Simple physical generation of quasicrystals

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18 August 1997 PHYSICS

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Physics Letters A 233 (1997) 110-l 14

Simple physical generation of quasicrystals C. Janot a, J. Patera b a Institut Lam-Langevin, B.P. 156, F-38042 Grenoble Cedex 9, France b Centre de Recherches Mathe’matiques, Universitt de Montrial, C.P. 6128, succ. Centre-ville, Mont&al. Qukbec, Canada H3C 357 Received 5 May 1997; accepted for publication 28 May 1997 Communicated by J. Flouquet

Abstract Quasiperiodic structures can be grown from a relevant seed by adding individual atoms one by one using a single local rule. These rules result from a frustration of the local point group of symmetries by a condition on the distances between pairs of atoms. This gives, for the first time, a physical growth mechanism which, in addition, fits the structure of real quasicrystals. 0 1997 Elsevier Science B.V.

Quasicrystals (QCs) are a new type of solids [l] which defy previous standard classifications. They are neither periodically ordered like ordinary crystals, nor are they disordered like amorphous solids. However, they have a well defined discrete symmetry group, but one which is incompatible with threedimensional periodic order (e.g. exhibiting five-fold symmetry axes). Instead, they possess a novel kind of aperiodic long range order apparently close to quasiperiodicity [2]. In a periodic crystal, the mass density can be expressed as a Fourier series of density waves, with a set of wavevectors G which define a discrete reciprocal lattice. Each wavevector in the sum is a linear combination of three basis vectors a,? which are linearly independent over the integers. The description in terms of density waves is still valid for quasiperiodic structure. But the number of integer linearly independent basis vectors required to “span” the reciprocal space exceeds the spatial dimension. For example, six basis vectors are re-

quired to span the reciprocal space with three-dimensional structures having icosahedral symmetry. Thus

6=

5I m,ar

i=

(1)

in which the a: can be selected to point along the five-fold axes of an icosahedron. This comes from the fact that any quasiperiodic function can be generated via an irrational cut of a higher-dimensional periodic function 13-51. Indeed, Eq. (1) is nothing else than a projection on the physical space of a six-dimensional periodic lattice. Then, the atomic structure of QCs can be best approached with methods currently applied to regular crystals via the specification of a high-dimensional periodic image of the actual structure [6,7]. For icosahedra1 QCs such periodic images are six-dimensional cubic Bravais lattices (whose six edges project on the five-fold axes of an icosahedron in the physical space) deco-

037%9601/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00420-9

III

C. Janor, J. Patera / Physics Letters A 233 (1997) I 10-I 14

cluster grows, this requirement imposes arbitrary long-range interactions, which is physically implausible. Matching rules, particularly well exemplified with the Penrose tiling [lo], would then seem to offer a potential mitigating factor to these growth problems. The classical edge-matching rules are typically indicated by placing different arrows on the edges of tiles that constrain the way two tiles must match edge-to-edge. Penrose clearly showed that the only plane-filling tiling consistent with the matching rules is a perfect Penrose tiling. Do these edge-matching rules also represent viable local rules for growing a tiling by adding one tile at a time to a random chosen edge? Unfortunately not. Mistakes are made which are not revealed at once and the catastrophe can be appreciated only after many further building steps. Removing and repositioning tiles for another try is obviously a dismal failure of the edge-matching rules as a growth procedure [ 121. Relaxing the matching rules (random tiling models) or replacing edge-matching rules by “forced vertex-matching rules” has certainly smoothed down part of the difficulties but the basic drawbacks remain the same [12]. Recently, Moody and Patera [13] have described a mathematical procedure to grow aperiodic ordered structures via strictly local rules in which a point is added to the growing patch if and only if (i) an ideal configuration is not violated and (ii) the point phase in the physical space remains within a chosen range of values. But, still, local phases are correlated to each other and are not exactly direct space parameters. However, it is possible to keep the spirit of the method and derive a purely local growth procedure that, moreover, seems to be consistent with structure and properties of real quasicrystals. Among the many properties of quasicrystals observed so far [2,8] two of them deserve to be selected for the purpose of the present paper: (i) their struc-

