RESEARCH PROBLEM Some conjectures on quasicrystals having fivefold

symmetry

Tibor Ódor Math. Inst. of Hung. Acad. of Sci. odor@math-inst.hu Baiyu Zhang Lab. für Kristall., ETH-Zentrum baiyu.zhang@kristall.erdw.eth.ch

November 13, 2000

Abstract

Some simple constructions for quasicrystals using the Penrose lattice (the vertice set of Penrose tiling) in the plane having fivefold symmetry are given. We pose conjectures which “essentially” state that they exhaust all the possible one and they are generalized for higher dimensions. One can regard the presented conjectures as generalization of Biebarbach’s theorem for quasicrystals not only orbits of discrete isometry groups.

1 Introduction

The discovery of solids whose X-ray diffraction pattern exhibiting symmetries forbidden by classical crystals (invariant under discrete space groups with compact fundamental domains) was first reported by [26] in Al − Mn and Al − Mn − Si alloys. These findings grasped the imagination of several physicists, chemists and mathematicians and became a strongly developing area of crystallography.

In studing of quasicrystals many different branches of mathematics and physics were involved, such as the investigation of the Penrose tiling, almost periodic functions (measures) and Fourier analysis.

Although the fundamentals of almost periodicity for different class of functions were well known from the papers of Harald Bohr and others, the work of Hao Wang, Raphael Robinson, Robert Berger, Roger Penrose, Robert Amman, N. G. de Bruijn etc on non-periodic tilings became an important tool to understand the atomic structure of quasicrystals.

Crystallographers realized the connections with Penrose tilings—as a model— through Martin Gardner’s very enjoyable article [13], which was written for the public, not to mathematicians. This paper had some influence even on N. G. de Bruijn [2], whose work was probably the most important at the beginnig from mathematical point of view.

The International Union of Crystallography has redefined (tentatively) the term crystal to mean “any solid having an essentially discrete diffraction diagram”. They also defined quasicrystals as “crystals with diffraction pattern having crystallographically forbidden symmetries”.

As far as the authors know, there is no agreement in the precise mathematical model of crystals and quasicrystals. To fulfill our needs we have to give our own definitions.

The aim of this paper is to pose a conjecture, which characterize quasicrystals by the discreteness of their Fourier transform and essentially to show that under certain symmetry assumptions only certain subsets of finitely many different Delone sets constructed by the cut and project method [37] can have discrete spectra.

This conjecture can be regarded as a generalization of Biberbach’s theorem (in every dimension n ∈ N up to conjugacy there are only finitely many different discrete subgroups of the isometry group of R^{n}) [51].

There were several attempt to prove Biberbach type theorems for quasicrystals in the Fourier space by D. Mermin and his school [33], [46] and by M. Baake [30], [31] using local deriveability, and other concepts, but, as far as the authors know, without success.

2 Quasicrystals

Let the term crystal in this paper be a Delone set (or, in an other language of discrete geometry, a (r, R) point set)—that is a discrete point set D ⊂ R^{n} for which any ball with radius R contains at least one point and the distance of any two different points is greater than (a fixed) r > 0—, which Fourier transform is discrete in the de Bruijn sense [1], [3], that is a de Bruijn type distribution which is a linear combination of countable many Dirac deltas (in this context usually called Dirac combs) concentrating in the points λ ∈ Λ. That is

D̃(t) = ∑_{x∈D} exp(i⟨x, t⟩) = ∑_{λ∈Λ} a_{λ}δ_{λ}(t),

where D is a Delone set,˜ is the Fourier transform on the de Bruijn distributions S′, a_{λ} ∈ C, a_{λ} ≠ 0 and Λ = Λ(D) is countable. (We call this set Λ the discrete support of D̃.)

Unfortunately, D̃ is not a measure (that is a continuous functional on continuous functions having compact support), unless the crystal is a locally finite periodic Delone set [38].

In X-ray crystallography we are not able measure D̃, but its “square absolute value”. In our case, for distributions it is not meaningful, but we can define it as a symbolic sum given by the following formula:

|D̃|^{2} = ∑_{λ∈Λ} |a_{λ}|^{2}δ_{λ}.

