ISQCS, June 23-27, 2019, Sendai, Tohoku University
Introduction to higher dimensional description of quasicrystal structures
Hiroyuki Takakura
Division of Applied Physics, Faculty of Engineering, Hokkaido University ISQCS, June 23-27, 2019, Sendai, Tohoku University
Outline
• Diffraction symmetries & Space groups of iQCs • Section method • Fibonacci structure • Icosahedral lattices • Simple models of iQCs • Real iQC structures • Cluster based model of iQCs • Summary ISQCS, June 23-27, 2019, Sendai, Tohoku University
Crystal Amorphous
Their diffraction patterns ISQCS, June 23-27, 2019, Sendai, Tohoku University
Diffraction symmetries and space groups of iQCs ISQCS, June 23-27, 2019, Sendai, Tohoku University
X-ray transmission Laue patterns of iQC
2-fold 3-fold 5-fold
i-Zn-Mg-Ho F-type ISQCS, June 23-27, 2019, Sendai, Tohoku University
Electron diffraction pattern of iQC i-AlMn
1
The arrangement of the diffraction spots is not periodic but quasi-periodic.
D.Shechtman et al., Phys.Rev.Lett., 53,1951(1984). ISQCS, June 23-27, 2019, Sendai, Tohoku University Symmetry of iQC
2 Point group 31.72º 5 Order : 120 37.38º 2 5 3 20.90º 3 2 2
Asymmetric region:
6 +10 +15 + m + center ISQCS, June 23-27, 2019, Sendai, Tohoku University
X-ray diffraction patterns of iQCs
P-type i-Zn-Mg-Ho F-type i-Zn-Mg-Ho 2fy 2fy
5f 5f
3f 3f
2fx 2fx Liner plots ISQCS, June 23-27, 2019, Sendai, Tohoku University
X-ray diffraction patterns of iQCs
P-type i-Zn-Mg-Ho F-type i-Zn-Mg-Ho 2fy 2fy
5f 5f
3f 3f
2fx All even or all odd for 2fx No reflection condition Log plots ISQCS, June 23-27, 2019, Sendai, Tohoku University
Vectors used for indexing
6 Any vectors can be used if all the reflections can be indexed correctly. 1 2 5 Six vectors → 6D reciprocal lattice
The 6D reciprocal lattice must have at least icosahedral 3 symmetry. 4 6D hypercubic lattice is chosen usually. ISQCS, June 23-27, 2019, Sendai, Tohoku University
6D Icosahedral lattices
Lattice types Reflection condition
• Primitive No condition e.g. • Body-centered for e.g.
• Face-centered All even or all odd for e.g. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Centering translations in centered lattices
Face-centered Body-centered (F-type) (I-type)
32 centering translations 2 centering translations
• Body-centered lattice has yet to be observed experimentally!
Recently, an I-type structure has formed as a computer simulated structure: M. Engel et al., Nat. mat. 14, 109 (2015). ISQCS, June 23-27, 2019, Sendai, Tohoku University
The section method ISQCS, June 23-27, 2019, Sendai, Tohoku University
Fibonacci structure Recurrence formula
1D quasi-periodic structure
Direct space L S L S L L S L L S
Fourier transformation
Diffraction pattern
00 53 85 32 10 6 21 64 74 11
Reciprocal Intensity space 0 q Intensity distribution along the 1D reciprocal space ISQCS, June 23-27, 2019, Sendai, Tohoku University Diffraction pattern of the Fibonacci structure 1D complementary space Internal space 2D reciprocal space * r⊥
* r// External space 1D reciprocal space
00 53 85 32 10 6 21 64 74 11
1D reciprocal Intensity space 0 Intensity distribution along the 1D reciprocal space ISQCS, June 23-27, 2019, Sendai, Tohoku University Construction of the Fibonacci structure
Internal space r⊥ （1D complementary space） External space r// (1D direct space)
L = t S = 1 1 Length: 1 2D direct space
Occupation domain
The angle is determined by the formula: ISQCS, June 23-27, 2019, Sendai, Tohoku University Relationship between the direct space and the reciprocal space of Fibonacci structure Direct space Reciprocal space
Fourier transformation 2D crystal lattice 2D reciprocal lattice Periodic Periodic
Structure Diffraction pattern Fourier transformation 1D section of Projection onto 1D the 2D crystal along the other 1D Quasi-periodic Quasi-periodic Fibonacci structure ISQCS, June 23-27, 2019, Sendai, Tohoku University Construction of the Fibonacci structure
Internal space r⊥ （1D complementary space） External space r// (1D direct space)
L = t S = 1 1 Length: 1 2D direct space
Occupation domain
The angle is determined by the formula: ISQCS, June 23-27, 2019, Sendai, Tohoku University Approximant crystals of Fibonacci structure
A shear strain (liner phason strain) along r⊥ is applied. r⊥ r⊥ r r// //
1/1 approximant 2/1 approximant …LS… …LSL… ISQCS, June 23-27, 2019, Sendai, Tohoku University
Determination of the icosahedral lattice ISQCS, June 23-27, 2019, Sendai, Tohoku University
Determination of the unit vectors of iQCs
Diffraction pattern Icosahedral symmetry
6 indices: reciprocal lattice Lattice constant:
Icosahedral lattice in 6D reciprocal space
(external) (internal)
Icosahedral lattice in 6D direct space
Lattice constant: ISQCS, June 23-27, 2019, Sendai, Tohoku University
Unit vectors in 6D reciprocal space
: unit reciprocal lattice vectors : orthonormal base vectors
: span the external space (3D) : span the internal space (3D)
: lattice constant in reciprocal space ISQCS, June 23-27, 2019, Sendai, Tohoku University
Projection of the unit vectors (reciprocal space) * * External space (3D) r// Internal space (3D) r⊥ ISQCS, June 23-27, 2019, Sendai, Tohoku University
Unit vectors in 6D direct space
Reciprocal lattice vectors Direct lattice vectors
: icosahedral lattice constant ISQCS, June 23-27, 2019, Sendai, Tohoku University
Projection of the unit vectors (direct space)
External space (3D) r// Internal space (3D) r⊥ ISQCS, June 23-27, 2019, Sendai, Tohoku University
Description of icosahedral quasicrytal structures ISQCS, June 23-27, 2019, Sendai, Tohoku University
Penrose tiling
Prototiles
thin Sir Roger Penrose Mathematical physicist Mathematician fat Philosopher of science
A Penrose tiling is an example of quasiperiodic 2D tiling that consists of two types of prototiles. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Decoration of the Penrose tiling with atoms
A 2D quasiperiodic atom arrangement is realized.
