6.6. Smallest discriminants of 1-dimensional submodules of icosahedral modules. The second column gives the discriminant of the module and the later columns discriminants of sets of 1-dimensional submodules Module I0 (1)

I0

(1,σ,τ)

I0

I5

I5

I9

I9

Discriminant

I1

I4

125/16

5

20

25

25

45

45

125/64

5

20

25/4

25

45/4

45

125/256

5

20

25/16

25

45/16

45

Table 6.6 collects this information about discriminants of the sets of 1dimensional submodules of the icosahedral modules. The first column gives the name of the module, the second the discriminant of the module itself 3 (which can be calculated from its index in 12 O , which has discriminant 53 /212 ), and the later columns the discriminants of 1-dimensional submodules on the lines through the points in the various sets. (σ) (τ ) (1) The discriminants for I0 and I0 are the same as for I0 . The fact that the ratio of the two smallest 1-dimensional discriminants is different for each of the three modules listed confirms that no two of these modules are similar. Other points of interest are that the ordering of the discriminants of the axes (1,σ,τ ) from that for of rotational symmetry, I1 , I5 , and I9 , is different in I0 the other two modules, the 2-fold axes having the largest discriminant of the three types instead of the smallest, and that in I0 the rotation axes do not provide the three smallest discriminants, the 1-dimensional modules in the I4 directions having a smaller discriminant than those along the 5-fold or 3-fold axes and also the I5 directions giving the same discriminant as the 5-fold axes and the I9 directions the same discriminant as the 3-fold axes. In (1) (1,σ,τ ) I0 the I4 directions give the fourth smallest discriminant, but in I0 √ the fourth position is taken by the directions 12 (±σ 2 , ±τ 2 , 0)A , 12 (±1, ±1, ± 5)A , 1 A 2 (±2, ±τ, ±σ) , which give a discriminant 245/16. 6.7.3 2-Dimensional Submodules The densities of planes in icosahedral quasilattices depend on the discriminants of 2-dimensional submodules of icosahedral modules, which in turn can be found from the discriminants of 1-dimensional submodules of the dual module. The duals of the icosahedral modules are given by 2 (1,σ,τ ) 2 (υ) 2 (υ)∗ (1,σ,τ )∗ , I0 = √ I0 , I0 = √ I0 , I∗0 = √ I0 5 5 5 (1,σ,τ )

where υ = 1, σ, or τ . Since I0 and I0 are O-modules and K has class number 1, (6.28) can be applied to derive the discriminants of the 2-dimensional (1,σ,τ ) submodules of I0 and I0 orthogonal to the directions in Table 6.6, from

218

Peter A. B. Pleasants (1,σ,τ )

the discriminants of 1-dimensional submodules of I0 and I0 . The results (1) are given in Table 6.7 which also gives the discriminants for I0 , found by in(1,σ,τ ) (1) for a given plane h then h∩I0 = h∩I0 , but terpolation: if h∩I0 = h∩I0 (1,σ,τ ) (1) then it has index 2 in h ∩ I0 . We note that if h ∩ I0 has index 4 in h ∩ I0 in each case the ratios of the discriminants of 2-dimensional submodules are the same as the ratios of the discriminants of 1-dimensional submodules in (1,σ,τ ) , this follows immediately the dual module. For the O-modules I0 and I0 from (6.28). Table 6.7. Smallest discriminants of 2-dimensional submodules of icosahedral modules Module I0 (1) I0 (1,σ,τ) I0

I1

I4

I5

I5

I9

I9

125/4

125

625/64

625/4

1125/64

1125/4

125/16

125/4

625/64

625/16

1125/64

1125/16

125/64

125/16

625/64

625/64

1125/64

1125/64

6.7.4 Window Statistics Let M be one of the icosahedral modules, Ω a window in E3 , Q the icosahedral quasilattice Q := {q ∈ M | q ∈ Ω} derived from M and Ω, and S a subspace of E3 with a basis consisting of vectors from M . The normalized densities of the subquasilattice QS := Q ∩ S and its corresponding quotient quasilattice QS are each given as a product of a module factor, a window factor and a normalizing factor. According to (6.25) and (6.26), the module factors are √ disc(M ∩ S)/ disc M 1/ disc(M ∩ S) and for QS and QS , respectively. When S is 1-dimensional the window factor is the length of Ω ∩ S , for QS and the area of the projection of Ω on (S )⊥ for QS , and when S is 2-dimensional it is the area of Ω ∩ S , for QS , and the length of the orthogonal projection of Ω on (S )⊥ , for QS . Finally, the normalizing factor is (disc M )1/6 /(vol Ω)1/3 or (disc M )1/3 /(vol Ω)2/3 according as the dimension of the quasilattice whose density is being calculated is 1 or 2. Table 6.8 provides the geometrical statistics necessary to calculate the normalized densities on subspaces parallel to or orthogonal to the icosahedral rotation axes for a variety of windows with icosahedral symmetry; namely, the sphere, the dodecahedron, the icosahedron, and the rhombic triacontahedron.


219

(Naturally, the windows are chosen so that their symmetry axes are those of the conjugate module I0 , not of I0 .) For each window and each axis we list the length of the intersection of the axis with the window, the area of the central cross section of the window orthogonal to the axis, and the area of the projection of the window orthogonal to the axis. In all these cases, the orthogonal projection of the window on an axis is the same as the intersection of the window with the axis, so there is no need to list the lengths of the projections separately. The window sizes are chosen so that the 2-fold axis has length 2 in each case. We also include, in the first four rows of the table, the volumes of the windows. Table 6.8. Axis lengths and areas of cross sections and projections orthogonal to the axes for the rotation axes of various windows with icosahedral symmetry. Windows are normalized so that the 2-fold axes have length 2 in each case. Additionally, the first four rows give the volumes of the windows 2-fold

5-fold

3-fold

Volume

2 √ 2 3/τ √ 2τ / 3 √ 2 3/τ

4π/3 √ 4 5/τ 2

Length of intersection with axis Sphere

2

2

Dodecahedron

2

csc(π/5)

Icosahedron

2

4 sin(π/5)

Triacontahedron

2

4 sin(π/5)

20/3τ 20/τ 3

Cross-sectional area orthogonal to axis Sphere

π

Dodecahedron

π √ 2 5/τ

Icosahedron

2τ

5 sin(π/5)

Triacontahedron

10 − 4τ

10 tan(π/10)

5 sin(π/5)

π √ 3 3/2 √ 2 3 √ 2 3

Area of projection orthogonal to axis Sphere

π

Dodecahedron

π √ 2 5/τ

Icosahedron

2τ

10 tan(π/10)

Triacontahedron

10 − 4/τ

10 tan(π/10)

10 tan(π/10)

π √ 2 3(3 − τ )/τ √ 2 3 √ 2 3

As examples, we take the commonly used triacontahedral window and, for each of the three types of icosahedral module, calculate the densities of the densest lines and the densest planes, together with the densities of the quotient quasilattices. In each case the densest line and the normal to the densest plane are rotation axes of the module and their images under the

220


-map are rotation axes of the window of the same order. We present the information in Table 6.9. The second column gives the axis (specified by its rotation order) which is the densest line or is orthogonal to the densest plane of the quasilattice. The next three columns, headed M , W , and N , give the module, window and normalizing factors, derived from Tables 6.6, 6.7, and 6.8. The sixth column is the maximum density of points on lines and planes in the given direction, equal to the product of the previous three columns and given as a decimal approximation. The final column is the density of the corresponding quotient quasilattice, also given as a decimal approximation, √ calculated from (6.19) and Table 6.8. In the fifth column l := 3 20/τ is the cube root of the volume of the triacontahedron, or, equivalently, the side length of a cube of the same volume. Table 6.9. Densest lines and planes in icosahedral quasilattices with a triacontahedral window, giving normalized densities (column 6) and normalized densities of quotients (column 7). The preceding three columns, headed M , W , and N , give the module factor, the window factor, and the normalizing factor. In column 5, √ l := 3 20/τ is the side length of a cube with the same volume as the triacontahedron. In column 2, the axis (that is, the line or the normal to the plane) is specified by its rotation order Axis

M

W

Line

2

√ 1/ 5

2

Plane

5

8/25

10 tan(π/10)

2

√ 1/ 5 √ 4/5 5

2

I0

N √

Densities

5/22/3 l

0.7510

1.9899

5/24/3 l2

0.7331

2.2072

0.5961

2.5071

0.5606

2.6658

(1)

I0

Line Plane

2

10 − 4τ

(1,σ,τ) I0

Line

5

Plane

2

4/5 √ 8/5 5

4 sin(π/5) 10 − 4τ

√

5/2l

5/4l √

2

5/24/3 l

0.9949

1.6263

5/28/3 l2

0.7063

2.1158

6.7.5 Relation to Root Lattices Finally, we should say something about the relation between this construction of the icosahedral modules and more familiar constructions. In a√similar manner to the 10- and 12-fold cases √ in the previous section, I0 / τ is √ negative -map on it. The also a module over K and α/ τ → τ α is a √ squared length of the vector corresponding to α/ τ in its associated lattice


221

is dif(n(α)/τ ). This lattice is, up to scale, D6 , with 60 minimum vectors of ±1/2 length 51/4 corresponding to the quaternions I1 . It has index 4 in the √in τ lattice associated with the module I(1,σ.τ ) / τ over K, which is D6∗ and has 12 minimum vectors of length 2−1/2 51/4 corresponding to the quaternions in √ I5 /2 τ . The√lattices corresponding to the other three icosahedral modules between shrunk by 1/ τ are, of course, the unique three lattices intermediate √ D6 and D6∗ . As before, shrinking the modules by a factor 1/ τ is essentially equivalent, in its effect on the associated lattices, to expanding the internal space by a factor τ . The modules most commonly used in the literature are the expansions of these by a factor 51/4 which, as explained in Sect. 6.5.1, have positive -maps. This method of associating standard lattices with modules in I is derived from the construction of the lattice E8 from the icosian ring in [2]√ Sect. 8.2.1 5. In fact using a Euclidean norm of α defined as a + b, where n(α) = a + b √ dif(n(α)/τ ) = 5(a − b), so the Euclidean norm of α is a constant times the squared lenght of the lattice vector corresponding to α .

6.8 Quasilattices from Higher-Degree Fields The methods of this chapter are readily extended to modules over algebraic number fields of degree greater than 2. In this section we outline how this is done and illustrate the methods with the example of 2-dimensional quasilattices with 14-fold symmetry. Let K = Q(θ) be an algebraic number field of degree d, where θ satisfies an irreducible polynomial equation of degree d with coefficients in Q. A number α ∈ K, instead of having a single conjugate α different from itself, has d − 1 conjugates different from itself, α(2) , . . . , α(d) , say. The trace of α is the sum of all its conjugates (including itself) and the norm of α is the product of all its conjugates. A finitely generated subgroup M of En is a module over K if v · w ∈ K for all v, w ∈ M , and it is full if it has dn vectors linearly independent over Q (this number being the maximum possible). A -map for M is a map M → E(d−1)n = En ⊕ · · · ⊕ En given by v → (v (2) , . . . , v (d) ) , where v (i) ·w(i) = ±(v ·w)(i) for all v, w ∈ M and i = 2, . . . , d. A -map gives a lattice L := {(v, v (2) , . . . , v (d) ) | v ∈ M } ⊂ Edn with disc L = | discM | and a cut-and-project scheme (En ⊕ E(d−1)n , L). The density of the quasilattice Q derived from a window Ω for this scheme is given by vol Ω . dens Q = | discM |