For Penrose lattice and other Delone sets constructed by the cut and project method using “nice” windows it is known [38] that |D̃|^{2} is a locally summable measure. In the next part of the paper we always assume this.

A finite subgroup G of O(n)—the group of isometries leaving invariant the origin O—is regarded as nondegenerate, if only the origin O is invariant with respect to G. In this paper G-quasicrystals are those crystals D, for which |D̃|^{2} is G-invariant with respect to a nondegenerate G ≤ O(n). (Observe, that we did not assume that G is crystallographically forbidden.) We say that D is quasicrystal if it is G-quasicrystal for a nondegenerate G.

A Delone set D is taken as repetitive if it is a Delone set for every finite substructure, that is for every finite set D ⊂ D there exists a real number R = R(D) that any ball with radius R contains a congruent copy of D. A Delone set D̄ is a refinement of the Delone set D if D ⊂ D̄.

Conjecture 1 If D ⊂ R^{2} is a repetitive crystal and the measure |D̃|^{2} has fivefold symmetry then it can be covered by finitely many translated copies of a crystal similar to the Penrose lattice P.

For generalized Penrose lattices constructed by the cut and project method [2], [4], [9], [19] it is easy to prove our conjecture.

Let P_{0} ⊃ … P_{i} ⊃ P_{i+1} … be (different) Penrose lattices. Then it is easy to see that

Q = ∑_{i=0}^{∞} ∑_{p_{i}∈P_{i}}(−1)^{i}δ_{p_{i}}

is well defined (because the above sum is locally finite), it is Delone set and |Q̃|^{2} has fivefold symmetry. Crystals constructed by the above method we call Penrose like. It is clear that they are not necessarily repetitive in general. (But they can be, even if P_{i} ≠ ∅ for every i ∈ N. An example for this when there are “huge gaps” between P_{i} and P_{i+1} for every i ∈ N.)

On the other hand Penrose like sets have refinements which are similar to the Penrose lattice. It is clear that if P_{i} = ∅ for i ≥ j then Q is repetitive.

The following conjecture may characterize the Penrose like quasicrystals.

Conjecture 2 Assume that Q ⊂ R^{2} is a quasicrystal admitting fivefold symmetry, and Λ(Q) ⊂ Λ(P). Then Q is a finite union of similar copies of Penrose like quasicrystals.

Our conjectures suggest that being a quasicrystal is a much stronger assumption than being a crystal. In some sense they show that “essentially” only the Penrose lattice can have diffraction pattern having fivefold symmetry.

It would be interesting to generalize our conjectures for three or higher dimensions or for rank 1 Riemannian symmetric spaces of non compact type. (But for the latter, if the space is not Euclidean, we cannot expect Biberbach type theorem, because there is no such even for orbits of groups acting discretely and properly discontinuously admitting compact fundamantal domain.) We say that the Delone set D is irreducible if from D = D_{1} ×D_{2} follows that D_{1} = D or D_{2} = D, where × is the Cartesian product of sets, and a crystal D is strongly irreducible if there is no crystal refinement D̄ of D which is not irreducible.

The finite family of quasicrystals {Q_{1}, …, Q_{k}} is said to be complete, if any strongly irreducible repetitive quasicrystal D has a refinement which is periodic or can be covered by finitely many translated copy of a crystal similar to Q_{i} for a certain fixed i ∈ N.

Conjecture 3 For any dimension n ≥ 3 there exists a finite complete family of quasicrystals.

We note here that if we do not assume that G in nondegenerate then there are simple counterexamples to Conjecture 3. (For example in dimension three the direct product of the Penrose lattice and an arbitrary one dimensional O symmetric Fibonacci set. This gives us continuum many different classes Q_{ω} for ω ∈ S^{1}.)

From this conjecture follows the next one, which would be interesting to prove independently.

Conjecture 4 Refinement is an equivalence relation on quasicrystals.

Refinement is almost obviously an equivalence relation on quasicrystals (with fixed symemtry group G) constructed by the cut and project method.

