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A cluster approach to random Penrose tilings
Materials Science and Engineering 294–296 (2000) 250–253
A cluster approach to random Penrose tilings Petra Gummelt∗ , Christoph Bandt Institut für Mathematik und Informatik, Arndt-Universität Greifswald, D-17487 Greifswald, Germany Received 1 September 1999; accepted 2 November 1999
Abstract An experimentally relevant subclass of the full random Penrose pentagon tiling ensemble is generated using a single cluster. Local matching rules based on cluster overlaps are equivalent to relatively weak axioms on minimal cluster distances. Energy and entropy arguments and recent results on constituent atomic clusters in decagonal quasicrystals are discussed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystals; Cluster models; Penrose tiling; Random tiling
1. Introduction Many recent experimental observations indicate typical atom clusters determining the structure of quasicrystalline phases. These clusters are nearly congruent copies of a single constituent prototype. Thus there is increased interest to describe quasiperiodic ordered structures with crystallographically forbidden symmetry in terms of a single building block. In the covering approach, relevant classes of quasiperiodic tilings are shown to be fully covered by a single covering patch of tiles. This property is satisfied for several examples of 8, 10 and 12-fold patterns [4,6,7,14,17] and is also discussed for icosahedral tilings [14,18]. The cluster- or ‘quasi-unit cell’ approach [12,23] is concerned with the generation of coverings by a single cluster. Analogous to the ‘puzzle principle’ of tilings, cluster matching rules (overlap rules) are defined. Up to now, decagonal Penrose tilings are special in the sense that only a decagon with aperiodic markings was proven to force perfectly ordered structures of Penrose type [9,10]. Although decagonally shaped atomic clusters have been observed in many decagonal quasicrystals, most of these phases have randomly ordered quasiperiodic layers [13]. Currently only d-Al–Ni–Co [20–23] and d-Zn–Mn–rare-earth [1,2] are regarded as nearly perfect decagonal structures. In particular, d-Al70 Ni19 Co11 and d-Al72 Ni20 Co8 are very close to a stacking of coverings by decagons whose centres define ideal Penrose patterns ∗ Corresponding author. Tel.: +49-3834-864637. E-mail address: [email protected] (P. Gummelt).
[20–23]. However, recent studies show that a replacement of geometrically defined aperiodic matching rules by a clever atomic decoration of the cluster does not work without a little contribution of disorder [3,5,21–24,27]. Thus, it seems useful to combine the random tiling model with the cluster approach. In the special case of Penrose-like structures, we relax the aperiodic decagon rules in order to create periodic approximants, random quasiperiodic as well as perfect quasicrystalline patterns by the same unit. An interpretation of energetically or entropically dominated stability mechanisms can be given for coverings of a single building block in a similar way as for tilings. In an idealised picture, perfect patterns describe energetically stabilised quasicrystals, assuming that energetical effects reflected by the tiling matching rules form the main component of free energy. Random tilings are related to entropically stabilised phases with energetically indistinguishable configurations at high temperatures [19].
2. Random decagon coverings Suppose, we have a set of prototiles P1 , . . . , Pk for some class of truly quasiperiodic tilings — like Penrose rhombs, or kite and dart. These are shapes together with certain arrows, vertex stars, or other decorations which force quasiperiodicity. If we remove the markings and require only edge-to-edge matching rules, we obtain a much larger class of tilings which is called the corresponding random tiling ensemble. The prototiles without markings are denoted P1? , . . . , Pk? . Here we want to do the same for the aperiodic regular decagon P introduced in [9,10]. In order to determine P ? ,
0921-5093/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 1 9 7 - 7
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251
Fig. 1. Aperiodic decagon P (subsets of type A and B and rocket-like decoration).