222


All r-dimensional subspaces in which Q is relatively dense derive from subschemes whose “physical” space is generated by a full r-dimensional submodule of M and whose “internal” space is the (d − 1)r-dimensional subspace of E(d−1)n generated by the image of this module under the -map. 6.8.1 14-Fold Quasilattices The module M in E2 with Z-basis {(cos(2mπ/7), sin(2mπ/7)) | m = 0, 1, 2, 3, 4, 5} is invariant under the dihedral group D14 and is a full module over K := Q(η), where η := 2 cos(2π/7) satisfies the equation η 3 + η 2 − 2η − 1 = 0 of degree 3. Its discriminant is 75 /26 and it is an O-module, where O := Z[η] is the ring of integers of K. Having degree 3, η has two conjugates, η (2) = cos(4π/7) and η (3) = cos(6π/7), and a -map for M is given by taking the image of (cos(2mπ/7), sin(2mπ/7)) to be (cos(4mπ/7), sin(4mπ/7), cos(6mπ/7), sin(6mπ/7)) . The field K has two fundamental units, which can be taken as η and 1 + η, and every unit of K is uniquely of the form ±η r (1 + η)s with r, s ∈ Z. As a subset of C, M is the ring of integers Z[ξ7 ] of the seventh cyclotomic field Q(ξ7 ), and the -map is given by ξ7 → (ξ72 , ξ73 ) from Z[ξ7 ] to Z[ξ7 ]2 . Because K has class number 1 [11], any 1-dimensional O-module in M has the form Om for some m ∈ M , and disc(Om) = 49 norm(m · m) = 49 Norm α , where α is the complex number corresponding to m and “Norm” is the norm from Q(ξ7 ) to Q. Hence the minimum discriminant of 1-dimensional submodules is 49 and is achieved by the units of Q(ξ7 ), which have the form ξ7r times a unit of K, with 0 ≤ r ≤ 6. So there are 7 different 1-dimensional submodules of discriminant 49, corresponding to the 7th roots of unity. The integers of Q(ξ7 ) of next smallest norm are 1−ξ7 and its unit multiples, which have norm 7. These give 7 1-dimensional submodules of discriminant 343, which bisect the angles between adjacent submodules of discriminant 49. The next smallest norm is given by η 2 − 1 − ξ7 , its complex conjugate, and their unit multiples, which have norm 8, giving 14 1-dimensional submodules of discriminant 392. The cyclotomic field Q(ξ7 ) has class number 1 [11] and its complex splitting primes are the primes congruent to 1, 2, or 4 (mod 7). Primes p ≡ 1 (mod 7) have 6 irreducible factors in Z[ξ7 ] of norm p in three complex conjugate pairs, and primes p ≡ 2 or 4 (mod 7) have 2 complex conjugate irreducible factors of norm p3 . The irreducible factors of 7 are 1 − ξ7 and its associates, and (1 − ξ7 )2 = ξ7 (η − 2) is a root of unity times an integer of O. Consequently, the discriminants of 1-dimensional full submodules of M over K have the form


223

d = 7r ps11 . . . pskk q13t1 . . . ql3tl : r = 2 or 3, pi ≡ 1 (mod 7), qi ≡ 2 or 4 (mod 7) and the number of submodules with discriminant d is g(d) = 7 × 2k+l × (2s21 + 1) × · · · × (2s2k + 1) . We have the Dirichlet series generating function ∞ g(d) d=1

ds

= 71−2s

ζF (s) , ζK (2s)

where F := Q(ξ7 ), and the average number of 1-dimensional submodules with a given discriminant is √ 7 7 2 2 2 log η − log(2 − η ) log(η + η − 1) . 8π 3 The module factors for the direction α in a√ quasilattice √ √ derived from M are 1/7 Norm α for subquasilattices, and 8 Norm α/7 7 for quotient quasilattices. With the 4-dimensional unit sphere as a window, the window factor is π, the area of the unit circle, (since the -map takes lines to planes), and the normalizing factor is 75/4 (75 /26 )1/4 . = 2 1/2 2π (π /2) So, for the 14-fold quasilattice with the unit 4-sphere as window, the maximum density of 1-dimensional subquasilattices in the direction α is 71/4 √ 2 Norm α and the density of the corresponding quotient quasilattice is √ 4 Norm α . 71/4 As usual, these normalized densities are independent of the radius of the window.

6.9 Summary This chapter began by reviewing the densest lines and planes of some 3dimensional crystals. The density of a 3-dimensional lattice is the reciprocal of the square root of its discriminant, the densest lines in the lattice lie along its shortest vectors, and the densest planes are orthogonal to the shortest vectors of its dual lattice. A quasilattice is derived from a lattice in a higher-dimensional space that projects onto a module in a subspace, the quasilattice being the subset of

224


the module consisting of the points that project into a certain window in a orthogonal subspace. At this level of generality quasilattices do not usually have alignment, in the sense of infinitely many points of the quasilattice lying in a lower dimensional subspace. So we concentrate attention on the situation where the high dimensional lattice is derived from an algebraic number field, especially a quadratic number field, and the orthogonal subspaces are related by algebraic conjugation in this field – the so-called -map. All the commonly studied quasicrystals are of this form. In this case a discriminant can be defined for the module, generalizing the notion of the discriminant of an algebraic number field, which is equal in absolute value to the discriminant of the derived lattice. This concept of discriminant is central to this chapter and is the key to calculating densities, the density of the quasilattice being the product of the reciprocal of the square root of the absolute value of the module discriminant and the volume of the window. Of these two factors, the one involving the discriminant is the most significant contributor to the relative density, the window largely having a simple scaling effect, particularly as the windows we consider are rather similar in shape, being close to intervals, circles, and spheres. Also in the number field case, the quasicrystal has subspaces of every lower dimension on which it is relatively dense. Such a subspace corresponds to a submodule of the module and thence to a sublattice of the lattice which (in the quadratic case) has twice the dimension of the submodule. Finding dense lines in our quasilattice amounts to finding 2-dimensional sublattices of the lattice with a small discriminant; but this is equivalent to finding 1-dimensional submodules of the module with a small discriminant, which can be achieved though the arithmetic of cyclotomic fields, for 2-dimensional quasilattices, or of quaternions, for 3-dimensional quasilattices. This arithmetic enables us to identify and count all 1-dimensional submodules of a given discriminant – a process which, surprisingly, is closely related to identifying and counting coincidence site submodules of the module. Finding dense planes in a 3-dimensional quasilattice is tantamount to finding 4-dimensional sublattices of small discriminant in the corresponding 6-dimensional lattice, but this reduces to finding 2-dimensional submodules of small discriminant in the module. Our main idea for dealing with this situation was that, under certain conditions, duality can be shown to apply to modules over an algebraic number field just as it does to lattices, so that finding 2-dimensional submodules of small discriminant in a 3-dimensional module is equivalent to finding 1-dimensional submodules of small discriminant in its dual module. In the final section we gave an example to show how a similar analysis could be made to work for modules over an algebraic number field of degree greater than 2, at the cost having a window space of higher dimension than the physical space and a -map that increases dimension.


225

Acknowledgments I should like to thank the editors for their encouragement to prepare this chapter for this volume and especially Zorka Papadopolos for much helpful discussion and correspondence. I also thank Takashi Soma for preparing Fig. 6.1 and Ludwig Danzer for useful comments and, in particular, the notational suggestion of using two types of star. As well, I am indebted to the University of the South Pacific for employment and support during the initial stages of writing the chapter.

References 1. Z. I. Borevich, I. R. Shafarevich: Number Theory (Academic Press, New York 1966) 198, 199 2. J. H. Conway, N. J. A. Sloane: Sphere Packings, Lattices and Groups, 3rd edn. (Springer, New York 1999) 188, 189, 190, 193, 213, 221 3. G. Kasner, Z. Papadopolos, P. Kramer, D. B¨ urgler: “Surface structure of i-Al68 Pd23 Mn9 : an analysis based on the T ∗(2F ) tiling decorated by Bergman polytopes”. Phys. Rev. B 60, 3899–3907 (1999) 185 4. P. Kramer, Z. Papadopolos, H. Teuscher: “Tiling theory applied to the surface structure of icosahedral AlPdMn quasicrystals”. J. Phys.: Condens. Matter 11, 2729–2748 (1999) 185 5. R. V. Moody: “Meyer sets and their duals”. In: The Mathematics of LongRange Aperiodic Order, NATO ASI Series C 489, ed. by R. V. Moody (Kluwer, Dordrecht 1997) pp. 403–441 192 6. Z. Papadopolos, G. Kasner, J. Ledieu, R. McGrath, R. D. Diehl, T. A. Lograsso, D. W. Delaney: “Bulk termination of the quasicrystalline five-fold surface of Al70 Pd21 Mn9 ”. Preprint cond-mat/0111479 185 7. Z. Papadopolos, P. Kramer, G. Kasner, D. B¨ urgler: “The Katz–Gratias–de Boissieu–Elser model applied to the surface of icosahedral AlPdMn”. In: Quasicrystals, Materials Research Society Symposium Proceedings, Vol. 553, ed. by J.-M. Dubois, P. A. Thiel, A.-P. Tsai, K. Urban (Materials Research Society, Pittsburgh 1999) pp. 231–236 185 8. P. A. B. Pleasants: “Designer quasicrystals: cut-and-project sets with preassigned properties”. In: Directions in Mathematical Quasicrystals, ed. by M. Baake, R. V. Moody, CRM Monograph Series (American Mathematical Society, Providence 2000), pp. 93–138 186, 192, 194, 200, 207 9. P. A. B. Pleasants, M. Baake, J. Roth: “Planar coincidences for N -fold symmetry”. J. Math. Phys. 37, 711–717 (1996) 187, 204, 205 10. D. S. Rokhsar, N. D. Mermin, D. C. Wright: “Rudimentary quasicrystallography: the icosahedral and decagonal reciprocal lattices”. Phys. Rev. B 35, 5487–5495 (1987) 214 11. L. C. Washington: Introduction to Cyclotomic Fields, 2nd edn. (Springer, New York 1996) 199, 222

7 Thermally-Induced Tile Rearrangements in Decagonal Quasicrystals – Superlattice Ordering and Phason Fluctuation Keiichi Edagawa

7.1 Introduction The structure of a quasicrystal is characterized by quasiperiodic translational order and by a point group symmetry that is not allowed in conventional crystallography [1, 2, 3]. Much theoretical and experimental work has been done to determine the atomic structure in this new class of ordered state. Experimentally, in addition to X-ray diffractometry, which is a standard tool for crystal structure determination, microscopic techniques such as highresolution transmission electron microscopy (HRTEM) have been utilized effectively; HRTEM gives direct information about the local structure, which is complementary to that obtained from X-ray diffractometry. The formation of a quasicrystalline phase was first reported in a metastable phase in an Al–Mn alloy in 1984 [4]. Subsequent studies have revealed that quasicrystalline phases are formed in many alloy systems (for a review see [5]), some of which are thermodynamically stable. So far, two types of quasicrystalline phase have been found in a stable state: icosahedral and decagonal phases. The former have a structure with a 3-dimensional icosahedral point group symmetry while the latter have a structure consisting of a periodic stacking of atomic layers with a 2-dimensional decagonal symmetry. HRTEM has been applied more effectively to decagonal phases than to icosahedral ones because HRTEM images a projected structure along the incident beam direction; the interpretation of the image is simpler and more direct for the decagonal phases. In most cases, the structure of a decagonal quasicrystal can be regarded as consisting of a quasiperiodic arrangement of some specific columnar atomic cluster whose column axis is parallel to the 10-fold periodic direction [6, 7, 8, 9]. By means of HRTEM with the incident beam parallel to the 10-fold direction, we can observe the arrangement of the columnar clusters, which forms a 2-dimensional tiling structure. In this chapter, we focus on the 2dimensional tiling structure: in particular, on its dynamic rearrangements induced thermally at high temperatures. In general, the structure of a quasicrystal can be described as a section of a high-dimensional periodic structure [2, 3, 10, 11]. This fact allows us to extend the concept of superlattice order in ordinary crystals to quasicrystals. The superlattice order in a quasicrystal is defined as the superlattice order in the high-dimensional periodic lattice. In a previous paper [12], we have reported the formation of a decagonal phase having such a kind of superlatP. Kramer (Ed.): Convering of Discrete Quasiperiodic Sets, STMP 180, 227–256 (2002) c Springer-Verlag Berlin Heidelberg 2002

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tice order. In subsequent studies [13, 14, 15, 16], we have characterized the superlattice structure and investigated the order–disorder transformation by electron diffraction, X-ray diffraction, and HRTEM. By means of HRTEM, we have observed qualitatively different tiling structures and shown that they can be interpreted as superlattice-ordered and disordered structures [15]. We have also observed the process of superlattice ordering as a change in the tiling structure [16]. Reflecting the quasiperiodic translational order, quasicrystals have additional elastic degrees of freedom not found in conventional crystals, which are termed phason degrees of freedom [18]. Quasicrystals have a phason elastic field u⊥ (r) in addition to the conventional elastic field u (r). Elastic excitations associated with u⊥ (r) are called phasons, in contrast to phonons, which are associated with u (r). While the gradient of u (r), i.e. ∇u (r), gives the conventional strain, the gradient ∇u⊥ (r) is called the phason strain. Phason strain has been observed experimentally as shifts and broadenings of diffraction peaks [19, 20, 21, 22], which correlate with the phason momentum G⊥ , as jogs in atomic rows in HRTEM images [19, 64], and as defects in the tile arrangement in HRTEM images [23, 24, 25]. In all of these studies, the phason disorders studied were either grown in or quenched. Recently, we have succeeded in observing tile rearrangements in motion by in situ high-temperature HRTEM experiments; these experiments are interpreted as thermally induced phason fluctuations [17]. The subject of this chapter is the tile rearrangements observed in HRTEM images of decagonal quasicrystals. The two types of tile rearrangements described above are described: superlattice ordering and phason fluctuation. In Sect. 7.2, the basic concepts related to quasicrystalline structural order, the 4-dimensional description of the 2-dimensional decagonal quasicrystalline structure, and the phason elastic degrees of freedom are briefly reviewed; these are necessary for the analysis of the experimental results described in the following sections. After the experimental procedures have been described in Sect. 7.3, the experimental results and a discussion of them are presented in Sect. 7.4. Finally, in Sect. 7.5, the results obtained are summarized.