In the plane the situation is quite different and it would be interesting to prove the following conjecture:

Conjecture 5 For any integer m ≥ 2 there exist a finite family of repetitive quasicrystals Q_{m} = {Q_{1}^{m}, …, Q_{k}^{m}} (with diffraction patterns) having m-fold symmetry that any quasicrystal having m-fold symmetry has a refinement which is periodic or it can be covered by finitely many translated copies of a crystal similar to Q_{i} for a certain i ∈ N.

It is not known that for which m holds that k = 1. This question is naturally the most challenging for m = 5.

Our previous two conjectures can be regarded as the generalization of Biberbach theorem for quasicrystals. (For rank 1 Riemannian symmetric spaces of non compact type, if they are non Euclidean we can have only 5 type of characterization, if we have any.)

D. Mermin has shown example for an m ∈ N that in the Fourier space there are infinitely many different Z-module admitting the same point group C_{m}. But they technique does not (and not able to) take into account the geometric structure of the Delone set D [46]. But this approach give essential information only in the Fourier space and unfortunately, it is not able to connect this information directly to the geometric structure of the Delone set D.

Note 1 If we do not assume that D is locally finite, then Conjecture 1 and 3 is false as the following informal sketch shows. Unfortunately, the details, because of the heavy usage of the Helgason-Fourier transform on hyperbolic spaces is beyond the scope of this paper.

Let H^{n+k} be the (n+k) dimensional (real) hyperbolic space, let H^{n} be an n dimensional horosphere and Γ be a Fuchsian subgroup of the isometry group of H^{n+k} with compact fundamental domain. As it is well known, the restriction of the metric of H^{n+k} on H^{n} is isometric to the standard metric of R^{n}.

Assume that k ≥ 2 and Γ and H^{n} are in general position with respect to eachother, which means that for any x ∈ H^{n+k} its orbit Γx with respect to Γ has at most one intersection point with every parallel copy of H^{n}. Let H_{ϵ}^{n} be an ϵ neighborhood of H^{n}, and the Delone set D is defined as the orthogonal projection of points of H_{ϵ}^{n} ∩ Γx for a fixed x ∈ H^{n+k}.

Similar to the classical Euclidean caseone can easily show that the Helgason-Fourier transform [15], (in the de Bruijn sense [4]) of D is discrete and if Γ and H^{n} are “well positioned” with respect to eachother, then it admits non trivial symmetry. On the other hand in general D is obviously not repetitive.

We can regard the previuous construction as an extension of the cut and project method to hyperbolic spaces. If we apply it to n planes in hyperbolic geometry insted of n horospheres we can construct quasicrystals in hyperbolic geometry.

One has to note here that if G/K is a Riemannian symmetric space and Γ\G/K is non compact, but has finite volume, then the Helgason-Fourier transform of D (in the de Bruijn sense) always has non discrete part. So we cannot generalize the previuous construction for this type of Γ’s or we have to allow the continuous part of the spectra, which is still compatible with the— intentionally—not precise definition of quasicrystal given by the ICU.

3 Why we think that the above conjectures are true?

The reader may can ask, why we think that the above conjectures are true? Well. Naturally the best argument would be a proof. But we try to collect some argument.

First of all, because there are not known counterexamples, although there were a thorough search for them among existing constructions of quasicrystals and we consulted a lot of experts. For quasi crystals which is a subset of a set constructed by the cut and project method the statement is true. No locally finite quasicrystal (or even crystal) is known for us, which is not a subset of a crystal, constructed by the cut and project method.

Second, because we think that the concept of refinement is very robust. Lot of different construction with substantially different Z-modules may admit the same refinement.

4 Acknowledgements

The first author would like to thank (in random order) the valuable discussion and information to Robert V. Moody, Michael Baake, Majorie Senechal, Ron Lifshitz, Ludwig Danzer, Peter Schmitt, Nikolai Dolbilin, Le Tu Quoc Thang, Jacek Miekisz, Gábor Gévay, Bert Hof and Toshiyuki Kobayashi. The authors would like to thank to the unknown referee the valuable suggestion and information which substantially improved the paper. Some details of 1 were discussed by Toshiyuki Kobayashi.