we have to consider the decoration of P in the covering context. The role of edges is replaced by six subsets of P with nonempty interior. They are congruent to A- or B-shaped sets and their symmetries are restricted by two rocket-like dark markings and a dark half-star (Fig. 1). In Fig. 2, the structure of the four “oriented quasi-edges” of type A (dark grey) and the two remaining ones of type B (light grey) is highlighted (this is not an equivalent aperiodic decoration!). Following the tiling picture, we now erase the arrows of all generalized edges of P . This results in the “unoriented” version P ? . To cover the plane by copies of P ? , we modify the overlapping rule [9,10] of the aperiodic decagon P . Neighbours are now assembled according to the “undirected quasi-edges” of P ? . Generation rule. Every copy of P ? has to be surrounded by neighbours which intersect P ? in either A1 , A2 , A3 , A4 , B1 or B2 (see Fig. 1) such that the boundary of P ? is covered by neighbour tiles. Proposition 1. Up to isometry, there are only five prototypes N1 , . . . , N5 of nearest-neighbour configurations of a tile P ? which fulfil the generation rule. Except for the orientation of the decagons, these configurations are congruent to the A–B neighbourhoods [9,10] of the aperiodic decagon P (positions of potential overlaps are not changed). The neighbourhoods N1 up to N5 are shown in Fig. 3. At the top, a pentagonal arrangement of decagons is fixed. Black dots represent the centres of P ? . Long edges
Fig. 2. Directed “quasi-edges” of P (left) and structure of the generalized prototile P ? (right).
Fig. 3. Nearest-neighbour configurations of P ? .
correspond to type-A overlaps, short edges connect decagons intersecting in a B-set. Proposition 1 is due to the small number of possible positions of a neighbour of P ? with A- or B-overlap (see Fig. 4). While the neighbours in Figs. 4a–d satisfy the aperiodic covering rules of P , new positions of neighbours are shown in Figs. 4e–h. However in cases (e) and (f), any further covering of the neighbour will contradict the generation rule. Moreover, the B-overlap of case (g) does not lead to new tilings. Only the remaining configuration (h) of two 180◦ -rotated decagons is responsible for an extension to random structures.
3. Hexagon-boat-star (HBS)-tilings The class of coverings generated by the aperiodic decagon cluster P is in one-to-one correspondence to perfect Penrose tilings. Moreover, it was shown in [10] that the pattern of decagon centres defines an ideal tiling consisting of the Penrose pentagon tiles (marked pentagons, star, ship, rhombus, cf. [8, p. 531]). One could expect that the centres of our modified decagon P ? determine the full random tiling ensemble generated by pentagon tiles without markings.
Fig. 4. Possible A- and B-neighbours of P ? .
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Fig. 5. Hexagon, boat, and star supertiles (grey).
Fig. 7. Forbidden linkages in aperiodic HBS-tilings.
However, we only obtain a special subclass, the so-called hexagon-boat-star (HBS)-tilings [16].
pattern of hexagon supertiles drawn in Fig. 6 illustrates the orthorhombic Al3 Mn phase [15,16], other approximants are reviewed in [26]. Various quasiperiodic HBS-tilings explaining the structure of decagonal Al–Mn, Ga–Mn and highly perfect Al–Ni–Co are derived in [11,16,21,25]. HBS-tilings are also interesting from a theoretical point of view. The number of different r-patches (consisting of translational P ? copies) grows exponentially for every radius r > 0, thus the configurational entropy of the system is positive. Perfect Penrose pentagon tilings (PPT) form a proper subset of HBS-tilings which are a subclass of the random Penrose pentagon (RPP) ensemble. A simple RPP-pattern which is not contained in the HBS-subclass is the periodic rhombus lattice. To reduce RPP-tilings to the HBS-subclass, star, ships and rhombus tiles have to be completely surrounded by pentagons, as stated in Proposition 2. Ideal order of Penrose type is forced for HBS-tilings if two special pentagon configurations are banned (Fig. 7). These forbidden patches consist of an N5 -neighbourhood and a 180◦ -rotated copy either of N5 or N2 . We conjecture that minimization of these “bad linkages” leads to (nearly) perfect quasiperiodic structures. This claim is supported by the analysis of several examples used to describe high-resolution electron micrographs of decagonal quasicrystals [16,20–23,25].
