A group-theoretic approach of quasicrystals

A Group-Theoretic Approach to Quasicrystals NICOLAE COTFAS Faculty of Physics, University of Bucharest, Bucharest, Romania Abstract A quasicrystal can be regarded as a packing of G-invariant atomic clusters, where the finite group G is the symmetry group of its diffraction pattern. We prove that a large variety of packings of G-invariant clusters can be obtained in an elegant way for any finite group G by using the strip projection method and some results from group theory. 1. Introduction The diffraction pattern of a quasicrystal contains a discrete set of intense Bragg peaks invariant under a finite group G, and the high-resolution electron microscopy suggests that a quasicrystal can be regarded as a packing of (partially occupied) copies of a well-defined G-invariant atomic cluster. A method to generate such a model, based on the existence of some inflation rules, has been presented by Janot and de Boissieu1 . The basic cluster, called pseudo-Mackay icosahedra (PMI), is made of 42 atoms (12 vertices of an icosahedron plus 30 vertices of an icosidodecahedron) and an inner shell of 8 or 9 atoms distributed on sites of a small dodecahedron. The structure is generated recursively by starting from a PMI. At each step √ the pattern is inflated τ 3 = 2 + 5 times and each of its points is replaced by a PMI having the orientation of the starting PMI. In order to fill the gaps between these clusters some ‘connecting units’ must be added and this leads to a complicated enough geometry. These ‘interfaces’ connecting the PMI are pieces of PMI arranged in shells having the same density as PMI, and they also obey the inflation rules of the PMI. Our purpose is to present a method to generate quasiperiodic packings of G-clusters based on the strip projection method and group theory. It represents an extended version of a model proposed by Katz & Duneau2 and independently by Elser3 for the icosahedral quasicrystals. In the case of the Katz-Duneau-Elser model there is an orbit C of the icosahedral group Y (the elements of C are the vertices of a regular icosahedron) such that the arithmetic neighbours of each point x are distributed on the sites of the translation x + C of C. More than that, the vertex set of this model can be regarded as a union of interpenetrating partially occupied translations the cluster C, that is, a packing of icosahedrons. The construction of the Katz-Duneau-Elser model starts from the Y cluster C formed by the vertices of a regular icosahedron. The same construction performed by starting from another orbit of Y, namely, a dodecahedron leads4 to a model in which the arithmetic neighbours of each point are distributed on the sites of a regular dodecahedron. It can be regarded as a packing of dodecahedrons. We prove that the method used in the case of the Katz-Duneau-Elser model works even if the regular icosahedron is replaced by a union of orbits of Y. If we start from the Y -cluster C formed by the vertices of a regular icosahedron and a regular dodecahedron we get a pattern in which the arithmetic neighbours of each point x are distributed on the sites of the translation x + C, that is, a packing of partially occupied copies of the starting cluster. A similar result is obtained if we add a new shell (for example, an icosidodecahedron) or arbitrarily modify the radii of the shells. A very large variety of packings of G-clusters can be obtained in this way5−7 for any finite group G. 2. Packings of G-clusters obtained by projection In this section we review the mathematical bases5−7 of our method. Let {g : IEn −→ IEn | g ∈ G } be an orthogonal IR-irreducible faithful representation of a finite group G in the usual n-dimensional Euclidean space IEn = (IRn , <, >), and let S ⊂ IEn be a finite non-empty set which does not contain the null vector. Any finite union of orbits of G is called a G-cluster. Particularly, C= S r∈S Gr ∪ S r∈S G(−r) = {e1 , e2 , ..., ek , −e1 , −e2 , ..., −ek } (1) is the G-cluster symmetric with respect to the origin generated by S. For each g ∈ G, there exist the numbers sg1 , sg2 , ..., sgk ∈ {−1; 1} and a permutation of the set {1, 2, ..., k} denoted also by g such that, gej = sgg(j) eg(j) for all j ∈ {1, 2, ..., k}. (2) In the case of the Katz-Duneau-Elser model, starting from the representation  a(x, y, z) = 21 x − τ2 y + b(x, y, z) = (−x, −y, z) τ −1 