7.2 Basic Concepts The diffraction intensity function I(q) ≡ | ρ(r) exp(−2πiq·r) dr|2 (where ρ(r) is the atomic density function in real space) observed experimentally for a quasicrystal has the following characteristics [1, 2, 3]: (1) It consists of δ-functions. (2) The number of basis vectors necessary for indexing the positions of the δ-functions exceeds the number of dimension. (3) It shows a rotational symmetry forbidden in conventional crystallography.

7 Thermally-Induced Tile Rearrangements

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Conversely, we define a quasicrystal as a material with a ρ(r) which gives a diffraction intensity function I(q) satisfying these conditions. The condition (1) implies that the atomic arrangement in this material has a kind of longrange translational order. The conditions (2) and (3) indicate that the order is not periodicity. The translational order defined by the conditions (1) and (2) is called quasiperiodicity. It can be shown mathematically that every quasiperiodic structure can be described as a section of an appropriate high-dimensional periodic structure. As an example, Fig. 7.1 shows a typical 1-dimensional (1D) quasiperiodic structure known as a Fibonacci lattice, which is described here as a 1D section of a 2D periodic structure. Here, E and E⊥ denote the physical space and the complementary space perpendicular to it, respectively. The 2D periodic structure consists of a periodic arrangement of line segments extending in the direction of E⊥ . These line segments are called atomic surfaces. A point sequence is obtained on the E section, comprising an arrangement of two spacings labeled L and S. Here, the slope of E with respect to the 2D lattice √ is an irrational number τ (the golden mean (1 + 5)/2). In this case, the resultant sequence of the two spacings becomes equivalent to the Fibonacci sequence and therefore it is called a Fibonacci lattice. The irrational slope indicates a lack of periodicity in the arrangement of L and S. However, we can prove that the diffraction intensity function I(q) for the Fibonacci lattice satisfies the conditions (1) and (2), and thus it has a quasiperiodicity. To describe the structure of a 2D decagonal quasicrystal with point group symmetry 10mm, we use a 4D periodic structure having the same point group symmetry [11, 26]. The basis vectors di (i = 1, . . . , 4) of the 4D periodic lattice are given by

Fig. 7.1. A Fibonacci lattice described as a 1-dimensional section of a 2-dimensional periodic structure

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di =

Mij ej ,  c1 s1 a  c 2 s2 M = √   c 3 s3 2 c4 s4 j

c2 c4 c1 c3

 s2 s4  , s1  s3

(7.1)

where cj = cos(2πj/5), sj = sin(2πj/5) and a is the lattice constant. ei (i = 1, . . . , 4) are the orthonormal unit vectors of the 4D space; e1 and e2 span the physical space E while e3 and e4 span the complementary space E⊥ perpendicular to E . Fig. 7.2 shows the vectors pi (i = 1, . . . , 4) and q i (i = 1, . . . , 4) that are defined by the projections of di (i = 1, . . . , 4) onto E and onto E⊥ , respectively. As for pi (i = 1, . . . , 4) and q i (i = 1, . . . , 4) in the figure, an explanation will be given later. The group 10mm is generated by a 10-fold rotation C10 and a mirror σ. The actions of these operators on di (i = 1, . . . , 4) are given by 4 × 4 integer matrices Γ (R):     0 0 −1 0 0 0 0 1  0 0   0 −1   , Γ (σ) =  0 0 1 0  . (7.2) Γ (C10 ) =   1 1   1 1 0 1 0 0  −1 0 0 0 1 0 0 0 The actions of these operators on ei are given by real matrices Γ (R) = M −1 Γ (R)M :     0 0 −c3 s3 1 0 0 0  −s3 −c3  0 −1 0 0  0 0    Γ (C10 ) =   0 0 −c1 s1  , Γ (σ) =  0 0 1 0  . (7.3) 0 0 −s1 −c1 0 0 0 −1

p

q

d

Fig. 7.2. The projections i (i = 1, . . . , 4) and i (i = 1, . . . , 4) of i (i = 1, . . . , 4) onto E and E⊥ , respectively, and the projections i and i of i onto E and E⊥ , respectively

p

q

d


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These matrices show that (1) the two operations leave E and E⊥ invariant, (2) the rotation C10 acts on E and E⊥ as 2D rotations of π/10 and 3π/5, respectively, and (3) the mirror σ acts on E and E⊥ as mirrors. By placing atomic surfaces that spreadout in E⊥ (in this case they are 2D objects) at appropriate positions in the 4D lattice, a 2D decagonal quasicrystalline structure is obtained as a section on E . To preserve the 10mm symmetry, each of the atomic surfaces must satisfy the site symmetry of the position on which the atomic surface sits. For example, the position (0,0,0,0) has a site symmetry of 10mm and therefore the atomic surface at (0,0,0,0) must satisfy the 10mm symmetry. The site symmetries of other positions are lower than the full symmetry of 10mm. Reflecting the high dimensionality of the structure, quasicrystals have special types of elastic degrees of freedom, not found in conventional crystals, which are termed phason degrees of freedom [18]. In Fig. 7.3a, the Fibonacci lattice is redrawn. Here, let the 2D periodic structure be translated by a vector U with respect to the origin of the physical space E (Fig. 7.3b). The two structures in E before and after the displacement U , shown in Figs. 7.3a,b are related in the following manner: the two structures can be overlapped, out to arbitrarily large finite distances, by a finite translation in E . Two structures satisfying this condition are said to belong to the same local isomorphism class (LI class) [1, 2, 3]. Thus, we find that the displacement U represents the degrees of freedom that generate a series of structures belonging to the same LI class. Obviously, from the definition of the LI class, a series of structures in the same LI class are geometrically indistinguishable on any finite scale and thus they are also physically indistinguishable. They should show the same diffraction intensity I(q) and also should have the same energy. The vector U can be decomposed into u in E and u⊥ in E⊥ : U = u + u⊥ .

(7.4)

Here, u represents the degrees of freedom of translation in physical space, which conventional crystals also possess. In contrast, u⊥ represents the degrees of freedom characteristic of a quasiperiodic system. As shown in Figs. 7.3c,d, while u results in a translation of a Fibonacci lattice in E , u⊥ generates a rearrangement of L and S. While elastic excitation related to u are called phonons, those related to u⊥ are called phasons [3, 18]. Accordingly, the two kinds of degrees of freedom are called phonon and phason degrees of freedom, and u and u⊥ are called phonon and phason displacements, respectively. When these displacements change as a function of the position r in E , the rate of change gives the strain. More specifically, while the gradient ∂u /∂r gives the conventional elastic strain, the gradient ∂u⊥ /∂r is called the phason strain. Introduction of phason strain generates local rearrangements of points (atoms) such as LS↔SL, which are called phason flips. It should be noted that the structure becomes periodic when the magnitude of the phason strain takes one of a particular set of values and is

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Fig. 7.3. A Fibonacci lattice (a), and a structure resulting from a displacement of (b), of (c) and of ⊥ (d)

U

u

u

constant in r . A phase having such a periodic structure is called a crystal approximant to a quasicrystal. Investigating phason disorder is important because it could give an answer to the fundamental question as to why the quasicrystalline structural order is realized in materials. At present, two distinct models have been proposed for the physical origin of quasicrystalline structural order. One is the perfectly ordered quasiperiodic model, in which the quasicrystal is assumed to be energetically stabilized [1, 28]. The other is the random-tiling model, in which the quasicrystal is assumed to be stabilized by a configurational entropy related to the phason disorder (for a review, see [31]). At present, there is still controversy as to which of the two models better describes quasicrystals in real material; it has also been suggested that a quasicrystal may make transitions between the two states with changing temperature.

7.3 Experimental Procedures Alloys with the compositions Al70 Ni17 Co13 , Al70 Ni22 Co8 , and Al65 Cu20 Co15 were prepared from the elemental constituents by arc melting under an argon atmosphere. The alloy ingots of Al70 Ni17 Co13 and Al65 Cu20 Co15 were remelted at 1433 K, slowly cooled down to 1173 K at a controlled cooling rate of 10 K/ h, and then furnace-cooled to room temperature. By means of this heat treatment, columnar grains of the decagonal phase 3–5 mm in


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length and 0.2–0.5 mm in diameter were grown for both of the two alloys. For the Al70 Ni22 Co8 alloy, melt-spun ribbon samples were prepared. The Al–Ni–Co single grains and ribbon samples were used for experiments on superlattice ordering, while the Al–Cu–Co single grains were used for experiments on phason fluctuations. The samples were sealed in evacuated quartz tubes and annealed at various temperatures for various periods, followed by water quenching. The samples were used for HRTEM experiments with a Hitachi H-9000 and a JEOL JEM-2010F transmission electron microscope. For the Al–Cu–Co samples, in-situ high-temperature HRTEM observations were made using a double-tilting heating stage in the temperature range between room temperature and 1203 K. The detailed procedures have been presented in previous papers [12, 13, 14, 15, 16, 17].

7.4 Results and Discussion 7.4.1 Superlattice Ordering Figs. 7.4a,b show HRTEM images of the Al70 Ni22 Co8 melt-spun ribbon sample annealed at 1123 K for 4 h and the Al70 Ni17 Co13 single-grain sample annealed at 923 K for 72 h, respectively. In the images, we can see clearly ringlike contrast features with a diameter of about 1.2 nm, which are believed to correspond to columnar atomic clusters. In general, the contrast in HRTEM images changes greatly, with the sample thickness and the defocus value. In the present experiments, we used the ring-like contrast features as the most recognizable structural unit to construct the tiling pattern. This was always possible by choosing regions with an appropriate sample thickness and by adjusting the defocus value. In the present experiment, a special technique was additionally used to enhance the ring contrast; the size of the aperture that restricted the number of diffraction spots contributing to the image was reduced [15]. A close inspection of the images reveals that not all the ring contrast features have a perfect ring shape; some of them are separated into parts and others are considerably distorted. In Fig. 7.4, only the ring contrast features with comparatively high perfection have been selected, and their centers have been connected by the five basis vectors pi (i = 0, . . . , 4), with a length of about 2.0 nm, shown in the figure. It should be noted that changing the threshold for selecting the rings has been confirmed not to alter the results of the analyses described below. In Figs. 7.5a,c, the tiling patterns constructed from the images of a larger area of the same samples as used in Figs. 7.4a,b, respectively, are presented. In Figs. 7.5b, the tiling pattern constructed from an image of the Al70 Ni17 Co13 single-grain sample annealed at 1123 K for 3 h is also presented. The meaning of the different colors will be explained later. In the pattern in Fig. 7.5a, we can see overlapping of the tiles at many places, reflecting the high density of the ring contrast features. This pattern contains many regular pentagons.