Proposition 2. For a covering constructed by the generation rule of P ? , the decagon centres define a tiling where all ship, star and rhombus tiles are surrounded by pentagons and thus define the HBS-supertiles of Fig. 5. All HBS-tilings can be obtained by coverings of P ? in this way. To prove this, we consider all prototypes of P ? coronas starting with our nearest-neighbour configurations N1 , . . . , N5 (Fig. 3). N1 forces a star surrounded by 10 pentagons (supertile S), N2 and N4 form a ship with eight pentagons around (supertile B) and N3 determines six pentagons and a rhombus within (supertile H). N5 appears as neighbourhood type of all decagon centres at the top of a star, ship or rhombus. Fig. 6 shows a simple periodic and a perfect quasiperiodic P ? covering, the corresponding pentagon tiling and the pattern of HBS-supertiles. In the case of ideal Penrose order, the decagon covering implies another HBS-tiling of smaller scale which is directly given by the aperiodic markings (pure rockets determine H-tiles, “half stars” lead to boats and dark stars define S-tiles, see also [24]). HBS-tilings represent an experimentally relevant subclass of the full random Penrose pentagon ensemble. Periodic as well as ideal or nearly perfect quasiperiodic HBS-structures are often used to describe decagonal quasicrystals and closely related approximant phases. For example, the
Remark. For the random tiling ensemble of thick and thin rhombi, maximization of the density of a certain decagonally shaped patch is essentially equivalent to Penrose’s aperiodic matching rules [12]. A main reason for this result is that the rhombic cluster decoration excludes many neighbourhoods which are responsible for periodicity. In the case of HBS-tilings, a simple maximization of P ? decagons or vertices of pentagon tilings, respectively, is not sufficient to force aperiodicity! For example, a pentagon-ship pattern consisting of translates of the boat supertile B has a vertex density higher than ideal PPTs by a factor of 1.021359. Recently, this tiling was used to describe the structure of new orthorhombic approximants of decagonal Ga–Mn and Ga–Fe–Cu–Si quasicrystals [26].
4. Minimal cluster distances
Fig. 6. Periodic and ideal quasiperiodic P ? structures.
Surprisingly, exactly the same class of decagon coverings we have defined by local matching rules can be generated using only some weak axioms on minimal distances of decagon
P. Gummelt, C. Bandt / Materials Science and Engineering 294–296 (2000) 250–253
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Drawing a decagon of diameter τ l = s + l around all x in Λ, we obtain a P ? covering according to our generation rule. The points x represent the centres of P ? clusters, l- and s-distances correspond to intersections of type A and B. Fig. 8. Nearest-neighbour triangles defined by Axiom 1.
References centres. Information on the specific local decoration of the decagonal cluster is not needed. This approach might lead to a simpler explanation of recent experimental observations where a (nearly) perfect quasiperiodic tiling of nearest neighbours of cluster centres was obtained only by various rather complex decorations of the clusters [5,23,24,27]. Let us consider a discrete plane point set Λ satisfying the Delone property and the following two additional conditions. Axiom 1. The distance d(x, y) of two points x, y ∈ Λ is either s, l or√d(x, y) ≥ s + l, where s and l are in ratio 1:τ , τ = 21 (1 + 5).
Axiom 2. Let y1 , y2 , . . . , yn be the nearest neighbours of x ∈ Λ in clockwise order. Then the angle ∠ yi xyi+1 between successive neighbours of x does not exceed 108◦ . Already the first axiom restricts the prototypes of nearest-neighbour triangles to configurations which are typical for pentagonal and decagonal symmetry (Fig. 8). Triangular arrangements of s-distances contradict a further covering of the plane. However, the lattice of equilateral triangles with edge length l must be excluded. This can be done by forcing the existence of short distances s. Nevertheless, gaps of arbitrary radius as well as strips and islands would still be possible. To avoid this, a kind of covering condition is given by Axiom 2. Under these conditions, the “beetle” of s- and l-neighbours of any point x in our set Λ must be congruent to one of the configurations N1 , N2 , N3 , N4 or N5 of Fig. 3. The proof will be given in a forthcoming paper. Minimal cluster distances of type s and l are physically justified by experimental observations. Density arguments can explain the second axiom.
