Report

Quasicrystals from Higher Dimensional Lattices Mehmet Koca Department of Physics College of Science Sultan Qaboos University Muscat-OMAN [email protected] Bangalore conference, 16-22 December, 2012 1 Crystallography Modern crystallography started in 1912 with the seminal work of von Laue who performed the first x-ray diffraction experiment. The crystals von Laue studied were ordered and periodic, and all the hundreds of thousands crystals studied during the 70 years from 1912 till 1982 were found to be ordered and periodic. Crystals are the 1D, 2D and 3D lattices invariant under the translation and the rotational symmetries of orders 2,3,4,6. In 2D and 3D translational invariance is not compatible with rotations of orders such as 5, 7, 8, 10, 12, 30, 36. Bangalore conference, 16-22 December, 2012 2 But in 1982 Daniel Shechtman has observed a crystal structure in Al-Mn alloy displaying 10 fold symmetry not invariant under translational symmetry. Shechtman's is an interesting story, involving a fierce battle against established science, ridicule and mockery from colleagues and a boss who found the finding so controversial, he has been asked to leave the lab. (Daniel Shechtman: Nobel Prize in Chemistry, 2011), Israel Institute of Technology (Technion) Bangalore conference, 16-22 December, 2012 3 A new definition for Crystal “… By crystal we mean any solid having an essentially discrete diffraction diagram, and by aperiodic crystal we mean any crystal in which three dimensional lattice periodicity can be considered to be absent.” Bangalore conference, 16-22 December, 2012 4 Electron diffraction from the Icosahedral Phase has five-fold rotational axes and it is not periodic. The ratio of distances between the central spot and other spots is the Fibonacci Number τ known also as the “Golden Mean”. 1 5 1.618 2 Bangalore conference, 16-22 December, 2012 5 Are QCs rare? QCs are not rare there are hundreds of them Bangalore conference, 16-22 December, 2012 6 Mathematical Modelling Penrose Tiling of the plane with 5-fold symmetry Bangalore conference, 16-22 December, 2012 7 Penrose tiling can be obtained from the lattice B 5 and A4 by orthogonal projection technique Affine A4 , Quaternions, and Decagonal Quasicrystals Mehmet Kocaa), Nazife O. Kocab) Department of Physics, College of Science, Sultan Qaboos University P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman and Ramazan Kocc) Department of Physics, Gaziantep University, 27310, Gaziantep, Turkey Bangalore conference, 16-22 December, 2012 8 The projected point set of the root lattice displays a generalized Penrose tiling with a point dihedral symmetry D5 of order 10 which can be used for the description of the decagonal quasicrystals. The projection of the Voronoi cell of the root lattice of A4 describes a framework of nested decagrams growing with the power of the golden ratio recently discovered in the Islamic arts. Note that the root and weight lattices of A3 correspond to the face centered cubic (fcc) and body centered cubic (bcc) lattices respectively Bangalore conference, 16-22 December, 2012 9 A3 Lattices Root Lattice (FCC Lattice) The root system of (e1 e 2 ),(e 2 e3 ),(e3 