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Fig. 7.4. HRTEM images of Al70 Ni22 Co8 melt-spun ribbon sample annealed at 1123 K for 4 h (a) and of the Al70 Ni17 Co13 single-grain sample annealed at 923 K for 72 h (b). Arrangements of ring-like contrast features are seen. The tiling patterns were constructed by connecting the centers of the ring contrast features by vectors i (i = 0, . . . , 4) with a length of about 2.0 nm

p

The pattern in Fig. 7.5b also contains regular pentagons, but the density of them is lower than in Fig. 7.5a. The major element in the pattern in Fig. 7.5b is a hexagon, which can be divided into a rhombus with a vertical angle of π/5 and two rhombuses with a vertical angle of 2π/5. The tiling pattern in Fig. 7.5c consists mostly of three elements: the two kinds of rhombus and a hexagon which can be divided into a 2π/5 rhombus and two π/5 rhombuses. In this pattern, we can detect almost no regular pentagons. Similar HRTEM observations of the change of structure in an Al–Ni–Co alloy have been reported by Hiraga et al. [27]. In the work of these authors, the arrangements of ring contrast features similar to those in Fig. 7.5a,c have been observed in Al70 Ni15 Co15 alloys annealed at 1073 and 823 K, respectively. In Hiraga et al.’s image for the 1073 K annealed sample, the density of the rings is relatively low and there are few overlaps of the tiles, in contrast to our observation for the 1123 K annealed sample. The difference is considered to


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Fig. 7.5. The arrangements of the centers of the ring contrast features for the Al70 Ni22 Co8 melt-spun ribbon sample annealed at 1123 K for 4 h (a), for the Al70 Ni17 Co13 single-grain sample annealed at 1123 K for 3 h (b), and for the Al70 Ni17 Co13 single-grain sample annealed at 923 K for 72 h (c). The five colors represent the five kinds of sites Am (m = 0, ±1, and ±2) (see text)

be mainly due to the difference in the observation conditions, especially the size of the aperture described above. In our experiment, the density of the ring contrast features tends to become high owing to the enhancement of the contrast. Because the ring positions r are connected by the basis vectors pi (i = 0, . . . , 4), we can assign indices (n0 , . . . , n4 ) to every r so as to satisfy r = 4 i=0 ni pi . However, this indexing has an ambiguity in the sense that (n0 +h, n1 +h, n2 +h,n3 +h, n4 +h), where h is an arbitrary integer, designate the same 4 position as (n0 , n1 , n2 , n3 , n4 ) because i=0 pi = 0. To ensure uniqueness of the indexing, we must adopt four of the five vectors as the basis vectors,

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e.g. pi (i = 1, . . . , 4). The basis vectors pi (i = 1, . . . , 4) correspond to those shown in Fig. 7.2 and can be regarded as the projection of the basis vectors di (i = 1, . . . , 4) of the 4D lattice defined by (7.1). This indicates that the indices (n1 , . . . , n4 ) designate a 4D lattice point. Now we consider a lattice transformation [12, 13, 26, 29, 30] defined by Tij dj , di = j   1 0 −1 0 (7.5)  0 1 0 −1  . T =  1 1 2 1  −1 0 0 1 The projections pi and q i of di onto E and onto E⊥ , respectively, are shown in Fig. 7.2, together with the projections pi and q i of di . Here, the following points should be noted: (1) T is an integer matrix, (2) the determinant of T is 5, and (3) pi and pi are geometrically equivalent on E , and the same is true for q i and q i on E⊥ . By “geometrically equivalent” it is meant that the set of pi or q i can be transformed into pi or q i by only a rotation and a scale change in the respective space. Points (1) and (2) indicate that the lattice spanned by di (the di -lattice) consists of five equivalent sublattices, one of which coincides with the di -lattice (see Fig. 7.9a). Point (3) indicates that the 10mm. The five sublattices are di -lattice also has the point group symmetry classified according to the sum of the indices i ni (mod 5), i.e. they can be written as Am = {r = i ni di | i ni = m (mod 5)} (m = 0, ±1, and ±2). In Fig. 7.5, different colors are assigned to the five kinds of sites. We find that in the patterns of Figs. 7.5a,b all five sets of points are homogeneously distributed. In contrast, they appear to be highly inhomogeneous in the pattern in Fig. 7.5c and we could divide the pattern into domains. The homogeneous distribution of points in the pattern in Fig. 7.5a is related to the fact that the tiling has many regular pentagons; each vertex of a regular pentagon belongs to each of the five sites. In contrast, an abundance of rhombuses in the pattern in Fig. 7.5c leads to an inhomogeneous distribution of the five sets of points. A similar discussion of the difference between a pentagonal tiling and a rhombic one has been given by Hiraga et al. [47]. To visualize more clearly the domain structure in the pattern of Fig. 7.5c and also the different natures of the patterns of Fig. 7.5a–c, we show a 2D color map in Fig. 7.6, in which the brightness changes in the radial direction and the hue changes as the polar angle about the center changes. The five positions ci (i = 0, ±1, and ±2) give the five colors representing the five kinds of sites. Using the color map of Fig. 7.6, we constructed the patterns shown in Fig. 7.7 from those in Fig. 7.5 through the following procedures. First, discs of radius r0 , which were painted with the colors corresponding to ci (i = 0, ±1, and ±2), were placed at the ring positions. If two or more disks overlapped, the overlapping area was painted with the color at the center of gravity (in the color map) of the positions ci giving the colors of the overlapping disks.


237

Fig. 7.6. The 2D color map used for constructing the painted patterns shown in Fig. 7.7 and Fig. 7.11. The five positions i (i = 0, ±1, and ±2) give the five colors representing the five kinds of sites Am (m = 0, ±1, and ±2) (see text)

c

We expect that (1) a position that has approximately the same numbers of points for all five sites in its neighborhood is given a dark color, that (2) a position that has very different numbers of points for the five sites in its neighborhood is given a bright color, and that (3), in the case of (2), the color indicates which site has a large population around the position. The patterns in Fig. 7.7 were constructed using a disk radius r0 = 4.0 nm. The pattern in Fig. 7.7a consists entirely of dark colors. In the pattern in Fig. 7.7b, we can see some moderately bright colors in some areas. In contrast, the whole pattern consists of bright colors in Fig. 7.7c. In addition, we find a domain structure in Fig. 7.7c, as shown in the figure. It should be noted that in any individual domain the point densities for the five sites differ grately; an individual domain consists mostly of only three of the five kinds of sites (see Fig. 7.5c). In the framework of the high-dimensional description of the quasicrystalline structure described in Sect. 7.2, the point density in E is determined by the size of the atomic surface (see Fig. 7.1). Thus, the different densities for the five sublattices indicate that the average sizes of the atomic surfaces for the five sublattices are different. This indicates nothing but a superlattice order. To make this point clearer, we have calculated r ⊥ = i ni q i for the indices (n1 ,n2 ,n3 ,n4 ) of the ring positions r , where q i are the projections of the basis vectors di of the 4D lattice onto E⊥ (Fig. 7.2). In Fig. 7.1, we can see that the position r of a point on E is given by the intersection between an atomic surface and E . Here, the vector pointing from the lattice point to the intersection point is given by −r⊥ . Thus, for an ideal quasicrystal, the set of points {−r⊥ i } in E⊥ deduced from {r i } in E should be distributed within a bounded domain corresponding to the atomic surface. In Fig. 7.8, the point distributions {r⊥i }, calculated separately for the five sublattices are shown; they have been deduced from the entire patterns shown

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Fig. 7.7. Painted patterns constructed from the tiling patterns in Fig. 7.5 using the 2D color map shown in Fig. 7.6. The brightness of the color corresponds to the degree of superlattice order. Different colors correspond to different variants of the superlattice structure. In the pattern in (c), we find a domain structure

in Figs. 7.5a,b and 7.7a,b, and from the two parts of the pattern shown in Figs. 7.5c and 7.7c denoted by A and B. We notice a striking contrast regarding the equivalence of the five sets; in Fig. 7.8a, the numbers and the spatial extensions of the points are almost the same for all the sets, while they are different in Fig. 7.8b, and the difference is much more pronounced in Figs. 7.8c,d. These facts clearly indicate that the phase of Figs. 7.5c and 7.7c is in a superlattice-ordered state and that the phase of Figs. 7.5a and 7.7a is in a disordered state. The phase of Figs. 7.5b and 7.7b is considered to correspond to an intermediate state between the two preceding phases. In Figs. 7.8c,d, the five sites appear to be interchanged cyclically, indicating that the two regions A and B in Fig. 7.7c correspond to two of the five pos-


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r

Fig. 7.8. The point distributions { ⊥i }, calculated separately for the five kinds of sites Am (m = 0, ±1 and ±2). (a), (b) Distributions deduced from the entire patterns of Figs. 7.5a,b (or 7.7a,b), respectively; (c), (d) distributions deduced from the two parts of the pattern of Fig. 7.5c (or 7.7c) denoted by A and B, respectively. The numerals indicate the numbers of points

sible variants in the present ordered phase. This type of domain structure of variants is commonly observed in conventional ordered crystalline phases. In Fig. 7.9, the relation between the superlattice-ordered structure and a disordered structure revealed by the analyses above is explained schematically using a low-dimensional example. The 2-dimensional square lattice spanned by d1 and d2 consists of five equivalent sublattices, one of which coincides with the square lattice spanned by d1 and d2 (Fig. 7.9a). When different atomic surfaces are assigned to the five sublattices, the resultant structure has a superlattice order. In Fig. 7.9b, only three of the five sublattices have atomic surfaces. When these atomic surfaces sit randomly on the di -lattice points as in Fig. 7.9c, the superlattice order is lost and the structure becomes disordered. In this case, the distributions of the five sets of points {r⊥i } deduced from {r i } in E should be indistinguishable, as shown in

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Fig. 7.9. Schematic illustrations showing the relation between a superlatticeordered structure and a disordered structure. In (a), the superlattice–sublattice relation in the high-dimensional space is shown. The 2-dimensional square lattice spanned by 1 and 2 consists of five equivalent sublattices, one of which coincides with the square lattice spanned by 1 and 2 . In (b), by assigning different atomic surfaces to the five sublattices, a superlattice-ordered structure is constructed in E . In (c), those atomic surfaces are placed randomly on the five sublattices, resulting in a structure without superlattice order on E

d

d

d

d

Fig. 7.8a. In this way, the superlattice-order-to-disorder transformation in a quasicrystal can be interpreted as resulting from a rearrangement of the atomic surfaces. The atomic surfaces have a 2D shape for a 2D quasicrystal and contain information about the positions of an infinite number of atoms


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(or atomic clusters). This is in contrast to the case in conventional crystals, in which each lattice point is decorated with a limited number of atoms. A rearrangement of the atomic surfaces can induce a global rearrangement of the atoms (or atomic clusters) and thus induce a qualitative change in the tiling pattern, as shown in Fig. 7.5. This is a special type of superlattice ordering characteristic of quasicrystals. Superlattice structure has also been characterized by electron diffraction [12, 14, 15, 43] and X-ray diffraction [13, 14, 33], and order-disorder transformation has been investigated by X-ray diffraction [13, 14, 34]. The results of these diffraction experiments have indeed shown the order–disorder relation in the 4D lattice given by (7.5), although a different interpretation on the basis of a structural modulation in real space has also been proposed [32]. In the present case of superlattice order, there exist two independent order parameters [13, 30]. Accordingly, the superlattice reflections have been classified into two groups, which are indicated by peaks referred to as S1 and S2. In the case of the composition Al70 Ni17 Co13 , we obtain a completely ordered state showing both S1 and S2 peaks at low temperatures. The sample of Al70 Ni17 Co13 annealed at 923 K in the present experiment corresponds to the completely ordered state. On the other hand, at high temperatures, we obtain a partially disordered state for which the S2 peaks disappear almost completely for the composition Al70 Ni17 Co13 . This state is called the S1 state. The sample of Al70 Ni17 Co13 annealed at 1123 K in the present experiment is in this state. We have confirmed that for the composition Al70 Ni17 Co13 , the completely disordered phase for which both the S1 and the S2 peaks completely disappear is not formed at any temperature below the melting point. The sample of Al70 Ni22 Co8 annealed at 1123 K in the present experiment is in the completely disordered state. Detailed investigations of the phase diagram of the Al–Ni–Co system have been performed [35, 36, 37, 38, 42]. According to the phase diagram that has been constructed, a complete order–disorder transformation takes place in a narrow composition range around Al71.2 Ni18 Co10.8 . It should be noted that in addition to the superlattice structure, a variety of structures related to the decagonal quasicrystal have been found to form in this system: a different type of superlattice structure called the type-2 superlattice structure [43], a one-dimensionally periodic 5-fold quasicrystal [39, 46], a one-dimensional quasicrystal [40, 41], and two different types of the basic decagonal quasicrystal. One of the basic decagonal quasicrystals (the Ni-rich basic phase) has a high structural perfection and has been subjected to many structural studies [44, 45, 48, 49, 50, 51, 52], although this phase is in a disordered state in the sense of the absence of superlattice order. Finally, in this section, we present the results of our observations of the superlattice ordering process with the passage of time. First, we prepared the S1 phase by annealing a single-grain sample of Al70 Ni17 Co13 at 1173 K for 1.5 h, followed by water quenching. The annealed grains of the S1 phase