[1] E. Abe, T.J. Sato, A.P. Tsai, Philos. Mag. Lett. 77 (1998) 205–211. [2] E. Abe, T.J. Sato, A.P. Tsai, Phys. Rev. Lett. 82 (1999) 5269–5272. [3] E. Abe, K. Saitoh, H. Takakura, A.P. Tsai, P.J. Steinhardt, H.-C. Jeong, Phys. Rev. Lett. 84 (2000) 4609–4612. [4] S.I. Ben-Abraham, F. Gähler, Phys. Rev. 60 (1999) 860–864. [5] E. Cockayne, M. Widom, Phys. Rev. Lett. 81 (1998) 598–601. [6] M. Duneau, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1998, pp. 116–119. [7] F. Gähler, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1998, pp. 95–98. [8] B. Grünbaum, G.C. Shephard, Tilings and Patterns, Freeman, New York, 1987. [9] P. Gummelt, Geometriae Dedicata 62 (1996) 1–17. [10] P. Gummelt, Aperiodische Überdeckungen mit einem Clustertyp, Shaker-Verlag, Aachen, 1999. [11] M. Hirabayashi, K. Hiraga, Mater. Sci. Forum 22–24 (1987) 45–54. [12] H.-C. Jeong, P. Steinhardt, Phys. Rev. B 55 (1997) 3520–3532. [13] D. Joseph, S. Ritsch, C. Beeli, Phys. Rev. B 55 (1997) 8175–8183. [14] P. Kramer, J. Phys. A 32 (1999) 5781–5793. [15] X.Z. Li, D. Shi, K.H. Kuo, Philos. Mag. B 66 (1992) 331–340. [16] X.Z. Li, Acta Cryst. B 51 (1995) 265–270. [17] R. Lück, Private communication, Stuttgart, 1999. [18] S. Ranganathan, K. Ramakrishnan, U.D. Kulkarni, N.K. Mukhopadhyay, Mater. Sci. Eng. A 294–296 (2000) 429–433. [19] C. Richard, M. Höffe, J. Hermisson, M. Baake, J. Phys. A 31 (1998) 6385–6408. [20] S. Ritsch, C. Beeli, H.-U. Nissen, T. Gödecke, M. Scheffer, R. Lück, Philos. Mag. Lett. 74 (1996) 99–106. [21] K. Saitoh, K. Tsuda, M. Tanaka, K. Kaneko, A.P. Tsai, Jpn. J. Appl. Phys. 36 (1997) L1400–L1402. [22] K. Saitoh, K. Tsuda, M. Tanaka, J. Phys. Soc. Jpn. 67 (1998) 2578– 2581. [23] P.J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A.P. Tsai, Nature 396 (1998) 55–57. [24] R. Wittmann, Z. Krist. 214 (1999) 501–505. [25] J.S. Wu, K.H. Kuo, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1998, pp. 215–218. [26] J.S. Wu, S.P. Ge, K.H. Kuo, Philos. Mag. A 79 (1999) 1787–1803. [27] Y. Yan, S.J. Pennycook, A.P. Tsai, Phys. Rev. Lett. 81 (1998) 5145– 5148.
A cluster approach to random Penrose tilings Petra Gummelt∗ , Christoph Bandt Institut für Mathematik und Informatik, Arndt-Universität Greifswald, D-17487 Greifswald, Germany Received 1 September 1999; accepted 2 November 1999
Abstract An experimentally relevant subclass of the full random Penrose pentagon tiling ensemble is generated using a single cluster. Local matching rules based on cluster overlaps are equivalent to relatively weak axioms on minimal cluster distances. Energy and entropy arguments and recent results on constituent atomic clusters in decagonal quasicrystals are discussed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystals; Cluster models; Penrose tiling; Random tiling
1. Introduction Many recent experimental observations indicate typical atom clusters determining the structure of quasicrystalline phases. These clusters are nearly congruent copies of a single constituent prototype. Thus there is increased interest to describe quasiperiodic ordered structures with crystallographically forbidden symmetry in terms of a single building block. In the covering approach, relevant classes of quasiperiodic tilings are shown to be fully covered by a single covering patch of tiles. This property is satisfied for several examples of 8, 10 and 12-fold patterns [4,6,7,14,17] and is also discussed for icosahedral tilings [14,18]. The cluster- or ‘quasi-unit cell’ approach [12,23] is concerned with the generation of coverings by a single cluster. Analogous to the ‘puzzle principle’ of tilings, cluster matching rules (overlap rules) are defined. Up to now, decagonal Penrose tilings are special in the sense that only a decagon with aperiodic markings was proven to force perfectly ordered structures of Penrose type [9,10]. Although decagonally shaped atomic clusters have been observed in many decagonal quasicrystals, most of these phases have randomly ordered quasiperiodic layers [13]. Currently only d-Al–Ni–Co [20–23] and d-Zn–Mn–rare-earth [1,2] are regarded as nearly perfect decagonal structures. In particular, d-Al70 Ni19 Co11 and d-Al72 Ni20 Co8 are very close to a stacking of coverings by decagons whose centres define ideal Penrose patterns ∗ Corresponding author. Tel.: +49-3834-864637. E-mail address: [email protected] (P. Gummelt).