e1 ) m11 m22 m33 Crystallography books use the generating vectors: Weigh Lattice e1 e 2 ,e 2 e3 ,e3 e1 a11 a22 a33 (a1 , a2 , a3 ) a1 2 1 0 m1 a 1 2 1 m 2 2 , a3 0 1 2 m 3 A3 The weight vectors i (i 1, 2,3) m1 3 2 1 a1 1 m 2 4 2 a 2 4 2 . m 3 1 2 3 a3 1 2 1 2 1 (e1 e 2 e 3 ), 2 e 3 , 3 (e1 e 2 e3 ) . The primitive vectors of the bcc lattice in text books 1, 1 2 , 2 3 Bangalore conference, 16-22 December, 2012 10 Orbits of W(A3) Tetrahedron: (1,0,0)A3 Octahedron : (0,1,0)A3 Cube : (1,0,0)A3 + (0,0,1)A3 Cuboctahedron : (1,0,1)A3 Rhombic Dodecahedron Wigner-Seitz Cell Truncated Octahedron : (1,1,1)A3 Wigner-Seitz Cell for BCC Bangalore conference, 16-22 December, 2012 11 Construction of the affine Coxeter group A4 in terms of quaternions 0 1 2 3 4 Extended Coxeter-Dynkin diagram of 1 2 , 2 3 2e1 , 4 A4 1 (1 e1 e 2 e 3 ) , 2 1 (e1 e 2 e 3 ) 2 1 5 1 5 2 2 , W (A4 ) {[ p , cpc ] [ p ,cpc ]} Bangalore conference, 16-22 December, 2012 12 1 ( 5 e 2 e 3 ), 10 1 3 ( 5e1 2e 2 2e 3 ), 10 1 (2 e 2 2 e 3 ), 10 1 2 4 (2e 2 2e 3 ) c, 10 5 1 H 1 2 3 4 1 4 ( 1 e 2 e 3 ) 2 Root Lattice: ( FCC in 4D) 1 2 m11 m 22 m33 m 44 Generalization of Rhombic Dodecahedron to 4D (WignerSeitz Cell in 4D) (1,0,0,0)A4 (0,1,0,0)A4 (0,0,1,0)A4 (0,0,0,1) A4 Each Face is a rhombohedron Weight Lattice (BCC in 4D) a11 a22 a33 a44 Generalization of Truncated Octahedron to 4D: (1,1,1,1)A4 Bangalore conference, 16-22 December, 2012 13 Orthogonal projection of the lattices onto the Coxeter plane and the decagonal quasicrystallography 1 1 1 (1 3 ), 2 ( 2 4 ) 2 2 xˆ x 1 (2 ) 2 1 (1 2 ), y (1 2 ) 2(2 ) 2 [a1 a4 (a3 a2 )], y 2(2 ) [a1 +a 4 (a2 +a3 )] Bangalore conference, 16-22 December, 2012 14 (a) (b) Projected 5-cells (a) the polytope polytope (0,0,0,1) (1,0,0,0)A4 and A4 (a) Projected polytopes (a) (b) the (b) (0,0,1,0) A4 and (b) Bangalore conference, 16-22 December, 2012 (0,1,0,0) A4 15 The root system of A4 projected onto the Coxeter plane (a) root system (b) the polytope The orthogonal projection of the Voronoi cell of the root lattice onto the Coxeter plane (a) points (b) dual of the polytope (1,0,0,1) Bangalore conference, 16-22 December, 2012 16 Decagram point distributions Bangalore conference, 16-22 December, 2012 17 Orthogonal projection of the polytope (1,1,1,1)A4 Bangalore conference, 16-22 December, 2012 18 Orthogonal projection of the polytope (0,1,1,0)A4 Bangalore conference, 16-22 December, 2012 19 Electron diffraction pattern of an icosahedral Ho-Mg-Zn quasicrystal and A4 prediction Bangalore conference, 16-22 December, 2012 20 Icosahedral symmetry 5 e1 1 e1 e2 e3 e2 2 The Coxeter diagram of H 3 with quaternionic simple roots W (H 3 ) is a maximal subgroup of the group W (D6 ) D 6 lattices and their projections Projection onto the Coxeter Plane Let i be the simple roots Let i be the weight vectors (i , j ) ij , i , j 1, 2,...,6 Bangalore conference, 16-22 December, 2012 21 A general technique for every Coxeter-Weyl Group Organize the generators of the Coxeter-Weyl group such that R1 r1r3...rn 1 , R 2 r2r4 ...rn ri commute with each other in each generator R i , i 1, 2 which generate the Coxeter group I 2 (h ), h Coxeter number Let 1 x i i , 2 x i i i 1,3 i 2,4 be the