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were then crushed into pieces, which were subsequently annealed at 973 K for 1 min, 5 min, 30 min, and 180 min to investigate the structural change that took place at this temperature. Each sample, sealed in an evacuated quartz tube, was inserted into an electric furnace, the temperature of which had been set at 973 K in advance; the sample was held at this temperature for a period and then quenched in water. Figs. 7.10a–e show the tiling patterns deduced from HRTEM images of samples annealed for 0, 1, 5, 30 and 180 min, respectively. Five colors are assigned to the five sublattices, as in Fig. 7.5. In the pattern in Fig. 7.10a, all the five colors appear to be distributed homogeneously. In the patterns in Figs. 7.10b–e, we can see that the additional annealing at 973 K introduces inhomogeneity into the distribution of the five sites, indicating that the superlattice ordering proceeds in the form of the evolution of a domain structure. The evolution of the domain structure may be seen more clearly in the painted patterns in Fig. 7.11. The painted patterns were constructed from the tiling patterns in Fig. 7.10 by the same method as for Fig. 7.7. In Figs. 7.11a,b, we can detect no definite domains, and dark colors cover the whole of the patterns, indicating a low degree of superlattice order. In Fig. 7.11c, the domains are formed in many places, and they cover the whole of the patterns in Figs. 7.11d,e. It is noteworthy that the degrees of order, which are measured by the brightness of the colors, and the domain sizes observed in Figs. 7.11d,e are approximately the same, indicating that the ordering becomes saturated at around 30 min for the annealing at 973 K. 7.4.2 Phason Fluctuations In this section, we present the results of our observations of thermally induced tile rearrangements interpreted as phason fluctuations; these observations were done by the means of in situ high-temperature HRTEM experiments. Fig. 7.12 shows an HRTEM image of a single-grain sample of Al65 Cu20 Co15 observed at 1123 K, in which we can recognize an arrangement of white spots. In these high-temperature experiments, ring-like contrast features such as that seen in Fig. 7.4 were not always observed clearly. Instead, a pattern of white spots could be obtained in a relatively thick region at any temperature by adjusting the defocus value. It has been confirmed by room-temperature observation that the positions of the white spots coincide with those of the centers of the ring contrast features. Besides the white spots, we can see diffuse clouds in between the spots. Phason flips were observed to occur in the following way: from time to time an embryo of a spot emerges in a diffuse cloud, and gains intensity to become a spot, and at the same time a spot nearby becomes weak and becomes buried in a diffuse cloud. In the image shown, the white spots are connected by the five basis vectors pi (i = 0, . . . , 4) with a length of about 2.0 nm. The tiling pattern constructed from an image of a larger area of the same sample is presented in Fig. 7.13a. The pattern consists of many elements: the major ones are a regular pentagon, a rhombus with a vertical angle of π/5 (π/5 rhombus), a 2π/5


243

Fig. 7.10. The arrangements of the centers of the ring contrast features for the sample annealed for 0 min (a), 1 min (b), 5 min (c), 30 min (d) and 180 min (e). The five colors represent the five kinds of sites Am (m = 0, ±1, and ±2)

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Fig. 7.11. Painted patterns constructed from the tiling patterns in Fig. 7.10 using the 2D color map in Fig. 7.6. The brightness of the color corresponds to the degree of superlattice order. Different colors correspond to different variants of the superlattice structure


245

Fig. 7.12. HRTEM image taken from an Al65 Cu20 Co15 single-grain sample at 1123 K. An arrangement of white spots can be seen. Tiling patterns were constructed by connecting the white spots by vectors i (i = 0, . . . , 4) with a length of about 2.0 nm

p

rhombus, a hexagon which can be divided into a π/5 rhombus and two 2π/5 rhombuses, and a hexagon which can be divided into a 2π/5 rhombus and two π/5 rhombuses. Comparing the pattern in Fig. 7.13a with those in Fig. 7.5, we find that the pattern of Fig. 7.13a contains all the elements that constitute the three patterns in Fig. 7.5 and that the pattern of Fig. 7.13a is qualitatively different from any of the three patterns in Fig. 7.5. We have applied the same analyses as those performed in the cases of Fig. 7.5 and 7.7 to the pattern Fig. 7.13a and found no definite domains of superlattice structure, although the five sites are not distributed very homogenously compared with the pattern in Fig. 7.5a. Electron diffraction experiments have shown that no superlattice reflections occur in this phase. It is concluded from these facts that the present phase has no superlattice order. Instead, local periodic arrangements of tiles are conspicuous in the pattern in Fig. 7.13a, indicating that this pattern contains a large amount of phason disorder. The presence of a large amount of phason disorder can be confirmed by inspecting the point distribution {r⊥ i } in E⊥ as shown in Fig. 7.13b. The circle in the figure indicates the size of the atomic surface to yield the same density of points in E as in the observed pattern. The average density of the tile vertices in the observed pattern was evaluated as 2.86 × 10−1 nm−2 . The corresponding circle radius was calculated to be 1.02|qi |, which is approximately the same as the center-to-vertex distance of the decagon atomic surface for the pentagonal Penrose tiling [15, 64], which is equal to |q i |. We notice that the point distribution is considerably scattered and extends a large way beyond the boundary of the atomic surface. This fact indicates the presence of a large

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Fig. 7.13. (a) Tiling pattern constructed from an HRTEM image of the Al65 Cu20 Co15 single-grain sample at 1123 K; (b): the point distributions { ⊥i } deduced from the tiling pattern. The circle in (b) indicates the size of the atomic surface required to yield the same density of tile vertices as in the observed pattern

r

amount of phason disorder. This can be interpreted as spatial phason fluctuations, in contrast to the temporal phason fluctuations described later. Figure 7.14 presents an example of the changes in the HRTEM image observed at 1123 K. In these images, we notice a change in the arrangement of the white spots as follows. In Fig. 7.14a, we have a hexagon on the left-hand side and a pentagon on the right-hand side. In the image of Fig. 7.14b where the elapsed time is 5 s, the spot at the position A becomes blurred while a new spot emerges at the position B. In Fig. 7.14c, the spot at A completely disappears while that at B becomes intense with the result that the positions of the hexagon and the pentagon are exchanged. After that, essentially no change was observed until an elapsed time of about 110 s, when the spot at the position B became blurred and that at A became visible again, as shown in Fig. 7.14e. Finally, in the image of Fig. 7.14f with an elapsed time of 115 s, essentially the same spot arrangement as that in Fig. 7.14a is observed. The


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Fig. 7.14. An example of the changes in the HRTEM image observed at 1123 K. Elapsed times for (a)–(f ) are 0, 5, 8, 110, 113 and 115 s, respectively. Transitions between two configurations are observed

distance between A and B is |pi |/τ 1.2 nm. During our observation over a period of about 20 min, a transition between the two configurations was observed to occur several times, irregularly at intervals ranging from a few tens of seconds to about ten minutes. Figure 7.15 presents a different example of the changes in the HRTEM image observed at 1123 K. The tile rearrangement observed is illustrated in the lower parts of the figure. Here, the three spots A, B, and C become weak and vanish simultaneously, and at the same time the three spots D, E, and F become intense; the reverse process also occurs. The fact that the transition was observed to take place in both directions indicates that the tile rearrangements observed are not due to any phase transition. The frequency of the transition is roughly the same as in the case shown in Fig. 7.14. In Figs. 7.16a,b, the point distributions {r⊥ i } on E⊥ calculated from the spot positions {r i } in the images of Figs. 7.14 and 7.15, respectively, are shown. Here, r ⊥ points calculated from the neighboring spots are also included. The r ⊥ points for A and B in Fig. 7.14 and those for A–F in Fig. 7.15 are indicated by the same letters in Figs. 7.16a,b, respectively. The large circle indicates the size of the atomic surface deduced from the average density of the tile vertices (the same atomic surface as in Fig. 7.13b). We

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Fig. 7.15. An example of the changes in the HRTEM image observed at 1123 K. The two configurations between which the transitions were observed are illustrated in the lower part

r

Fig. 7.16. (a) and (b) Point distributions { ⊥i } deduced from the spot positions in the images of (a) Figs. 7.14 and (b) 7.15. The ⊥ points calculated from the neighboring spots are also included. The ⊥ points for A and B in Fig. 7.14 and those for A–F in Fig. 7.15 are indicated by the same letters in (a) and (b), respectively. The large circle indicates the size of the atomic surface required to yield the same density as that of the observed spots

r

r

find that the distance between the r⊥ positions of A and B in Fig. 7.16a is roughly comparable to the diameter of the atomic surface. The same is true for the distance between the two groups of (A, B, and C) and (D, E, and F) in Fig. 7.16b. These facts suggest that the tile rearrangements in Figs. 7.14 and 7.15 are due to phason fluctuations. In the present experiments, changes in the tile configuration were observed happening at temperatures above 1073 K. As expected, the frequencies, both spatial and temporal, become higher with increasing temperature. In addition, we found that the frequencies have a tendency to be high in the regions


249

Fig. 7.17. An example of a tile change observed in an HRTEM image taken from a thin part of the sample at 1203 K. The elapsed times are also shown

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of small sample thickness near the edge of the sample. In Fig. 7.17, an example of a tile change observed in an HRTEM image at 1203 K in a thin region is presented. In contrast to the images at 1123 K of relatively thick regions, where tile changes are observed to occur at rather isolated positions, tile changes were observed almost everywhere in the image of the thin area at 1203 K. In addition, not only go-and-return transitions between two configurations but also successive transitions among many configurations were seen, as exemplified in Fig. 7.17. The temporal frequency was also much higher in the case of the results shown in Fig. 7.17. A detailed analysis of the tile changes in Fig. 7.17 will be reported elsewhere. Phason flips in atomistic models of decagonal quasicrystal have been discussed theoretically [9, 53, 54, 55, 56, 58], and their energetics have been investigated using computer simulation [54, 55, 56]. Thermally induced phason flips have been observed experimentally by neutron scattering [57, 58, 59, 61], by M¨ ossbauer spectroscopy [60], and by NMR [62, 63]. The timescale of the phason flips observed has been shown to be 1 ps–10ns at high temperatures above 1000 K, which is many orders of magnitude shorter than timescales observed in the present experiment. The origin of such a large difference can be explained as follows. What is observed by neutron scattering, Mössbauer spectroscopy, and NMR is an elemental phason flip which involves only a single atom (or a few atoms). On the other hand, what we observed by HRTEM is a large-scale phason flip which consists of a collective motion of elemental phason jumps, involving many atoms. Here, we note that what was observed in the present experiment correspond to phason flips in a 2 nm tiling and that HRTEM images a projection of the structure. These facts indicate that a large number of atoms are involved in the phenomena observed here. This explains the large difference in the timescale; the probability of a collective motion of many atoms is expected to be lower than the probability of an elemental motion by many orders of magnitude. This also explains why the frequency of the phason flips is high in thin regions; the number of atoms involved is small there. To make this point clearer, we present the results of a computer simulation using a simple model, which is illustrated in Fig. 7.18. Here, we assume two positions between which a white spot can make transitions in the HRTEM image. In each atomic layer, atoms can take either of the two configurations, i.e. the configuration contributing to a white spot at A or that contributing to a white spot at B. The ratio of the numbers of layers corresponding to the configurations A and B determines whether we observe a white spot at A or B in the HRTEM image. We assume an energy difference ε1 = εB − εA (> 0) between the configurations A and B in each layer and an interlayer mismatch energy ε2 which is imposed when we have an A–B or B–A sequence for two adjacent layers. In this model, we have two independent parameters: ε1 /ε2 and kT /¯ ε (¯ ε = (ε1 + ε2 )/2). We have calculated equilibrium states for various values of the two parameters by a Monte Carlo method.