[20–23]. However, recent studies show that a replacement of geometrically defined aperiodic matching rules by a clever atomic decoration of the cluster does not work without a little contribution of disorder [3,5,21–24,27]. Thus, it seems useful to combine the random tiling model with the cluster approach. In the special case of Penrose-like structures, we relax the aperiodic decagon rules in order to create periodic approximants, random quasiperiodic as well as perfect quasicrystalline patterns by the same unit. An interpretation of energetically or entropically dominated stability mechanisms can be given for coverings of a single building block in a similar way as for tilings. In an idealised picture, perfect patterns describe energetically stabilised quasicrystals, assuming that energetical effects reflected by the tiling matching rules form the main component of free energy. Random tilings are related to entropically stabilised phases with energetically indistinguishable configurations at high temperatures [19].
2. Random decagon coverings Suppose, we have a set of prototiles P1 , . . . , Pk for some class of truly quasiperiodic tilings — like Penrose rhombs, or kite and dart. These are shapes together with certain arrows, vertex stars, or other decorations which force quasiperiodicity. If we remove the markings and require only edge-to-edge matching rules, we obtain a much larger class of tilings which is called the corresponding random tiling ensemble. The prototiles without markings are denoted P1? , . . . , Pk? . Here we want to do the same for the aperiodic regular decagon P introduced in [9,10]. In order to determine P ? ,
0921-5093/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 1 9 7 - 7
P. Gummelt, C. Bandt / Materials Science and Engineering 294–296 (2000) 250–253
251
Fig. 1. Aperiodic decagon P (subsets of type A and B and rocket-like decoration).
we have to consider the decoration of P in the covering context. The role of edges is replaced by six subsets of P with nonempty interior. They are congruent to A- or B-shaped sets and their symmetries are restricted by two rocket-like dark markings and a dark half-star (Fig. 1). In Fig. 2, the structure of the four “oriented quasi-edges” of type A (dark grey) and the two remaining ones of type B (light grey) is highlighted (this is not an equivalent aperiodic decoration!). Following the tiling picture, we now erase the arrows of all generalized edges of P . This results in the “unoriented” version P ? . To cover the plane by copies of P ? , we modify the overlapping rule [9,10] of the aperiodic decagon P . Neighbours are now assembled according to the “undirected quasi-edges” of P ? . Generation rule. Every copy of P ? has to be surrounded by neighbours which intersect P ? in either A1 , A2 , A3 , A4 , B1 or B2 (see Fig. 1) such that the boundary of P ? is covered by neighbour tiles. Proposition 1. Up to isometry, there are only five prototypes N1 , . . . , N5 of nearest-neighbour configurations of a tile P ? which fulfil the generation rule. Except for the orientation of the decagons, these configurations are congruent to the A–B neighbourhoods [9,10] of the aperiodic decagon P (positions of potential overlaps are not changed). The neighbourhoods N1 up to N5 are shown in Fig. 3. At the top, a pentagonal arrangement of decagons is fixed. Black dots represent the centres of P ? . Long edges
Fig. 2. Directed “quasi-edges” of P (left) and structure of the generalized prototile P ? (right).
Fig. 3. Nearest-neighbour configurations of P ? .
correspond to type-A overlaps, short edges connect decagons intersecting in a B-set. Proposition 1 is due to the small number of possible positions of a neighbour of P ? with A- or B-overlap (see Fig. 4). While the neighbours in Figs. 4a–d satisfy the aperiodic covering rules of P , new positions of neighbours are shown in Figs. 4e–h. However in cases (e) and (f), any further covering of the neighbour will contradict the generation rule. Moreover, the B-overlap of case (g) does not lead to new tilings. Only the remaining configuration (h) of two 180◦ -rotated decagons is responsible for an extension to random structures.
3. Hexagon-boat-star (HBS)-tilings The class of coverings generated by the aperiodic decagon cluster P is in one-to-one correspondence to perfect Penrose tilings. Moreover, it was shown in [10] that the pattern of decagon centres defines an ideal tiling consisting of the Penrose pentagon tiles (marked pentagons, star, ship, rhombus, cf. [8, p. 531]). One could expect that the centres of our modified decagon P ? determine the full random tiling ensemble generated by pentagon tiles without markings.
Fig. 4. Possible A- and B-neighbours of P ? .
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P. Gummelt, C. Bandt / Materials Science and Engineering 294–296 (2000) 250–253
Fig. 5. Hexagon, boat, and star supertiles (grey).
Fig. 7. Forbidden linkages in aperiodic HBS-tilings.