generators of I 2 (h ) . The coefficients x i are eigenvectors of the incidence matrix (2I A )X cX , c=2cos h (largest eigenvalue) Bangalore conference, 16-22 December, 2012 22 Define the generators of I 2 (h ), (h 10) by R1 r1r3r5r6 , R 2 r2 r4 The simple roots of I 2 (10) 1 1 (1 2 3 5 6 ), 2 2 1 ( 2 4 ) 2 A general vector of the weight lattice D 6 (a11 a22 a33 a44 a55 a66 ) (a1 , a2 , a3 , a4 , a5 , a6 ), (ai Z, i 1, 2,..., 6) Define the orthogonal unit vectors in the Coxeter plane 1 1 xˆ ( 1 2 ), yˆ ( 1 2 ) 2(2 2) 2(2 2) Components of the vector x y (2 2 ) 10 (2 2 ) 10 (a1 ( 2 )a2 2a3 ( 2 )a4 a5 a6 ), (a1 ( 2 )a2 2a3 ( 2 )a4 a5 a6 ) define the locations of the projected lattice points Bangalore conference, 16-22 December, 2012 23 Some Examples Bangalore conference, 16-22 December, 2012 24 Some Examples (0,0,1,0,0,0)D6 (0,0,0,1,0,0)D6 Bangalore conference, 16-22 December, 2012 25 Some Examples (0,0,0,0,1,0)D6 Bangalore conference, 16-22 December, 2012 (0,0,0,0,0,1)D6 26 Some Examples (1,0,0,1,0,0)D6 (0,1,0,0,1,0)D6 Bangalore conference, 16-22 December, 2012 27 Some Examples (1,1,1,1,1,1)D6 (0,0,1,0,1,0)D6 Bangalore conference, 16-22 December, 2012 28 Projection into 3D space of H3 R1 r3r6 , R 2 r2 r4 , R 3 r1r5 1 1 1 ( 2 4 ) ( l 3 l 4 l 5 l 6 ) 2 2 1 1 2 ( 3 6 ) (l 2 l 3 l 4 l 5 ) 2 2 1 1 3 (1 5 ) (l 1 l 2 l 5 l 6 ) 2 2 Bangalore conference, 16-22 December, 2012 29 Using 1 1 2e1 , 2 ( e1 e 2 e 3 ), 3 2e 2 2 We obtain l1 a (e 2 e 3 ), l 2 a (e 2 e 3 ), l 3 a (e 3 e1 ), l 4 a (e 3 e1 ), l 5 a (e1 e 2 ), l 6 a (e1 e 2 ), 1 a 2(2 ) representing the vertices of an icosahedron (1,0,0,0,0,0)D6 (0,0,1)H 3 Icosahedron(1) Bangalore conference, 16-22 December, 2012 30 6D cube defined by the polytope (0,0,0,0,0,1)B6 (0,0,0,0,1,0)D6 (0,0,0,0,0,1)D6 1 ( l1 l 2 l 3 l 4 l 5 l 6 )(odd number of (-) sign) 2 1 ( l1 l 2 l 3 l 4 l 5 l 6 )(even number of (-) sign) 2 (0, 0, 0, 0,1, 0) D6 (0, 0, 0, 0, 0,1) D6 Bangalore conference, 16-22 December, 2012 31 Projection of (0, 0, 0, 0,1, 0) D 6 (1, 0, 0) H 3 (0, 0,1) H 3 Dodecahedron(1)+ Icosahedron(2) Dodecahedron(1) {a ( e1 e 3 ), a ( e 2 e1 ), a ( e 3 e 2 ), a ( e1 e 2 e 3 )} Icosahedron(2) {a ( e1 e 2 ), a ( e 2 e 3 ), a ( e 3 e1 )} Bangalore conference, 16-22 December, 2012 32 Projection of (0, 0, 0, 0, 0,1) D 6 (1, 0, 0) H 3 (0, 0,1) H 3 Dodecahedron(2)+ Icosahedron(3) Dodecahedron(2) {a ( e1 e 3 ), a ( e 2 e1 ), a ( e 3 e 2 ), a (e1 e 2 e 3 )} Icosahedron(3) {a ( e1 e 3 ), a ( e 2 e1 ), a ( e 3 e 2 )} Bangalore conference, 16-22 December, 2012 33 A. Rhombic Triacontahedron Icosahedron(2)+Dodecahedron(2) B. Rhombic Triacontahedron Icosahedron(1)+Dodecahedron(1) Bangalore conference, 16-22 December, 2012 34 C. Icosahedron(3)+ Dodecahedron(1) Bangalore conference, 16-22 December, 2012 35 Tiling three dimensional space with two rhombohedra Acute rhombohedron Obtuse rhombohedron Bangalore conference, 16-22 December, 2012 36 Conclusion The projection technique developed can be applied to any Coxeter group. Projected points have the dihedral symmetry Dh of order 2h. Quasicrystals possessing a dihedral symmetry of order 2h can be described by the appropriate Coxeter group. Bangalore conference, 16-22 December, 2012 37