251

Fig. 7.18. Illustration of the model of phason flips used in a computer simulation. The structure of a decagonal quasicrystal consists of a periodic stacking of atomic layers in the 10-fold direction. Two positions, between which a white spot can make transition in the HRTEM image, are assumed and are denoted by A and B. In each layer, atoms can take either of the two configurations, i.e. the configuration contributing to a white spot at A or that contributing to a white spot at B

Figure 7.19 shows snapshots of the equilibrium states at various temperatures for ε1 /ε2 = 0.05. At sufficiently low temperatures, we have the A-configurations in almost all the layers. As the temperature increases, the number of B configuration increases and at a certain temperature the fraction of the B configuration reaches roughly 50%. At this temperature, the density of interlayer mismatches is still low because a relatively high interlayer mismatch energy is assumed here. However, this density becomes high at a sufficiently high temperature of kT = 50¯ ε. Figure 7.20 shows the variation of the ratio between the numbers of configurations A and B with elapsed time (measured in Monte Carlo steps) for various thicknesses at kT = 0.9¯ ε, for ε1 /ε2 = 0.05. Here, the calculation was performed for a system of 1000 layers and the number ratios were evaluated for the whole system, for a certain fixed portion of 100 layers, and for that of 30 layers. For the sufficiently large thickness of 1000 layers, the fraction of the B configuration is almost constant at about 30%, which is the equilibrium value at this temperature. Here, we would always observe a white spot at the position A in an HRTEM image and detect no transitions of the spot. In contrast, for a small thickness of 30 layers, the value fluctuates greatly and from time to time it deviates by a large amounnt from the equilibrium value.

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Fig. 7.19. Snapshots of the equilibrium states for ε1 /ε2 = 0.05 at kT /¯ ε = 0.33 (a), 0.66 (b), 0.9 (c), 12.5 (d), 20 (e) and 50 (f ). Black and white indicate the A and B configurations, respectively

Fig. 7.20. Variation of the ratio between numbers of the configurations A and B with elapsed time (measured in Monte Carlo steps) for various thicknesses at kT = 0.9¯ ε (¯ ε = (ε1 + ε2 )/2), for ε1 /ε2 = 0.05


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This gives a qualitative explanation for the fact that we observe phason flips frequently in thin regions.

7.5 Summary We have presented two types of tile rearrangement observed by HRTEM in decagonal quasicrystals: superlattice ordering and phason fluctuations. In the Al–Ni–Co system, various qualitatively different tile structures have been found to form, which have been analyzed on the basis of the 4D description of the 2D decagonal quasicrystalline structure. The analysis has shown that the different tile structures can be attributed to decagonal quasicrystalline structures with and without superlattice order. The superlattice-order-todisorder structural change is interpreted as resulting from a rearrangement of the atomic surfaces, which induces a qualitative change in the tiling pattern. The superlattice ordering process has been observed as a function of time. It has been shown that the ordering process proceeds in the form of the evolution of a domain structure of variants. In the Al–Cu–Co system, go-and-return transitions between two local tile configurations have been observed by in situ high-temperature HRTEM. These transitions are interpreted as due to thermal phason fluctuations. The spatial and temporal frequencies of the tile configuration changes become higher with increasing temperature. In addition, the frequencies are found to have a tendency to be high in thin regions near the edge of the sample. At a high temperature and in a thin region, not only go-and-return transitions between two configurations at isolated positions but also successive transitions among many configurations have been observed almost everywhere. The timescale of the phason flips observed is of the order of a few seconds to a few tens seconds, which is many orders of magnitude larger than those previously found by neutron scattering and M¨ ossbauer spectroscopy. The origin of the difference can be explained as follows. The phason flips observed by neutron scattering and M¨ ossbauer spectroscopy are elemental ones involving only a single atom (or a few atoms), while those observed by HRTEM in the present experiment are large-scale phason flips which consists of a collective motion of the elemental phason flip, involving many atoms. The probability of such a collective motion of many atoms is expected to be lower than the probability of an elemental motion by many orders of magnitude. This also explains why the frequency of the phason flips is high in thin regions: the number of atoms involved is small there. A Monte Carlo simulation using a simple model has demonstrated this effect.

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Acknowledgments The author thanks S. Takeuchi for many valuable comments and discussions. He is also indebted to K. Suzuki and M. Ichihara for HRTEM experiments, H. Tamaru for data analysis, and Y. Kamimura for manuscript preparation.

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8 Tilings and Coverings of Quasicrystal Surfaces Rón´ an McGrath, Julian Ledieu, Erik J. Cox, and Renee D. Diehl

8.1 Introduction The tiling and covering of images derived from the experimental technique of high-resolution transmission electron microscopy (HRTEM) has been a fruitful avenue in studies of bulk quasicrystal structures. In particular, the quasiunit-cell model has been successfully applied to the 2-dimensional decagonal AlNiCo (d-AlNiCo) quasicrystal (see Sect. 8.3.1). However, HRTEM is a bulk technique and can only be used to observe the average structure over the thickness of the sample. On the other hand, scanning tunneling microscopy (STM) is a purely surface technique. Hence, if high-quality images of quasicrystal surfaces can be obtained, then the possibility arises of applying tiling and covering methodologies to these surfaces, with a view to elucidating their structure and determining whether or not they are terminations of the bulk. A corollary of this approach is that such studies would provide independent verification of current bulk tiling and covering models. In this chapter, we describe our attempts to apply tilings and coverings to images of surfaces of quasicrystals obtained using STM [1, 2, 3, 4, 5, 6, 7]. We begin with a short review of previous STM work on quasicrystal surfaces. The first STM study of a quasicrystal surface was reported in 1990 by Kortan et al. [8] for decagonal (d-)AlCuCo. These workers produced a pentagonal Penrose tiling model from their high-resolution studies. As this work preceded the quasi-unit-cell model, the latter paradigm was not available to apply to their results. Attention then passed to the study of the 5-fold, 3-fold and 2-fold surfaces of the icosahedral quasicrystal i-AlPdMn. Schaub et al. produced detailed images of the 5-fold surface, which they interpreted in terms of an Ammann pentagrid model with Fibonacci relationships between structural elements within the terraces and across steps on the surface [9, 10, 11, 12]. Later, these measurements were shown to be in correspondence with the Katz–Gratias–Elser model [13, 14] for the atomic positions [15, 16]. Shen et al. , using an autocorrelation analysis, showed that the surface structure is consistent with a bulk structure based on truncated pseudo-Mackay icosahedra or Bergman clusters [17]. However, the somewhat limited resolution of these measurements precluded their analysis using the tiling approach described in this chapter. Recently, a joint STM/low-energy electron diffraction (LEED) study of the 5-fold quasicrystalline surface of icosahedral AlCuFe (i-AlCuFe) has been reported [18]. This study produced images somewhat similar to those previP. Kramer (Ed.): Convering of Discrete Quasiperiodic Sets, STMP 180, 257–268 (2002) c Springer-Verlag Berlin Heidelberg 2002

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ously obtained for i-AlPdMn [9, 10, 11, 12, 17] and led to the conclusion that this surface does not have a perfect quasicrystalline nature, but has stacking faults, as evidenced by the observed order of step sequences and the appearance of screw dislocations. Finally, a recent STM study [19] has produced atomic-resolution images of the 10-fold and 2-fold surfaces of d-AlNiCo. The images of the 10-fold surface show 5-fold-symmetric features which have opposite orientations in successive planes. In this chapter, we describe some results from our recent studies of the 5-fold i-AlPdMn and 10-fold d-AlNiCo surfaces [1, 2, 3, 4, 5, 6, 7, 20]. We concentrate on the methodology we have employed for tilings and coverings of STM images of quasicrystal surfaces, rather than on the details of the preparation and characterization of the surfaces (these topics have been described elsewhere). We focus on the analysis methodology which we have developed to aid the interpretation of our results and to allow comparison with bulk structural models; specifically, we apply Penrose tiling and quasi-unit-cell covering models to STM images of the 5-fold surface of i-AlPdMn, and the 10-fold surface of d-AlNiCo.

8.2 The 5-Fold Surface of i-AlPdMn We first describe studies of the 5-fold surface of i-AlPdMn. Our initial studies produced results similar to those previously reported, with limited resolution and with some structural imperfections. However, these results led to incomplete tilings of the surface; the main findings are described in Sect. 8.2.1. Later results were of a more structurally perfect surface and led to more complete tiling patches; these are described in Sect. 8.2.2. 8.2.1 Initial Results The quasicrystalline surface phase of i-AlPdMn can be produced by cycles of sputtering and of annealing the surface to approximately 970 K. The LEED pattern obtained [1, 2] from such a preparation has sharp spots exhibiting 5-fold rotational symmetry and a low background intensity (Fig. 8.1a). The perfection of the quasicrystalline phase turns out to be very dependent on preparation conditions. Figures 8.1b,c show STM data obtained from such a surface. The 20 nm× 20 nm STM image of Fig. 8.1c shows various features including large protrusions and pentagonal depressions (average width 0.6 ± 0.2 nm). A 2dimensional autocorrelation pattern was calculated for the surface shown in Fig. 8.1c and reveals a relative strong spatial correlation over the entire STM image (see Fig. 8.2a). The pentagonal depressions themselves can be used to form the vertices of pentagonal tiles (average width 1.6 ± 0.2 nm), as shown in Fig. 8.2b. In order

8 Tilings and Coverings of Quasicrystal Surfaces

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Fig. 8.1. (a) LEED pattern (inverted contrast) recorded after annealing to 970 K. (b) 150 nm × 150 nm image of the surface; the box indicates the size of the image shown in (c); (c) 20 nm × 20 nm high-resolution STM image of a flat terrace (bias voltage 2.29 V, tip current 0.59 nA). After [21]

to more easily identify this tiling, a threshold filter (Fig. 8.2c) was applied to the images, making the pentagonal depressions more visible. Height values greater than 0.06 nm were discarded. The resulting tiling is shown in Fig. 8.2. A

B

Protrusions

pentagonal pattern

C

D

Fig. 8.2. (a) 20 nm×20 nm autocorrelation function of the STM image of Fig. 8.1c. (b) 10 nm × 10 nm region of the STM image shown on Fig. 8.1c. Large protrusions and a pentagon enclosing a 5-fold star are indicated. (c) Map obtained by applying a threshold filter where height values > 0.06 nm were discarded. (d) Map obtained by tiling the inverted contrast 20 nm × 20 nm STM image of Fig.8.1c (Vt = 2.29 V, It = 0.59 nA)

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Fig. 8.2d shows that the tiling is split into several different patches, with discontinuities between. There are two types of frustration present: (i) inherent frustration (see diamond shapes In Fig. 8.2d) and (ii) frustration due to protrusions. Nevertheless, there are many common features between the tiling obtained from this analysis and the random tiling described by Janot [22] (see Fig. 8.3). The random tiling described by Janot [22] is a network generated by combining clusters of atoms (or other structural units) having icosahedral symmetry and connected by atomic bonds or by atoms.