However, we only obtain a special subclass, the so-called hexagon-boat-star (HBS)-tilings [16].
pattern of hexagon supertiles drawn in Fig. 6 illustrates the orthorhombic Al3 Mn phase [15,16], other approximants are reviewed in [26]. Various quasiperiodic HBS-tilings explaining the structure of decagonal Al–Mn, Ga–Mn and highly perfect Al–Ni–Co are derived in [11,16,21,25]. HBS-tilings are also interesting from a theoretical point of view. The number of different r-patches (consisting of translational P ? copies) grows exponentially for every radius r > 0, thus the configurational entropy of the system is positive. Perfect Penrose pentagon tilings (PPT) form a proper subset of HBS-tilings which are a subclass of the random Penrose pentagon (RPP) ensemble. A simple RPP-pattern which is not contained in the HBS-subclass is the periodic rhombus lattice. To reduce RPP-tilings to the HBS-subclass, star, ships and rhombus tiles have to be completely surrounded by pentagons, as stated in Proposition 2. Ideal order of Penrose type is forced for HBS-tilings if two special pentagon configurations are banned (Fig. 7). These forbidden patches consist of an N5 -neighbourhood and a 180◦ -rotated copy either of N5 or N2 . We conjecture that minimization of these “bad linkages” leads to (nearly) perfect quasiperiodic structures. This claim is supported by the analysis of several examples used to describe high-resolution electron micrographs of decagonal quasicrystals [16,20–23,25].
Proposition 2. For a covering constructed by the generation rule of P ? , the decagon centres define a tiling where all ship, star and rhombus tiles are surrounded by pentagons and thus define the HBS-supertiles of Fig. 5. All HBS-tilings can be obtained by coverings of P ? in this way. To prove this, we consider all prototypes of P ? coronas starting with our nearest-neighbour configurations N1 , . . . , N5 (Fig. 3). N1 forces a star surrounded by 10 pentagons (supertile S), N2 and N4 form a ship with eight pentagons around (supertile B) and N3 determines six pentagons and a rhombus within (supertile H). N5 appears as neighbourhood type of all decagon centres at the top of a star, ship or rhombus. Fig. 6 shows a simple periodic and a perfect quasiperiodic P ? covering, the corresponding pentagon tiling and the pattern of HBS-supertiles. In the case of ideal Penrose order, the decagon covering implies another HBS-tiling of smaller scale which is directly given by the aperiodic markings (pure rockets determine H-tiles, “half stars” lead to boats and dark stars define S-tiles, see also [24]). HBS-tilings represent an experimentally relevant subclass of the full random Penrose pentagon ensemble. Periodic as well as ideal or nearly perfect quasiperiodic HBS-structures are often used to describe decagonal quasicrystals and closely related approximant phases. For example, the
Remark. For the random tiling ensemble of thick and thin rhombi, maximization of the density of a certain decagonally shaped patch is essentially equivalent to Penrose’s aperiodic matching rules [12]. A main reason for this result is that the rhombic cluster decoration excludes many neighbourhoods which are responsible for periodicity. In the case of HBS-tilings, a simple maximization of P ? decagons or vertices of pentagon tilings, respectively, is not sufficient to force aperiodicity! For example, a pentagon-ship pattern consisting of translates of the boat supertile B has a vertex density higher than ideal PPTs by a factor of 1.021359. Recently, this tiling was used to describe the structure of new orthorhombic approximants of decagonal Ga–Mn and Ga–Fe–Cu–Si quasicrystals [26].
4. Minimal cluster distances
Fig. 6. Periodic and ideal quasiperiodic P ? structures.
Surprisingly, exactly the same class of decagon coverings we have defined by local matching rules can be generated using only some weak axioms on minimal distances of decagon
P. Gummelt, C. Bandt / Materials Science and Engineering 294–296 (2000) 250–253
253
Drawing a decagon of diameter τ l = s + l around all x in Λ, we obtain a P ? covering according to our generation rule. The points x represent the centres of P ? clusters, l- and s-distances correspond to intersections of type A and B. Fig. 8. Nearest-neighbour triangles defined by Axiom 1.
References centres. Information on the specific local decoration of the decagonal cluster is not needed. This approach might lead to a simpler explanation of recent experimental observations where a (nearly) perfect quasiperiodic tiling of nearest neighbours of cluster centres was obtained only by various rather complex decorations of the clusters [5,23,24,27]. Let us consider a discrete plane point set Λ satisfying the Delone property and the following two additional conditions. Axiom 1. The distance d(x, y) of two points x, y ∈ Λ is either s, l or√d(x, y) ≥ s + l, where s and l are in ratio 1:τ , τ = 21 (1 + 5).
