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. - 2 Oct. 2010, La Valérane Carqueiranne, France
-2 2 Scale =" &'" e1( )10*+
Position !(0,0,0,0)
42 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Simple models of d-QC’s
43 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
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 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
45 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
46 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
47 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
48 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
49 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Modified Al-Mn model A.Yamamoto & K.N. Ishihara, Acta Cryst. (1988). A44, 707-714.
Space group
Al-Mn: 6 layers
52 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Modified Al-Fe model A.Yamamoto & K.N. Ishihara, Acta Cryst. (1988). A44, 707-714.
Space group
Al-Fe: 8 layers
53 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Symmetry & Space group of d-QC’s in a little more details
54 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
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 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
• In many cases, OD’s are located at the special position of the space group. • Those OD’s are invariant under the site- symmetry 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 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Decagonal point groups
Order Order
40 20
20 20
20 10
10
57 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Decagonal space groups with reflection conditions
Cf. D.A.Rabson et al., Rev. Mod. Phys. 63 (1990) 699-733.
58 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Space group P105/mmc Similar to
Point group :
Generators :
• tenfold screw :
• glide plane :
• inversion :
• lattice translations :
Non-primitive translation vector :
59 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France Matrix representation of the generators
of 5D space group P105/mmc
(hyper-) screw
(hyper-) glide plane
60 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
C10 and its action !/5 rotation
C10
61 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
!’and its action
!’
!’ !’
62 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
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 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Cluster based models of d-QC’s
64 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Cluster based model of QC’s
First applied to icosahedral quasicrystals by A.Yamamoto and K. Hiraga, ‘Structure of an icosahedral Al-Mn quasicrystal’, Physical Review B, 37 (1988) 6207-6214.
model + structure refinement decagonal quasicrystals by S. E. Burkov, ‘Structure model of the Al-Cu-Co decagonal quasicrystal’, Phys. Rev. Lett. 67 (1991) 614-617.
model, no refinement
65 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
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 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Structure model of d-Al-Cu-Co by Burkov
67 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
68 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
2 nm cluster in the Burkov model
! 2 nm
+ =
superposition
69 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
70 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
71 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
72 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
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 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Yamamoto’s 2 nm cluster model A.Yamamoto, Sci. Rep. RITU. (1996). A42, 207-212.
74 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
75 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Construction of cluster based model
•
•
•
76 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Refined structure of d-Al72Ni20Co8 QC
Space group P105/mmc
H.Takakura, A.Yamamoto, A.P.Tsai, Acta Cryst. (2001). A57, 576-585.
77 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Structure-factor formula for QC’s
78 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
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 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Provided that the occupation domain consists of $ independent triangles (or tetrahedra), it is given by
Rotational part of site-symmetry operator
80 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Fourier integral of a triangle
o Asymmetric part
81 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Fourier integral of a tetrahedron
o Asymmetric part 82 International School on Aperiodic Crystals 26 Sept. - 2 Oct. 2010, La Valérane Carqueiranne, France
Structural parameters for QC’s
•Position •ADP o •Occupancy o •Ratio of elements Asymmetric parts for th occupation domain
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