Fig. 8.3. (a) Map obtained by tiling pentagons on a high-resolution inverted (20 nm ×20 nm) STM image (Fig. 8.1c) of a flat terrace (Vt = 2.29 V, It = 0.59 nA). (b) Random tiling with forced edge orientational order [22]

It is apparent that the large protrusions that appear on the images from the surface (shown in Fig. 8.2b and in Fig. 8.3a in reverse contrast) limit the applicability of the tiling approach here. Such protrusions have also been observed by the other groups who have studied this surface [9, 10, 11, 12, 17]. Kishida et al. [19] have observed similar protrusions on 10-fold and 2-fold dAlNiCo surfaces. Kishida et al. interpret these protrusions as atoms in locally symmetric sites which are relatively strongly bound and have been left behind during terrace formation while the sample is being annealed after sputtering. Such a suggestion is consistent with our own analysis of the positioning of these protrusions, where we find that the distances and angles between them are not random but are consistent with what might be expected for highsymmetry sites [1]. 8.2.2 Higher-Resolution Studies In subsequent experiments, refinements to our polishing and preparation techniques (described in [4]) led to high-resolution STM images such as that shown in Fig. 8.4a. This image shows atom-sized features as well as larger features (0.4–0.6 nm wide), which probably correspond to clusters of a few


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atoms. The density of pentagonal depressions appears to be approximately three times lower than in Fig. 8.1c. A 5 nm × 5 nm STM image from the same surface is shown in Fig. 8.4b. Several pentagons having edge lengths equal to 0.80 ± 0.03 nm are outlined on this image. These pentagons have been generated by connecting protrusions on the image, rather than 5-fold depressions as in the work described in Sect. 8.2.1. The inset on Fig. 8.4b shows a fast Fourier transform calculated for the STM image of Fig. 8.4a, exhibiting rings of ten spots. A semiquantitative measurement of the difference in resolution between this image and that of Fig. 8.1(c) can be obtained by comparing radial distribution functions of the autocorrelation patterns of the two images. Such a comparison is shown in Fig. 8.5. It can be seen that there is a far greater level of detail in the data presented in Fig. 8.4a, which allows us to carry out a more complete tiling analysis. The identification of pentagons drawn by connecting STM protrusions, as shown in Fig. 8.4b can be extended over a larger area, and the result produces a tiling, as shown in Fig. 8.6a. In this tiling, several different shapes, namely rhombuses, boat-like shapes, and large 5-fold stars, are distinguishable. The tiling is not a perfect match to the STM image, in that only 93% of the 119 vertices shown match areas of high contrast. This is what we might expect for STM data, where the surface itself may not be perfect but may contain vacancies and possibly adsorbates such as H, which may produce contrast variations. Additionally, the STM technique maps charge density rather than atomic positions; on an aperiodic surface there will inevitably be contrast variations in the vertices of the tiling due to atomic-structure differences at these locations. Finally, small distortions due to piezoelectric drift cannot be ruled out. The perfection of HRTEM images, which are averages of the bulk

Fig. 8.4. (a) 10 nm × 10 nm STM image of 5-fold surface of i-AlPdMn. (b) 5 nm × 5 nm STM image (for both images, V = 1 V, I = 0.3 nA). Pentagons are outlined. Inset: fast fourier transform. After [4]

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Fig. 8.5. Radial distribution functions derived from 2-dimensional autocorrelation patterns from (a) the image of Fig. 8.1c and (b) the data shown in Fig. 8.4a

structure, cannot be expected for STM data, and this is one of the inherent limitations of the technique. Nevertheless, such tilings can be compared with those expected from bulk models. Figure 8.6b represents a theoretical tiling T ∗(p1)r derived from the Katz–Gratias–Elser tiling model T ∗(2F ) defining the quasiperiodic structure [4]. T ∗(p1)r is referred to as a random Penrose tiling because of the rules of its derivation; it clearly matches the geometry of the experimentally derived tiling of Fig. 8.6a. The experimental tiling can also be superimposed on one of the dense atomic planes perpendicular to the 5-fold axes of the Al70 Pd21 Mn9 quasicrystal described by Boudard’s bulk model [4, 23]. The conclusion that can be drawn is that the STM results are indicative of a truly bulk-terminated surface.

8.3 The 10-Fold Surface of d-AlNiCo There has been considerable recent interest in the bulk structure of the d-AlNiCo quasicrystal [24], because of the recent development of the quasiunit-cell model. This model is described in Sect. 8.3.1. In Sect. 8.3.2 we compare the 10-fold surface of d-AlNiCo the quasi-unit-cell model. 8.3.1 Quasi-Unit-Cell Covering Model The results of quantitative structure determinations of decagonal quasicrystals indicate that the 3-dimensional structure consists of quasiperiodic atomic


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Fig. 8.6. (a) Experimental tiling using pentagons of edge length 0.80 ± 0.03 nm on the STM image shown on Fig. 8.4a. (b) Left: the tiling T ∗ of the plane by the acute rhombus, pentagon, and hexagon, locally derived from T ∗(A4 ) . Center : the construction of the tiling T ∗(p1)r . Right: the tiling T ∗(p1)r without the content of golden triangles [4]

layers which are stacked according to different sequences. An equivalent structural description can be formulated in terms of columnar clusters parallel to the 10-fold axis [25]. One way of describing this columnar-cluster model is as a 3-dimensional extension of a 2-dimensional Penrose tiling. In the Penrose tiling picture, the atoms are arranged into clusters which are analogous to the rhombic Penrose tiles. The interactions connecting clusters can be compared to the Penrose matching rules for tiles. However, recently Steinhardt et al. [24] have proposed a different model for the quasicrystalline structure of d-AlNiCo: a single repeating “quasi-unit cell”, which is illustrated in Fig. 8.7. This picture utilizes identical clusters as repeating units. Unlike a periodic unit cell, however, these clusters can share atoms, i.e. they can overlap. Many theoretical overlapping-cluster models have since emerged, with overlap rules constraining the merging of neighboring clusters. These models do not force a unique structure – for example, Burkov’s model can be compared to a binary tiling with an infinite number of possible atomic arrangements [27]. Gummelt has shown that atomic clusters and overlap rules can be chosen so as to force a unique atomic arrangement which is isomorphic to a Penrose tiling [28]. Previously, the widely accepted view was that two types of cluster were necessary to force quasiperiodicity but Gummelt showed that instead of two incommensurate lengths arising from two different tile shapes, incommensurate lengths can also arise from overlap rules which allow only two nearest-neighbor distances between clusters. In two dimensions, the clusters are replaced by decagonal tiles which overlap, covering the two-dimensional plane. Gummelt showed that with the correct overlap rules, these decagonal tiles can force a perfect quasiperiodic tiling. Therefore, to determine the

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Fig. 8.7. A quasiperiodic tiling can be forced using marked decagons as shown in (a). Decagons may overlap only if the shaded regions overlap. This leads to two possibilities, where the overlap area is either small (A-type) or large (B-type), as shown in (b). If each decagon is inscribed with an obtuse rhombus as in (c), a tiling of overlapping decagons (d, left) is converted into a Penrose tiling (d, right). From [26]

Fig. 8.8. A candidate model for the atomic decoration of the decagonal quasi-unit cell. Large circles represent Ni and Co, and small circles Al. The structure has two distinct layers along the periodic c axis. Solid circles represent c = 0, open circles c = 1/2. After [29]

atomic structure, one needs only to determine the atomic distribution within a decagonal tile. On the basis of this paradigm, Steinhardt et al. have proposed a model for the atomic structure of d-AlNiCo, shown in Fig. 8.8.


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The resulting structure is consistent with results from HRTEM [30, 31, 32, 33] and X-ray diffraction, and provides a better fit than do previous models, including Penrose tilings [24]. The quasi-unit-cell and Penrose tile pictures are real-space descriptions of quasicrystals where the structure can be defined by an identical decoration of each quasi-unit cell or Penrose tile. In the hyperspace model, quasicrystals are viewed as projections from a higher-dimensional periodic, hypercubic lattice. The decoration of this lattice consists of atomic surfaces in this higherdimensional space (usually five or six dimensions). These surfaces project into point atoms in three dimensions. Thus the quasi-unit-cell picture is a simpler concept, since it is much easier to consider atomic arrangements within a single quasi-unit cell in real 3-dimensional space than by considering decoration of two or more tiles, or by dealing with 5- or 6-dimensional surfaces. 8.3.2 Experimentally Derived Covering Like the 5-fold surface of i-AlPdMn, the 10-fold surface of d-AlNiCo exhibits a crystalline phase after being sputtered, which progresses into a clustered surface structure upon annealing to relatively low temperatures, and finally forms a terraced phase at higher annealing temperatures (≥ 725 K). The quality of the surface obtained is dependent on the annealing temperature: as the temperature increases so do the number, intensity, and sharpness of the LEED spots, indicating increasing long-range order. The best-quality LEED patterns have been obtained after the sample has been annealed to 1125 K, as shown in Fig. 8.9. The patterns have 10-fold rotational symmetry, with the peak positions being related by the golden number, τ , which is indicative of quasicrystallinity in the surface region. Since d-AlNiCo is periodic perpendicular to the 10fold direction, a single interlayer spacing is expected. Gierer et al. [34] have deduced the interlayer spacing from their spot-profile-analysis LEED (SPA-

D C

a

B

A

b

Fig. 8.9. (a) LEED pattern obtained after annealing to 1125 K, taken at 79 eV, room temperature. (b) LEED pattern after annealing to 1125 K and cooling the sample to 100 K. The peak positions are related by the golden number τ . A/B = τ , C/D = τ

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LEED) data to be 0.204 nm. An average terrace width was also determined in that study to be 17.0 nm. The STM images after this treatment indicate a surface which is much flatter than those prepared at lower annealing temperatures. Figure 8.10a shows the evolution of flat terraces on the surface; Fig. 8.10b shows a 15 nm× 15 nm atomically resolved STM image [6]. A correspondence can be drawn between the overlapping tiling model described in Sect. 8.3.1 and the high-resolution STM images. Figure 8.11 shows an STM image decorated with overlapping decagons which have been chosen to coincide with the protrusions, which form rings 2 nm in diameter. The decagons overlap in the ways shown in Fig. 8.7. It is possible to see evidence of the atomic structure inside the decagons. It is apparent, at least on this length scale, that the quasi-unit-cell model may be applied to the surface of this material. These results suggest that the surface has the same quasiperiodic structure as in the bulk – i.e. that the surface has a bulk-like termination, in agreement with X-ray photoelectron diffraction (XPD) and reflection high-energy electron diffraction (RHEED) analyses [35].

a

b

Fig. 8.10. (a) 80 nm × 80 nm STM image after annealing to 1075 K – “Flat” terraces. (b) 15 nm × 15 nm STM image after annealing to 1125 K

Fig. 8.11. 5 nm × 5 nm STM image of d-AlNiCo showing a partial covering, obtained using the overlap rules described in [29]


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8.4 Conclusions In this chapter, we have given a brief overview of the tiling approach to the analysis of quasicrystal surfaces. The method provides a further indication that the sputtering/annealing methodology for surface preparation leads to surfaces of both i-AlPdMn and d-AlNiCo which are essentially terminations of the bulk structure. A corollary is the verification of the validity of bulk tiling and covering models. This work and work by other groups indicate that these surfaces are now well enough understood that there are possibilities for their exploitation in the technologically important area of nanotechnology. One possibility for the creation of such nanostructures and arrays which we are currently exploring is the use of quasicrystalline surfaces as templates for atomic and molecular adsorption [7, 20]. Acknowledgements We thank Prof. Pat Thiel, Dr. Cynthia Jenks, Dr. Tom Lograsso, Dr. Amy Ross, Dr. Nate Kelso, and Dr. Paul Canfield of the quasicrystals program at the Ames Laboratory, Iowa, for provision of samples. We acknowledge Dr. Zorka Papadopolos and Dr. Gerald Kasner of the University of Magdeburg for many stimulating discussions. The EPSRC (grant numbers GR/N18680 and GR/N25718) and NSF (grant number DMR9819977) are acknowledged for funding.