τ 2 z, 2 x + τ −1 2 y 1 τ − 12 z, τ −1 2 x + 2y + 2z  of the icosahedral group Y = 235 =< a, b | a5 = b2 = (ab)3 = e > and the set S = {(1, 0, τ )} one obtains C = {e1 , e2 , ..., e6 , −e1 , −e2 , ..., −e6 }, where e1 = (1, 0, τ ), e2 = (τ, −1, 0), e3 = (τ, 1, 0), e4 = (0, τ, 1), e5 = (−1, 0, τ ), e6 = (0, −τ, 1). For g = a and g = b the relation (2) can be written as a= e1 e2 e3 e4 e5 e6 e1 e3 e4 e5 e6 e2 ! e1 e2 e3 e4 e5 e6 e5 −e2 −e3 e6 e1 e4 b= ! . Theorem 1. The group G ocan be identified with the group of permutations n C −→ C : r 7→ gr g ∈ G and the formula g(x1 , x2 , ..., xk ) = (sg1 xg−1 (1) , sg2 xg−1 (2) , ..., sgk xg−1 (k) ) (3) defines an orthogonal representation of G in IEk . Theorem 2. The subspaces k IEk = { (< r, e1 >, < r, e2 >, ..., < r, ek >) | r ∈ IEn } IE⊥ k = n (x1 , x2 , ..., xk ) ∈ IEk Pk i=1 xi ei =0 (4) o k of IEk are G-invariant, orthogonal, and IEk = IEk ⊕ IE⊥ k. Let u1 = (1, 0, 0, ..., 0), u2 = (0, 1, 0, ..., 0), ..., un = (0, ..., 0, 1) be the canonical basis of IEn , and let ej = (ej1 , ej2 , ..., ejn ), for any j ∈ {1, 2, ..., k}. Theorem 3. The vectors v10 , v20 , ..., vn0 , where vi0 = (e1i , e2i , ..., eki ) form k an orthogonal basis of IEk , and ||v10 || = ||v20 || = ... = ||vn0 ||. k The orthonormal basis of IEk corresponding to {v10 , v20 , ..., vn0 } is formed 0 by the vectors v1 = %v1 , v2 = %v20 , ... vn = %vn0 , where % = 1/||v10 ||. k Theorem 4. The representation of G in IEk is equivalent with the representation of G in IEn , and the isomorphism k Ξ : IEn −→ IEk Ξr = (% < r, e1 >, % < r, e2 >, ..., % < r, ek >) (5) having the property Ξui = vi allows us to identify the two spaces. Theorem 5. The mapping π k : IEk −→ IEk  π k (x1 , ..., xk ) = %2 Pk i=1 < e1 , ei > xi , ... , %2 Pk i=1 < ek , ei > xi  (6) k is the orthogonal projector corresponding to the subspace IEk . Theorem 6. The ZZ-module L = κZZk ⊂ IEk , where κ = 1/%, is G-invariant, P k and in view of the identification Ξ : IEn −→ IEk , we have π k L = ki=1 ZZei . Using the strip projection method we can define a pattern n Q = πkx x ∈ L, π ⊥ x = x − π k x ∈ Ω o (7) for any ‘window’ Ω ⊂ IE⊥ k . The arithmetic neighbours of a point y ∈ Q belong to the set y + C, that is, Q is a union of interpenetrating partially occupied translations of the G-cluster C. View publication stats 3. Self-similarities and rational approximants It is well-known that τ 3 is an inflation factor for the Katz-Duneau-Elser model Q0 , that is, τ 3 Q0 ⊂ Q0 . By using the same method one can prove8 that Q0 admits an infinite number of independent inflation factors and an infinite number of inflation centres. More than that, the same method may allow4,7 us to find certain self-similarities of our packings of G-clusters. Some rational approximants of our patterns can be obtained by using the same technique5 as in the case of the Katz-Duneau-Elser model. Most of the methods used in the description of the physical properties9 of quasicrystals seem also to admit extensions to our models. 4. Conclusions In the case of any finite group G we can obtain, in an explicit way, an infinite number of quasiperiodic patterns by using the infinite number of the corresponding G-clusters. In order to obtain a mathematical model for a real quasicrystal it is sufficient to determine the corresponding symmetry group G and to approximate the local structure by using a G-cluster. The existing10 computer programs for the cut and project method allow one to compare the obtained model with the experimental data. If the agreement is not acceptable one has the possibility to look for a more suitable G-cluster describing the local structure. References 1. Janot, C., de Boissieu, M., Phys. Rev. Lett., 72, 1674-7 (1994). 2. Katz, A., Duneau, M., J. Physique, 47, 181-96 (1986). 3. Elser, V., Acta Cryst. A, 42, 36-43 (1986). 4. Cotfas, N., Z. Kristallogr., 213, 311-15 (1998). 5. Cotfas, N., Verger-Gaugry, J.-L., in Aperiodic’ 97, ed. M. de Boissieu, J.L. Verger-Gaugry and R. Currat, World Scientific, Singapore, 36-42 (1999). 6. Cotfas, N., Lett. Math. Phys., 47, 111-23 (1999). 7. Cotfas, N., J. Phys. A: Math. Gen., 32, 8079-93 (1999). 8. Cotfas, N., J. Phys. A: Math. Gen., 31, 7273-77 (1998). 9. Janssen, T., in The Mathematics of Long-Range Aperiodic Order, ed. R. V. Moody, Kluwer Acad. Publ., Dordrecht, 269-306 (1997). 10. Vogg, U., Ryder, P. L., J. Non-Cryst. Solids, 194, 135-44 (1996).