rated by three-dimensional density objects, so-called atomic surfaces (whose cut by the physical space gives point objects where atoms can be sited). Detailed description of these atomic surfaces is difficult, perhaps impossible, to achieve experimentally and, so far, only low resolution structures of QCs have been obtained from diffraction approaches [S]. This is nonetheless very useful for understanding properties of QCs [9]. The main drawbacks of the high-dimensional scheme are twofold: (i) the crude resulting physical structure in the three-dimensional space is concealed into a list of atomic positions without any clear guides on how to design straightforwardly space occupation and, even more disturbing, (ii) there is a total lack of how to grow the whole structure by adding atomic positions one by one or, at the least, cluster by cluster. Alternatively, the growth problem in QCs has been attacked by a variety of space tiling approaches. Actually, growing a piece of matter within certain rules for short and long range order is not an easy task. For regular periodic crystals, the sequence of atoms that exists in a seed cluster repeats exactly again and again; so it appears that the atom to be added must interact only with a small number of atoms at some places on the cluster surface. Moreover, there is a single ground state structure for a given space group which means that the structure is energetically stabilised and can be grown perfectly. The various mathematical procedures that have been used so far to generate quasiperiodic lattices are somewhat suggestive that growing a perfect QC would be a daunting task. The sequence of atoms never exactly repeats, so that atoms added to the surface of a cluster must interact with each atom in the seed cluster to ensure that it sticks at a site consistent with perfect quasiperiodic order. As the

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Fig. I. (a) A decagonal cluster of sites defining a ten-fold star of vectors. (b) A second decagon of sites has been added to the one shown in (a), with its centre on a vertex site of the decagon (a). (c) Same as in (b) except that the sites overly close to those of (a) have been removed.

C. Jatwt, J. Patera / Physics Letters A 233 11997) 110-I 14

112

Fig. 2. Aperiodic structure grown by computer using a ten-fold-star and shortest distances between sites being above 0.2 radius of the initial decagon

(growth

step).

ture appears as basically resulting from a “covering” of space by rigid atomic clusters [7,9,14] with “forbidden” symmetries and (ii) their shear modulus [151 is as large as those obtained with semiconductors, thus revealing strongly directional atomic bondings. One very simple way to preserve what can be preserved of that while growing the structure is to proceed as follows: (i) A “star” of atomic bonding is deduced from a given cluster of atoms. In the two-dimensional example of a decagonal centred cluster (Fig. la), the “star” is made of the ten radial vectors linking centre to vertices and placed at 27r/lO angles from each other (cluster requisite). .. .. . .. . .. .. . .. .. . .. .. . .. . . . . . ..*....*........ . . . . . . . . . . . . ..e. . _. . . . . . .* . . . . . . . . . . . . . . . . . . . . . . . . . . . :.::*.*.::~:. :, .: . . , . . . . . . . . . .* . . . . . . . . . . . . . . . . . . * . . . . . . .. . .. . .. .. . .. . .. .. . .. , .. *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*......a... :. .:.. :.:. .:.:. .:.::.:. . . . . . . . . . 1. . . . . . . . . . . . . . . . _. . . . . . . . . . . . . .. . .. .. 1 .. + .. .. . .. . .. . ..~.~..~.~*.‘.*~.~..~.**.~..~.~..-.~~. . . . . . . ... ..* . ... ... . ,.. . . . . . . .:,..:.. . *:.. . .:. *:.:. .: . . . . . . . . . . . . . , . . . . . . . 1. . . . . . . . . . . . . . . +.. :.. . . . . . . . . . . . . . . . . .:. . ::*:. ::.:.::.:. ::.::*:: . . . . . . . . . . . . . . . . . . . . . . . * . . . :..*.. . :.. . . . . . . . . . . . .:.. . .::. . . . . . . . . . . . . . . . . . . . :. :.::.::~: .. . .. . . .. .. . . .. . .. .. . .. (4

(ii) The above “star” of vectors defines the only possible translations, originating from an existing site at the surface of the growing structure, to create new sites directed at n. 27r/lO angles of the already existing radial bondings (directional bonding requisite). (iii) New sites that would introduce too short distances between pairs of atoms, with respect to the already existing sites, are rejected (finite density requisite). The point is illustrated with Fig. lb and lc. In this example, the threshold pair distance has been set to the length of the decagonal edges. Fig. 2 shows a piece of one structure that can be iteratively generated by pursuing again and again the above addition procedure with no constraints on the random choice of the surface sites. That this procedure is easily feasible and that it allows structure growth via purely single local rules is thus obvious. At least two remaining basic questions must be carefully addressed: 6) does this mechanism generate a single state structure or, conversely, a variety of “energetically” equivalent structures and (ii) does it result into quasiperiodicity? If combinations of two-, three-, four- and six-fold stars of vectors are used in the growing sequence, one trivially gets lattices of sites which are periodic crystal lattices and it has been well known for almost a century that one given star generates one and only one lattice. With pentagonal, decagonal, icosahedral . . . stars it is no longer possible to generate a@00 0 OS3(3 se @I30 00 00 0 00 BO E)OQ OQQQQOQ QQQQQQQQQQQQQQQQQQQ?bQQ*“dQQQ$ Q ‘$