Axiom 2. Let y1 , y2 , . . . , yn be the nearest neighbours of x ∈ Λ in clockwise order. Then the angle ∠ yi xyi+1 between successive neighbours of x does not exceed 108◦ . Already the first axiom restricts the prototypes of nearest-neighbour triangles to configurations which are typical for pentagonal and decagonal symmetry (Fig. 8). Triangular arrangements of s-distances contradict a further covering of the plane. However, the lattice of equilateral triangles with edge length l must be excluded. This can be done by forcing the existence of short distances s. Nevertheless, gaps of arbitrary radius as well as strips and islands would still be possible. To avoid this, a kind of covering condition is given by Axiom 2. Under these conditions, the “beetle” of s- and l-neighbours of any point x in our set Λ must be congruent to one of the configurations N1 , N2 , N3 , N4 or N5 of Fig. 3. The proof will be given in a forthcoming paper. Minimal cluster distances of type s and l are physically justified by experimental observations. Density arguments can explain the second axiom.
[1] E. Abe, T.J. Sato, A.P. Tsai, Philos. Mag. Lett. 77 (1998) 205–211. [2] E. Abe, T.J. Sato, A.P. Tsai, Phys. Rev. Lett. 82 (1999) 5269–5272. [3] E. Abe, K. Saitoh, H. Takakura, A.P. Tsai, P.J. Steinhardt, H.-C. Jeong, Phys. Rev. Lett. 84 (2000) 4609–4612. [4] S.I. Ben-Abraham, F. Gähler, Phys. Rev. 60 (1999) 860–864. [5] E. Cockayne, M. Widom, Phys. Rev. Lett. 81 (1998) 598–601. [6] M. Duneau, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1998, pp. 116–119. [7] F. Gähler, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1998, pp. 95–98. [8] B. Grünbaum, G.C. Shephard, Tilings and Patterns, Freeman, New York, 1987. [9] P. Gummelt, Geometriae Dedicata 62 (1996) 1–17. [10] P. Gummelt, Aperiodische Überdeckungen mit einem Clustertyp, Shaker-Verlag, Aachen, 1999. [11] M. Hirabayashi, K. Hiraga, Mater. Sci. Forum 22–24 (1987) 45–54. [12] H.-C. Jeong, P. Steinhardt, Phys. Rev. B 55 (1997) 3520–3532. [13] D. Joseph, S. Ritsch, C. Beeli, Phys. Rev. B 55 (1997) 8175–8183. [14] P. Kramer, J. Phys. A 32 (1999) 5781–5793. [15] X.Z. Li, D. Shi, K.H. Kuo, Philos. Mag. B 66 (1992) 331–340. [16] X.Z. Li, Acta Cryst. B 51 (1995) 265–270. [17] R. Lück, Private communication, Stuttgart, 1999. [18] S. Ranganathan, K. Ramakrishnan, U.D. Kulkarni, N.K. Mukhopadhyay, Mater. Sci. Eng. A 294–296 (2000) 429–433. [19] C. Richard, M. Höffe, J. Hermisson, M. Baake, J. Phys. A 31 (1998) 6385–6408. [20] S. Ritsch, C. Beeli, H.-U. Nissen, T. Gödecke, M. Scheffer, R. Lück, Philos. Mag. Lett. 74 (1996) 99–106. [21] K. Saitoh, K. Tsuda, M. Tanaka, K. Kaneko, A.P. Tsai, Jpn. J. Appl. Phys. 36 (1997) L1400–L1402. [22] K. Saitoh, K. Tsuda, M. Tanaka, J. Phys. Soc. Jpn. 67 (1998) 2578– 2581. [23] P.J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A.P. Tsai, Nature 396 (1998) 55–57. [24] R. Wittmann, Z. Krist. 214 (1999) 501–505. [25] J.S. Wu, K.H. Kuo, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1998, pp. 215–218. [26] J.S. Wu, S.P. Ge, K.H. Kuo, Philos. Mag. A 79 (1999) 1787–1803. [27] Y. Yan, S.J. Pennycook, A.P. Tsai, Phys. Rev. Lett. 81 (1998) 5145– 5148.