Decagonal quasicrystals

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1 Decagonal quasicrystals higher dimensional description & structure determination Hiroyuki Takakura Division of Applied Physics, Faculty of Engineering, Hokkaido University 1

2 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.

3 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

4 The section method 3

5 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 1 Length!! 1 +! 2 The gradient of the square lattice: tan " = 1! 4

$Diffraction pattern of the Fibonacci$ space External space 1D reciprocal

6 Diffraction pattern of the Fibonacci sequence Internal space 2D reciprocal space External space 1D reciprocal space Intensity

7 Direct space 2D crystal Fourier transform Reciprocal space 2D reciprocal lattice Structure Diffraction pattern 1D section of the 2D crystal Fourier transform Projection onto 1D along the other 1D. 6

8 Diffraction symmetry & Space groups of d-qc s 7

9 Diffraction patterns of a decagonal quasicrystal 2 y d-al 75.2 Ni 14.6 Ru a = nm c = nm x reciprocal lattice plane 8

10 x 2 y No reflection condition Reflection condition: 9

2 y 10 10 2 x 2 x 2 y along between All reflections can

11 2 y x 2 x 2 y along between All reflections can be indexed with five integer using these unit vectors. 10

$Feature of the diffraction pattern of the$ condition Order : 40 (Hyper-) glide plane

12 Feature of the diffraction pattern of the d-alniru QC Diffraction symmetry Reflection condition Order : 40 (Hyper-) glide plane Rotational symmetry Non-primitive translation vector 11

13 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)

14 Possible space groups of the d-alniru QC Order Point group Space group

15 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

16 Unit vectors in decagonal system 15

17 A flowchart representing a process for determining the unit vectors of d-qc s Point group symmetry Special reflection conditions (Extinction rules) Diffraction pattern 5D reciprocal lattice Indexing 5 indices: Lattice constants: Unit vectors in 5D reciprocal space """" Unit vectors in 5D direct space 16

18 Unit vectors in 5D reciprocal space! unit reciprocal lattice vectors : orthonormal base vectors : external #$D% : internal (2D) ( for 2D quasiperiodic plane ) : lattice constants 17

19 Projections of the unit vectors The projection of the unit vectors onto the external space. The projection of the unit vectors onto the internal space. 2!/5 4!/5 For Indexing 18

20 Unit vectors in 5D direct space Reciprocal lattice vectors Direct lattice vectors : lattice constants 19

21 Projections of the unit vectors The projection of the unit vectors onto the external space. The projection of the unit vectors onto the internal space. 20

22 Vectors for specifying atom positions in the 2D external space 2!/5 21

23 Vectors employed for defining occupation domains in the internal space 4!/5 22

24 Similarity transformation Choice of is not unique. 1 : transposed matrix of 2 2 The case

25 External space Internal space What to Remember !/5 1 4!/5 Reciprocal space Direct space

26 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

27 Description of d-qc structures 26

28 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

29 Penrose tiling Edge length 28

30 Decoration of the Penrose tiling with atoms Edge length Vertex decoration 29

$Diffraction pattern of the Penrose tiling Similar to plane of$

31 Diffraction pattern of the Penrose tiling Similar to plane of real d-qc Vertex decoration with point scatters 30

32 Construction of the Penrose tiling Occupation domains: A B C D The 4D unit cell: Lattice constant: 31

33 2D section of 4D structure Internal External 32

34 Projection of the 4D unit cell onto the 2D internal space 33

35 Generalized Penrose tiling 34

36 Point density 35

37 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

38 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

39 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

40 Point density of the Penrose tiling Point density Area of the occupation domains Point density: 39

41 Scaling of a tiling Scale =, &'e 1 ()10*+ Position!(0,0,0,0) 2a 5 40

42 Scale =" #1 &'" e 1 ( )10*+ Position!(0,0,0,0) 41

43 Scale =" -2 &'" 2 e 1 ( )10*+ Position!(0,0,0,0) 42

44 Simple models of d-qc s 43

45 Feature of simple models Simple models based on the Penrose tiling. explain diffraction intensity qualitatively. consider no atom shifts from ideal positions. can t explain density. 44