References

[1] de Bruijn, N. G., A theory of generalized function, with applications to Wigner distribution and Weyl correspondence, Nieuw Archief voor Wiskunke, 3 (XXI), (1973), 205–280.

[2] de Bruijn, N. G., Algebraic theory of Penroses’s non-periodic tilings of the plane, Proc. Kon. Nederl. Akad. Wetensch., A84, (=Indagationes Mathematicae, 43), (1981), 38–52 and 53–66.

[3] de Bruijn, N. G., Dualization of multigrids, In: Proceedings of the International Workshop Aperiodic Crystals, Les Houches 1986, Journal de Physique, 47, Colloque C3, Suppl. Colloq. C3, 9–18, July 1986.

[4] de Bruijn, N. G., Quasicristals and their Fourier transform, Proc. Kon. Nederl. Akad. Wetensch., A 89, (=Indagationes Mathematicae, 48), (1986), 123–152.

[5] de Bruijn, N. G., A riffle-shuffle card trick and its realtion to quasicrystal theory, Nieuw Archif voor Winskunde, 4(5), 285–301.

[6] de Bruijn, N. G., Modulated quasicrystals, Proc. Kon. Nederl. Akad. Wetensch., A90, (=Indagationes Mathematicae, 49), (1987), 121–132.

[7] de Bruijn, N. G., Updown generation of Betti sequences, Proc. Kon. Nederl. Akad. Wetensch., A92, (=Indagationes Mathematicae, 51), (1989), 385– 407.

[8] de Bruijn, N. G., Remarks on Beenker patterns, (peprint), (1989).

[9] de Bruijn, N. G., Updown generation of Penrose tilings, Indagationes Mathematicae, New Series, 1(2), (1989), 201–219.

[10] de Bruijn, N. G., Penrose patterns are almost entirely determined by two points, Discrete Math., 106/107, 97–104.

[11] de Bruijn, N. G., Remarks on Penrose tilings, In: The mathematics of Paul Erdős, Springer-Verlag, 1994.

[12] de Bruijn, N. G.—Senechal, M., On the uniform distribution of the projection of a lattice onto a subspace, (manuscript).

[13] Gardner, M., Mathematical Recreations, Scientific American, 236, (January 1977), 110-121.

[14] Gardner, M., Penrose Tiles to Trapdoor Ciphers, W.H.Freeman, New York, 1989.

[15] Helgason, S., Groups and Geometric Analysis, Academic Press, 1984.

[16] Penrose, R., The Emperor’s New Mind, Oxford University Press, New York, 1989.

[17] Stan Wagon, Mathematica in Action, W.H.Freeman, New York, 1991, chapter 4.

[18] Peterson, I., The Mathematical Tourist: Snapshots of Modern Mathematics, chapter 7, W.H.Freeman, New York, 1988.

[19] Senechal, M., Quasicrystals and Geometry, Cambridge University Press, 1995.

[20] Nelson, D. R., Quasicrystals, Scientific American, 255, (August 1986), 42-51.

[21] Nelson D. R.—Halperin B. I., Pentagonal and icosahedral order in rapidly cooled metals, Science, 229 (19 July 1985) 233-238.

[22] Penrose, R., Pentaplexity, Bulletin of the Institute for Mathematics and Applications, 10, (1974), 266–271.

[23] Penrose, R., Tilings and quasicrystals: an non-local growth problem, In: Jaric, M. (ed.), Introduction to Mathematics of Quasicrystals, Academic Press, San Diego, 1989.

[24] Harburn, G.—Taylor C.A.—Welberry T.R., Atlas of Optical Transforms, G. Bell & Sons, Ltd., London, 1975.

[25] Senechal, M.—Taylor, J., Quasicrystals: The View from Les Houches, The Mathematical Intelligencer, 12, No. 2, (1990).

[26] Schechtman D.—Blech, I.—Gratias, D.—Cahn, J., Metallic phase with long range orientational order and no translational symmetry, Physical Review Letters, 53, (1984), 1951–1954.