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C. Jmwt, J. Patera / Physics Letters A 233 (1997) 110-I 14

lattices and fully dense set of sites are obtained instead, if there is no restriction made on distances between pairs. The physical constraint on density via the rejection rule of overly close atoms is then usefully applied. But, clearly, which particular new sites have to be rejected strongly depends on which sites are already there, which in turn depends on the order chosen to explore the surface sites of the growing structure. Thus, this gives rise to an infinite number of very slightly different structures (Fig. 3). If one decides to circle the seed always in, say, a step by step clockwise exploration of the surface sites, this generates a more regular structure of the family with an overall symmetry axis in its centre. The class of structures obtained here is very reminiscent of what results from a random tiling procedure except that we are really packing atoms (not tiles) with a constantly preserved density and without the burden to explore the whole surface before deciding to place a new site. The question of whether the obtained structures are periodic, quasiperiodic or simply ordered aperiodically has not been completely answered at this stage. Of course, the trivial requirement is that vector stars that generate crystal lattices must be strictly avoided. This being said, the geometrical scheme described in the present paper is more related to the hyperspace description than it would seem at first sight [ 16,171. Indeed, specifying the high-dimensional Bravais lattice by its translation vectors corresponds to the selection of a star of vectors in the . . . . . . . . . . . . . . . .* . . . . . . . .

. . . . *. : *. * : . *. * * . - . . * . * *. *. : * . * : . * . * *. . . . . . . . . 6. . . . . . . . . . . . . . . . . *. ::. :. ::.::~:. ::. :. ::.: . . . ..*.* * .*.*.. .._._.. .**.** ._.*.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . : . - . : * . * . . * . - - . - . . - . * . . * . - * . * . . *. - : . * . : . . . . . . . . . . . . . .. . . . , . . . . . . . .-.:.. -. :.:. :* . * .:.. :.:. .:.. *. . .f. . . . . . . . . . . . . . . _.. . . . . . . . .*. . . . . . . . . . . . . . . . . . . . . . . . ..+.. .. . .. . .. .. . . . . * . . *. * . . - . : - . *. . * . * : . * . . *. * . . * . * * . .*. . .*. . . . .* . . . . . . .*. . . . .*I . . . . . . . . . . ._. . . *. . . . . . . . . . . . . . . . . . . :. . . . . .*. . . *.. . . . . . . . . . . . . . . . . . . . I , . . . . . . . .*. . . *. . .. . . . . . . . . . . . . . . . . . . . . .. . . :. ::. ::. :. ::. :.::. ::. :. . .. .I... ... ..... .... .... ..... ... ..... .. . . . . . .. . . . . . . . . . . . . . . . . . . . - . . *. - : . - . : - . *. . * . * * . * . . - . * : . - . : ’ . * . . * .. .. . .. . .. .. . .. .. . .. . .. ..

three-dimensional physical space. Adding to this hyperlattice some atomic surfaces is equivalent to define acceptance domains for permitted atomic pair distances and directions. But this is certainly not the case since quasiperiodicity definitely cannot be achieved adding individual atoms (or clusters) one by one using a single local rule; to assess the position of a new atom, information about the positions of all other atoms already there is needed. Then, we can at best expect to have generated long range ordered aperiodic structures, close to quasiperiodicity but containing defects. As a positive illustration of this statement, Figs. 4a and 4b present an interesting comparison between a structure obtained via the “star-short distance” scheme @SDS) and a five-fold planar cut of an atomic arrangement deduced from diffraction data with an AlPdMn real quasicrystal via the hyperspace method and within spherical approximations for the atomic surfaces [7]. This is strong support to the double suitability of the procedure to generate structures that can be aperiodically ordered at long distance and even describe real quasicrystal quite nicely. Even more interesting perhaps is the lift of the structures grown by the present method into the associated four-dimensional space followed by their projection into an acceptance window (the “star map” of Refs. [17,18]), see Fig. 5. The projected set of points is bounded and its size grows slowly with the increasing number of points of the structure. Thus our growth rule does not eliminate the phason defects but tends to mend possible tears rapidly and . . *. . 06 0 . . ..o~..q..o~...o*..po...e.o~~o. a0 .O.*