46 45

47 46

48 47

49 48

50 49

51 Modified Al-Mn model A.Yamamoto & K.N. Ishihara, Acta Cryst. (1988). A44, Space group Al-Mn: 6 layers 52

52 Modified Al-Fe model A.Yamamoto & K.N. Ishihara, Acta Cryst. (1988). A44, Space group Al-Fe: 8 layers 53

53 Symmetry & Space group of d-qc s in a little more details 54

54 Symmetry of d-qc s Symmetry reflected in diffraction patterns is that of 5D crystals. If a 5D crystal has a (hyper-) glide plane or (hyper-) screw axis, it cause a systematic extinction of reflections. Rotational symmetry Diffraction symmetry Non-primitive translation Reflection conditions 55

55 In many cases, OD s are located at the special position of the space group. Those OD s are invariant under the sitesymmetry group, which is a subgroup of the point group of the space group. The site symmetry restricts the shape of the OD, since the shape has to be invariant under the site-symmetry group. 56

56 Decagonal point groups Order Order

57 Decagonal space groups with reflection conditions Cf. D.A.Rabson et al., Rev. Mod. Phys. 63 (1990)

58 Space group P10 5 /mmc Similar to Point group : Generators : tenfold screw : glide plane : inversion : lattice translations : Non-primitive translation vector : 59

59 Matrix representation of the generators of 5D space group P10 5 /mmc (hyper-) screw (hyper-) glide plane 60

60 C 10 and its action!/5 rotation C 10 61

61 ! and its action!!! 62

62 What to Remember A Vector in the external or internal space is transformed into a vector in the same space by a symmetry operation. 63

63 Cluster based models of d-qc s 64

64 First applied to Cluster based model of QC s icosahedral quasicrystals by A.Yamamoto and K. Hiraga, Structure of an icosahedral Al-Mn quasicrystal, Physical Review B, 37 (1988) decagonal quasicrystals by model + structure refinement S. E. Burkov, Structure model of the Al-Cu-Co decagonal quasicrystal, Phys. Rev. Lett. 67 (1991) model, no refinement 65

65 Feature of the Burkov model Space group Two layered structure ~ 0.4 nm period Two types of atom clusters: decagon and pentagon 2 nm 10-fold atomic cluster 1.2 nm cluster linkage 66

66 Structure model of d-al-cu-co by Burkov 67

67 68

68 2 nm cluster in the Burkov model! 2 nm + = superposition 69

69 70

70 71

71 72

72 Feature of the Yamamoto 2nm model Space group Two layered structure ~ 0.4 nm period Three types of atom clusters: decagon, pentagon, and star 2 nm 10-fold atomic cluster 2 nm cluster linkage Pentagonal Penrose tiling 73

73 Yamamoto s 2 nm cluster model A.Yamamoto, Sci. Rep. RITU. (1996). A42,

74 75

75 Construction of cluster based model 76

76 Refined structure of d-al 72 Ni 20 Co 8 QC Space group P10 5 /mmc H.Takakura, A.Yamamoto, A.P.Tsai, Acta Cryst. (2001). A57,

77 Structure-factor formula for QC s 78

78 Structure factor formula Independent occupation domain Symmetry operators of space group which generate equivalent occupation domains in a unit cell from the independent occupation domain µ 79

79 Provided that the occupation domain consists of $ independent triangles (or tetrahedra), it is given by Rotational part of site-symmetry operator 80

80 Fourier integral of a triangle o Asymmetric part 81

81 Fourier integral of a tetrahedron o Asymmetric part 82

82 Structural parameters for QC s for Position ADP Occupancy Ratio of elements th occupation domain o o Asymmetric parts It is assumed that the shape of the OD s is known. One of the difficult problems of the structure determination is to determine the size and shape of OD s. 83

Quasicrystals. Materials 286G John Goiri

Quasicrystals. Materials 286G John Goiri Quasicrystals Materials 286G John Goiri Symmetry Only 1, 2, 3, 4 and 6-fold symmetries are possible Translational symmetry Symmetry Allowed symmetries: Steinhardt, Paul J, and Luca Bindi. In Search of