[27] Kenyon R., Self-Similar Tilings, Geometry Center Preprint, 21, (1990).

[28] Berman, S.—Moody, R. V., The algebraic theory of quasicrystals with five fold symmetries, J. Phys. A, 27, (1994), no. 1, 115–129.

[29] Hargittai, István (Ed.), Fivefold symmetry, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.

[30] Baake, M.—Schlottmann, M.—Jarvis, P. D., Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A, 24 (1991), no. 19, 4637–4654.

[31] Baake, M.—Schlottmann, M., Geometric aspects of tilings and equivalence concepts, In: Proc. ICQ 5, Avignon, 1995.

[32] Peng, Zhi Zhong, Deduction of quasilattice with five fold symmetry and particle fractal structure model, Sci. China Ser. B, 32 (1989), no. 2, 215– 226.

[33] Mermin, N. David—Rokhsar, Daniel S.—Wright, David C., Beware of 46 fold symmetry: the classification of two dimensional quasicrystallographic lattices, Phys. Rev. Lett., 58, (1987), no. 20, 2099–2101.

[34] Knupfer, A.—Kramer, P., On the relation between decagonal and icosahedral quasicrystals, Modern Phys. Lett. B, 5, (1991), no. 22, 1543–15.

[35] Hargittai, István (Ed.), Quasicrystals, networks, and molecules of fivefold symmetry, VCH Verlagsgesellschaft mbH, Weinheim, 1990.

[36] Kumar, Vijay—Sahoo, Debendranath—Athithan, G., Characterization and decoration of the two dimensional Penrose lattice, Phys. Rev. B (3), 34, (1986), no. 10, 6924–6932.

[37] R.V. Moody, Meyer sets and their duals, (toappear), 1996.

[38] Ódor, T., Characterization of lattices by their Fourier transform, (submitted).

[39] Lamadrid, Jesús Gil—Argabright, Loren N., Almost Periodic Measures, Memoirs of the AMS, Nr. 428, AMS, Providence, Rhode Island, USA, 1990.

[40] Bohr, Harald, Zur Theorie der fast periodischen Funktionen. I. Eine Verallgameinarung der Fourierreichen, Acta Math., 45, (1924), 29–127.

[41] Eberlein, W. F., The point spectrum of a weakly almost periodic function, Mich. Math. J., 3, (1955-56), 137–139.

[42] Bombieri, E.—Taylor J., Quasicrystals, tilings and algebraic number theory: some preliminary connections, Contemporary Mathematics, 64, (1987), 241–264.

[43] Hof, A., Quasicrystals, Aperiodicity and Lattice Systems, PhD. Thesis, Gröningen, The Neatherlands, 1993.

[44] Hof, A., On diffraction by aperiodic structures, Comm. in Math. Phys., (1993).

[45] Gelfand, I. M—Shilov, G. E., Generalized Functions I., Academic Press, 1964.

[46] Lifshitz, R., Personal communication, Crakow, July 1996.

[47] Le Tu, Q. T., Local rules for pentagonal quasicrystals, Discrete and Computational Geometry, (to appear), (1993).

[48] Le Tu, Q. T.—Piunikhin S., Local rules for multidimensional quasicrystals, Differential Geometry and its Applications, (toappear), (1993).

[49] Le Tu, Q. T.—Piunikhin S.—Sadov, V., Local rules for quasiperiodic tilings of quadratic 2-palnes in R^{4}, Comm. in Math. Phys., 150, (1992), 23–44.

[50] Le Tu, Q. T.—Piunikhin S.—Sadov, V., Geometry of quasicrystals, Russian Math. Surveys, (in Russian), 6, (1992), 41–102.

[51] Biberbach, L., Über die Bewegungsgruppen der n-dimensional en Euklidischen Räume mit einem endlichen Fundamental bereich, Göttinger Nachrichten, (1910), 75–84.

[52] Meyer, Y., Algebraic Numbers and Harmonic Analysis, Nort–Holland, Amsterdam, 1972.

[53] Pisot, C., Répatrition (mod 1) des puissances successives des nombres réels, Commentarii Mathematici Helvetici, 19, (1946), 153–160.