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C. Jatwt. J. Patera / Physics Letters A 233 (1997) 110-l I4

ation as so efficiently analysed in the one-dimensional case by Quemerais [ 181. But we cannot refrain from thinking that, finally, a real quasicrystal may be less difficult to describe than suggested so far and, also, that they can be grown via the simplest mechanism, i.e. adding one atom or a small cluster at a time. The work was partially supported by the National Science and Engineering Research Council of Canada and by the FCAR of Quebec.

References Fig. 5. The structure of Fig. 3a mapped into the acceptance window. More precisely, the star map [17,18] of all 1081 computed points of the structure is shown while Fig. 3a contains only the points of the structure visible through a rectangular window.

produces well ordered structures. Some of these structures can be simply degenerated into large patches of ordinary crystal, possibly assembled into a twinned textures. This occurs when too large a distance is selected for the nearest pairs, with respect to the star size. We are currently working on this particular point within a more rigorous approach of the requirement for the method to generate fully aperiodic structures. We are also systematically investigating how the resulting classes of structures are influenced by the choice of the star and, for a given star, by the ranges of the selected short distances. We have observed that with a decagonal star, one jumps from one class of structure to another each time the close pair distance become smaller than the successive powers of I/T. Differences between structures grown, with the same star and short distance, resulting from the order, in which the star points are added, are being analysed in terms of equivalence with the phason defects modifications of the hyperspace scheme. Finally, extension of the method to three-dimensional QCs with chemical order constraints is formally straightforward and is in progress. These will be subjects for a forthcoming longer paper. At this stage, one weak point of the present growth model is certainly to bypass energy consider-

[l] D. Shechtman, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [2] C. Janot, Quasicrystals: a primer (Oxford Univ. Press, Oxford, 1992; 2nd Ed. 1994). [3] P. Bak, Phys. Rev. B 52 (1985) 5764; A. Katz and M. Duneau, Phys. Rev. Lett. 54 (1985) 2688. [4] A. Janner and T. Janssen, Physica A 99 (1979) 47. [5] R. Moody and J. Patera, J. Phys. A 26 (1993) 2829. [6] D.P. Di Vincenzo and P.J. Steinhardt, eds., Quasicrystals: the state of the art (World Scientific, Singapore, 1991). [7] M. Boudard, M. de Boissieu, C. Janot, G. Heger, C. Beeli, H.U. Nissen, H. Vincent, R. Ibberson, M. Audier and J.M. Dubois, J. Phys. Condens. Matter 4 (1992) 10149. [8] F. Hippert and D. Gratias, teds.), Lectures on quasicrystals (Les Editions de Physique, France, 1994). [9] C. Janot and M. de Boissieu, Phys. Rev. Lett. 72 (1994) 1674; C. Janot, Phys. Rev. B 53 (1996) 181; J. Phys. Condens. Matter, to be published. [lo] R. Penrose, in: Introduction to the mathematics of quasicrystals, ed. M. Jaric (Academic Press, New York, 1989) p. 53. [ll] M. Gardner, Sci. Am. 236 (1977) 110. [12] G.Y. Gnoda, P.J. Steinhardt, D.P. di Vicenzo and J.E.S. Socolar, Phys. Rev. Lett. 60 (1988) 2653. [13] R.V. Moody and J. Patera, Len. Math. Phys. 36 (1996) 291. (141 M. de Boissieu, M. Boudard, R. Bellissent, M. Quilichini, B. Hennion, R. Currat, AI. Goldman and C. Janot, J. Phys. Condens. Matter 5 (1993) 4945; 7 (1996) 7299. [15] K. Tanaka, Y. Mitarai and M. Koiwa, Phil. Mag. A 73 (1996) 1715. [ 161 L. Chen, R. Moody and J. Patera, Noncrystallografic root systems, in: Quasicrystals and discrete geometry, Fields Institute Monograph Series (Amer. Math. Sot., 1997). [17] J. Patera, in hoc. of the NATO ASI, Mathematics of aperiodic order, ed. R. Moody (Kluwer, Dordrecht. 1997). pp. 443-465. [ 181 P. Quemerais, J. Phys. (France) 4 ( 1994) 1669.