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Decagonal Quasicrystals Higher Dimensional Description & Structure Determination

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Decagonal Quasicrystals Higher Dimensional Description & Structure Determination International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Decagonal quasicrystals higher dimensional description & structure determination Hiroyuki Takakura Division of Applied Physics, Faculty of Engineering, Hokkaido University 1 Disclaimer and copyright notice Copyright 2010 Hiroyuki Takakura for this compilation. This compilation is the collection of sheets of a presentation at the “International School on Aperiodic Crystals,“ 26 September – 2 October 2010 in Carqueiranne, France. Reproduction or redistribution of this compilation or parts of it are not allowed. This compilation may contain copyrighted material. The compilation may not contain complete references to sources of materials used in it. It is the responsibility of the reader to provide proper citations, if he or she refers to material in this compilation. International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Overview • The section method • Diffraction symmetry & Space groups of d-QC’s • Unit vectors in decagonal system • Penrose tiling • Description of d-QC structures • Point density • Simple models • Symmetry & Space group of d-QC’s in a little more details • Cluster based models • Example of structure determination of real d-QC’s • Structure factor formula 2 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France The section method 3 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Construction of a Fibonacci sequence Internal space 2D direct space L S L S L L S L L S External space 1D direct space 1 The gradient of the Length! ! square lattice: 1+! 2 1 tan" = 1 ! 4 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Diffraction pattern of the Fibonacci sequence Internal space 2D reciprocal space External space 1D reciprocal space 00 53 85 32 10 6 21 64 74 11 Intensity 5 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Direct space Reciprocal space Fourier transform 2D crystal 2D reciprocal lattice Structure Diffraction pattern 1D section of Projection onto 1D the 2D crystal Fourier along the other 1D. transform 6 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Diffraction symmetry & Space groups of d-QC’s 7 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Diffraction patterns of a decagonal quasicrystal 2y d-Al75.2Ni14.6Ru10.2 a = 0.2764 nm 21-1-20 c = 1.6523 nm reciprocal lattice plane -1-2-2-10 2x 8 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France 10 10 2x 2y -1-2-2-10 21-1-20 No reflection condition Reflection condition: 9 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France 2y 10 10 2x 2x 2y along between All reflections can be indexed with five integer using these unit vectors. 10 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Feature of the diffraction pattern of the d-AlNiRu QC Diffraction symmetry Reflection condition (Hyper-) glide plane Order : 40 Rotational symmetry Non-primitive translation vector 11 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Decagonal space groups in 5D space with highest symmetries along * between between along * The order of the point group is 40. There are 34 decagonal non-equivalent space groups. Cf. D.A.Rabson et al., Rev. Mod. Phys. 63 (1990) 699-733. 12 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Possible space groups of the d-AlNiRu QC Order Point group Space group 20 20 40 13 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France What to Remember Diffraction patterns of d-QC’s can be indexed with five integers. Reciprocal lattice in 5D space Crystal in 5D space 14 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Unit vectors in decagonal system 15 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France A flowchart representing a process for determining the unit vectors of d-QC’s Point group symmetry Diffraction pattern Indexing Special reflection 5 indices: conditions 5D reciprocal lattice (Extinction rules) Lattice constants: Unit vectors in """" 5D reciprocal space Unit vectors in 5D direct space 16 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Unit vectors in 5D reciprocal space ! unit reciprocal lattice vectors : orthonormal base vectors : external #$D% : internal (2D) ( for 2D quasiperiodic plane ) : lattice constants 17 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Projections of the unit vectors The projection of the The projection of the unit vectors onto unit vectors onto the external space. the internal space. 2!/5 4!/5 For Indexing 18 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Unit vectors in 5D direct space Reciprocal lattice vectors Direct lattice vectors : lattice constants 19 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Projections of the unit vectors The projection of the The projection of the unit vectors onto unit vectors onto the external space. the internal space. 20 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Vectors for specifying atom positions in the 2D external space 2!/5 21 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Vectors employed for defining occupation domains in the internal space 4!/5 22 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Similarity transformation Choice of is not unique. : transposed matrix of 1 2 The case 1 3 2 2 1 4 3 1 3 3 4 4 4 2 23 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France What to Remember External space Internal space 1 3 2 2!/5 1 4!/5 Reciprocal space 3 4 4 2 1 3 2 1 Direct space 3 4 4 2 1 3 2 1 3 4 4 2 24 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France • The first four vectors, , are relevant to the quasi-periodic plane of d-QC. • Choice of is not unique. • A similarity transformation does not change the size of the 5D unit cell, but changes the scale of the projected vectors in the external and internal spaces. 25 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Description of d-QC structures 26 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France D-QC structure • Periodic stacking of atom planes along the 10-fold axis. • Each plane has a quasiperiodic atomic order. A decorated Penrose tiling with atoms would give an example of such atomic plane. 27 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Penrose tiling Edge length 28 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Decoration of the Penrose tiling with atoms Edge length Vertex decoration 29 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Diffraction pattern of the Penrose tiling Similar to plane of real d-QC 21-1-2 -1-2-2-1 Vertex decoration with point scatters 30 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Construction of the Penrose tiling Occupation domains: A B C D The 4D unit cell: Lattice constant: 31 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France 2D section of 4D structure Internal External 32 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Projection of the 4D unit cell onto the 2D internal space 33 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Generalized Penrose tiling 34 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Point density 35 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France • The point density is the number of vertices per unit area (or volume) of a tiling. • Because the density of QC’s is a fundamental quantity, therefore it has to be explained by their models. • The density calculations are based on the fact that an OD at some lattice point intersects the external space at a point and the set of all possible cross points covers the OD homogeneously. 36 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Point density calculation Unit vectors of the nD lattice: The unit-cell volume: Sum of the area (volume) of the OD’s : Point density: 37 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Point density of the primitive 4D decagonal lattice Primitive 4-dimensional decagonal lattice: The unit-cell volume: Area of the unit cell in the internal space: Point density: 38 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Point density of the Penrose tiling Point density Area of the occupation domains Point density: 39 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Scaling of a tiling Scale =, &'e1()10*+ Position !(0,0,0,0) 2a 5 ! 40 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France #1 Scale =" &'" e1( )10*+ Position !(0,0,0,0) 41 International School on Aperiodic Crystals 26 Sept.
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