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Index

acceptance domain, 191 algebraic number field, 221, 224 – cyclotomic, 199, 202, 208, 210, 222 – quadratic, 198–199, 213 alignment, 185, 193–196 alternation condition, 65, 79, 85 Ammann bars, 207 aperiodic decagon, 64 aperiodic point set, 6 associate, 204 atomic positions, 15 atomic surface, 229 automorphism, 202 boundary, 4, 98 class number, 198 cluster, 63, 166, 169–171, 174, 176–183 – coding of, 170 – covering, 16, 17, 97, 100 – decagonal, 97 – Delone, see Delone cluster – filling, 107, 112, 144, 154 – G-, 16 – linkage, 14, 125 – patch, 182 – volume, 157 – Voronoi, 12, 16, 98, 100 – window, 100, 107 – window for, 18 cluster density maximization, 63 cluster-covering principle, 64 coincidence site modules, 204 complex splitting primes, 204 conjugation – algebraic, 186, 193, 198, 221, 224 – complex, 187 – quaternionic, 187, 213

covering, 1, 2, 165, 166, 169–172, 176, 177, 182, 183, 185 – by Delone clusters, 121 – decagon, 12 – Delone, 165, 166, 170 – – thickness of, 166, 171 – Delone CT ∗(D6 ) , 166, 176, 177, 180, 182 – – thickness of, 166, 180, 182 – Delone CTk ∗(A4 ) , 166, 170, 171, 182 – – thickness of, 170 – Delone CTs ∗(A4 ) , 165, 166, 169, 170, 172, 182 – – local derivation of, 169 – – thickness of, 166 – – window of, 170 – full, 166, 182 – local derivation of, 169 – of tile, 108, 160 – of vertex, 108, 160 – protoclusters, 169 – thickness, 2 – thickness of, 100, 166, 182 – window of, 166 Coxeter – cone, 136 – group, 7, 133 Coxeter group, 8 crystal, 185, 188–191, 223 crystal approximant, 232 crystallography – n-dimensional, 5, 7 cut-and-project – quotient scheme, 197 – subscheme, 196 cut-and-project scheme, 186, 191–198, 200

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Index

d-AlCuCo, 168, 257 d-AlNiCo, 257 decking, 166, 171, 180, 183 – double-, 166, 171 – fraction of, 180 – single-, 166, 171 – triple-, 166, 171 – zero-, 166, 183 deflation, 12 Delone cell, 165 – boundary of, 165 – – projected X∗ , 165 Delone cluster, 11, 16, 98, 100, 106, 152, 154, 166, 171, 176–180, 182 – T ∗(A4 ) – – Dx , 169–171 – – Dy , 169–171

– T ∗(D6 ) , 178–180 – – Da , 178 – – Db , 180 – – Dc , 179 – coding of, 176 – filling of, 176 – frequencies of, 180 – window of, 171, 177, 180, 181 – – generating code, 177 Delone cluster Da , 177, 178 – filling of, 177, 178 – total window of, 178 – – motif of, 178 – window of the filling, 178 – – generating code of, 177, 178 Delone cluster Db , 180, 181 – filling of, 180 – total window of, 180, 181 – – motif of, 180, 181 – window of the filling, 180 – – generating code of, 180 Delone cluster Dc , 179 – filling of, 179 – total window of, 179 – – motif of, 179 – window of the filling, 179 – – generating code of, 179 Delone cluster Dh , 176 density – center, 188 – of a lattice, 188

derivability radius, 67 different, 198 Dirichlet series, 204, 209, 211, 223 discriminant, 198, 223 – of a lattice, 188 – of a module, 188 dual – boundary, 102 duality, 8, 98 experimental techniques – HRTEM, 227 – RHEED, 266 – SPA-LEED, 266 Fibonacci lattice, 229 filling – unique, 100 for tiling – fundamental domain, 10 fractional ideal, 198 function – compatible with tiling, 9, 105 – periodic, 99, 105 – quasiperiodic, 9, 99, 105 fundamental domain, 4, 9, 16, 18, 100, 104, 125, 157 general position, 192, 193 golden ratio, 193 group – Coxeter, 7, 133 – general linear, 134 – icosian, 214 high-dimensional lattice L, 165, 177 hole – in lattice, 102, 110 – point symmetry, 111, 136 – representative, 136 HRTEM, 233, 257 i AlPdMn, 166, 257 i-AlCuFe, 166, 257 icosian, 213 icosian ring, 214 inflation, 12 internal space, 207, 222 Katz–Gratias–Elser model, 257

Index klotz construction, 8 lattice, 3, 98, 185, 188–191, 223 – basis, 132 – dual, 186, 188, 191, 223 – hole, 110, 132 – holohedry, 3, 109 – quotient, 186, 190 – reciprocal, 188 – root, see root lattice – sublattice, 186, 190, 224 LEED, 257 LI class, 231 local equivalence, 67 local isomorphism, 67 local rules, 63 M¨ ossbauer spectroscopy, 250 matching rules, 63, 68 maximal cluster-covering principle, 65 maximum-density principle, 15 Meyer set, 192 minimal embedding, 192 model surface, 166 module, 6, 7, 223 – dual, 201–202, 217 – full, 198, 199, 221 – icosahedral, 101, 132, 213, 220 – in O, 198 – O-basis of, 201 – O-module, 198 – over K, 199–202, 221 module factor, 200 Monte Carlo method, 250 mutual local derivability, 67 neutron scattering, 250 NMR, 250 noncrystallographic group, 165 – icosahedral Ih , 178–180 norm, 198, 221 – quaternionic, 213 – reduced, 213 normalizing factor, 205 ordering principles, 65 orthogonal space E ⊥ , 165, 177–180 packing, 1

271

– density of, 1 parallel space E , 165, 170, 177–180, 182 partly random tiling, 172–176 – T ∗(nr ) , 174–176 – – local derivation of, 175, 176 – T ∗(p1)r , 171–174 – – local derivation of, 173 Penrose quasilattice, 193 Penrose tiling, 11, 193, 194, 205, 257 phason – degrees of freedom, 231 – disorder, 228, 245 – displacement, 231 – elastic field, 228 – flip, 231, 250 – fluctuations, 228, 242, 246, 248 – strain, 228, 231 phason degrees of freedom, 231 physical space, 222 point group, 99, 106 polytope – Delone, 4, 98, 140 – klotz, 103 – Voronoi, 3, 98, 102 projection – compatible, 106 – icosahedral, 134 quasi-unit cell, 16, 97, 106 quasi-unit-cell, 257 quasicrystal, 6, 185 – icosahedral, 7, 98, 130 – lines in, 17 – planes, 17 – surface in experiment, 17 – surface in theory, 17 – with 5-fold symmetry, 7, 110 quasilattice, 185, 191, 224 – 10-fold, 202–207 – 12-fold, 210–212 – 14-fold, 222 – 8-fold, 208 – icosahedral, 213–220 – quotient, 185, 186, 196 – repetitive, 192 – subquasilattice, 196 – subquasilattices, 186 – uniform, 192

272

Index

quasiperiodicity, 229 – enforced, 15 quaternions, 213 random tiling model, 232 range of matching rules, 64 ring – icosian, 214 – of integers, 198 root lattice, 3, 109, 131, 177–181, 220 – A4 , 165, 170 – – hole x, 171 – – hole y, 171 – D6 , 165, 174, 176–182 – – hole a, 177, 178 – – hole b, 177, 180, 181 – – hole c, 177, 179 scaling symmetry, 137 shelling, 15 -map, 186, 199, 224 STM, 167, 257 subcovering, 166, 170–172, 182 – CTs ∗(A4 ) , 166, 170–172, 182 – – local derivation of, 170 – – thickness of, 170, 171 – – window of, 171, 172 subtiling, 166, 167 super-tile random tiling, 66 TEM, 167, 227, 233, 257 terrace, 185, 186 theta series, 15 thickness, 122 tile, 177 – X∗ , 165, 177 – – coded by X⊥ , 176 – – codings (coding windows) X⊥ , 165 – – windows of, 177 – golden tetrahedra, 165, 168, 174, 177–179 – – codings (windows) of, 181 – golden triangles, 165, 167–169, 171 – – codings (windows) of, 167 – window, 103 tiling, 2 – T ∗(A4 ) , 263 – T ∗ , 263 – T ∗(p1)r , 262

T ∗(2F ) , 262 canonical, 103 coloring, 162 Fibonacci, 8 icosahedral, 140, 143 partly random, see partly random tiling – Penrose, see Penrose tiling – quasiperiodic, see tiling (quasiperiodic) – triangle, 12, 111 – Voronoi, 4 – window, 97, 103 tiling (quasiperiodic), 97, 165–177, 180, 182, 183 – τ T ∗(z) , 169, 174, 176 – E , space of tiling, 165, 174 – E ⊥ , coding space, 165, 176 – T ∗(ms) , 183 – T ∗(2F ) , 166 – T ∗(A4 ) , 165–172, 174, 175, 182 – – vertex configurations of, 169 – – window of, 168, 169, 172, 174, 175 – T ∗(D6 ) , 165–168, 174, 176, 177, 180, 182, 183 – – vertex configurations of, 182 – – window of, 177 – T ∗(L) , 165, 177 – T ∗(n) , 172, 174, 175 – – window of, 174, 175 – T ∗(p1) , 174 – – window of, 174 – T ∗(z) , 167–175, 182 – – window of, 168, 169 – canonical, 165, 166 – decagonal, 165, 166, 182 – decking of, 183 – icosahedral, 165–167 – Niizeki star-, 172 – patch of, 177 – Penrose P 1, 166, 172, 182 – Penrose P 2, 165, 182 – prototiles, 165–169 – subtiling, 165 – window of, 166 trace, 198, 221 – reduced, 213 T¨ ubingen tiling, 205 – – – – – –

Index unit, 199 – fundamental, 199, 222 V -condition, 192 Voronoi cell, 165, 174 – boundary of, 165, 174 – – projected X⊥ , 165, 176 – projected V⊥ , 165, 174, 178–180 W -condition, 192

Weyl reflection, 134 window, 9, 186, 191, 224 – criterion, 108 – for tiling, 99 – total, 108 window factor, 200 X-ray diffractometry, 227 XPD, 266

273

Springer Tracts in Modern Physics 160 Physics with Tau Leptons By A. Stah12000. 236 figs. VIII, 315 pages 161 Semiclassical Theory of Mesoscopic Quantum Systems By K. Richter 2000.50 figs. IX, 221 pages 162 Electroweak Precision Tests at LEP By W. Hollik and G. Duckeck 2ooo. 60 figs. VIII, 161 pages 163 Symmetries in Intermediate and High Energy Physics Ed. byA. Faessler, T.S. Kosmas, and G.K. Leontaris 2000.96 figs. XVI, 316 pages 164 Pattern Formation in Granular Materials By G.H. Ristow 20oo. 83 figs. XIII, 161 pages 165 Path Integral Quantization and Stochastic Quantization By M. Masujima 2ooo. o figs. XII, 282 pages 166 Probing the Quantum Vacuum Pertubative Effective Action Approach in Quantum Electrodynamics and its Application By W. Dittrich and H. Gies eooo. 16 figs. XI, 241 pages 167 Photoelectric Properties and Applications of Low-Mobility Semiconductors By R. K/~nenkamp 2ooo. 57 figs. VIII, loo pages 168 Deep Inelastic Positron-Proton Scattering in the High-Momentum-Transfer Regime of HERA By U.E Katz 2ooo. 96 figs. VIII, 237 pages 169 Semiconductor Cavity Quantum Electrodynamics By Y. Yamamoto, T. Tassone, H. Cao 2ooo. 67 figs. VIII, 154 pages 170 d-d Excitations in Transition-Metal Oxides A Spin-Polarized Electron Energy-Loss Spectroscopy (SPEELS) Study By B. Fromme 2ool. 53 figs. XII, 143 pages 171 High-Tc Superconductors for Magnet and Energy Technology By B. R. Lehndorff 2OOL 139 figs. XII, 2o9 pages 172 Dissipative Quantum Chaos and Decoherence By D. Braun 2OOL 22 figs. XI, 132 pages 173 Quantum Information An Introduction to Basic Theoretical Concepts and Experiments By G. Alber, T. Beth, M. Horodecki, E Horodecki, R. Horodecki, M. R/~tteler, H. Weinfurter, R. Werner, and A. Zeilinger 2ool. 60 figs. XI, 216 pages 174 Superconductor/Semiconductor]unctions By Thomas Sch~ipers20OL 91 figs. IX, 145 pages 175 Ion-Induced Electron Emission from Crystalline Solids By Hiroshi Kudo 2002. 85 figs. IX, 161 pages 176 Infrared Spectroscopy of Molecular Clusters An Introduction to Intermolecular Forces By Martina Havenith 20o2. 33 figs. VIII, 12o pages 177 Applied Asymptotic Expansions in Momenta and Masses By Vladimir A. Smirnov 2oo2.52 figS. IX, 263 pages 178 Capillary Surfaces Shape - Stability - Dynamics, in Particular Under Weighflessnes By Dieter Langbein 2oo2. 182 figs. XVIII, 364 pages 179 Anomalous X-ray Scattering for Materials Characterization Atomic-Scale Structure Determination By Yoshio Waseda 2002. 55 figs. XIV, 214 pages

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