Vertex decoration model ISQCS, June 23-27, 2019, Sendai, Tohoku University
Diffraction pattern of the Penrose tiling Similar to diffraction plane of real dQCs
21-1-2
-1-2-2-1
Vertex decoration with point scatters ISQCS, June 23-27, 2019, Sendai, Tohoku University
3D Ammann-Kramer-Neri tiling
• A 3D analog of Penrose tiling. • The AKN tiling consists of two primitive rhombohedra: the acute rhombohedron (AR) and the obtuse rhombohedron (OR). • Space-filling structure with icosahedral symmetry.
AR OR ISQCS, June 23-27, 2019, Sendai, Tohoku University
Vertex decoration of the rhombohedra
AR OR
Put atoms at every vertex of the AKN tiling. ISQCS, June 23-27, 2019, Sendai, Tohoku University A naïve model of iQC (vertex decoration model of AKN tiling)
Projected view of the structure along a 5f axis. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Construction of AKN tiling by the section method (Generate the vertices of the AKN tiling)
Occupation Domain Corner vectors Position
Site-symmetry group:
Basis:
This rhombic triacontahedral OD corresponds o to the projection of the 6D unit cell onto the 3D Asymmetric part internal space. ISQCS, June 23-27, 2019, Sendai, Tohoku University
2D section of the 6D structure including 5f axes
5f i 5f e
The line segment is the 1D section of the OD along a 5f axis in the internal space.
Note that any direction in the plane has 5-fold rotational symmetry. ISQCS, June 23-27, 2019, Sendai, Tohoku University
2D section of the 6D structure including 3f axes
3f i
The line segment is the 1D e section of the OD along a 3f 3f axis in the internal space. ISQCS, June 23-27, 2019, Sendai, Tohoku University
2D section of the 6D structure including 2f axes
2f i
The line segment is the 1D section of the OD along a 2f axis in the internal space. 2f e ISQCS, June 23-27, 2019, Sendai, Tohoku University
Simple decoration model of icosahedral QC based on the AKN tiling ISQCS, June 23-27, 2019, Sendai, Tohoku University Atomic decoration of two rhombohedra in the case of simple decoration model
AR OR • Vertices • Mid edges • Body diagonal two positions in ARs
This model was proposed as the structure model of i-Al-Cu-Li iQC (Bergman type cluster) at the early stage. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Simple decoration model of iQC in 6D
5f i 5f e
Three large occupation domains characterize the 6D structure. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Site-symmetry group
RT Independent occupation domains and their shapes in the internal space.
V: 5f e
FD E B
B: V
RI
E: The symmetry of the occupation domain is equal to or higher than the site symmetry. ISQCS, June 23-27, 2019, Sendai, Tohoku University
How to obtain the OD at E
5f i 5f e
Rhombic icosahedron ISQCS, June 23-27, 2019, Sendai, Tohoku University
How to obtain the OD at B (step no.1) Acute rhombohedral-shaped OD 3f i
Center:
3f e
Length of the diagonal: ISQCS, June 23-27, 2019, Sendai, Tohoku University
(step no.2) 3f i 20AR shift Center: + shift along E-space
FD
3f e Center: ISQCS, June 23-27, 2019, Sendai, Tohoku University
3f i
3f e
The result is equivalent to put an OD with FD shape on (1,1,1,1,1,1)/2 instead of putting ODs with AR shape at the original positions in the 6D unit cell. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Site-symmetry group
RT
Simple decoration model V: 5f e
FD E B
B: V
RI
E: ISQCS, June 23-27, 2019, Sendai, Tohoku University
What is important to remember
• iQCs can be described as 6D periodic crystals. • Occupation domain (OD) can be considered as atom in the 6D periodic crystal. • Each OD has a specific size and shape. • The symmetry of OD is equal to or higher than the site-symmetry. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Structure factor formula for QCs ISQCS, June 23-27, 2019, Sendai, Tohoku University
Structure factor formula
Independent occupation domain Symmetry operators of space group which generate the equivalent occupation domains in a unit cell from the independent occupation domain
Atomic scattering factor
s.o.f
ADP ISQCS, June 23-27, 2019, Sendai, Tohoku University
Provided that the occupation domain consists of independent triangles (or tetrahedra), it is given by
Rotational part of the site-symmetry operator ISQCS, June 23-27, 2019, Sendai, Tohoku University
Real structure of icosahedral quasicrystals ISQCS, June 23-27, 2019, Sendai, Tohoku University
Phase problem in QCs
Unavoidable loss of phase information in the diffraction intensity.
The phase cannot be obtained by ordinary diffraction experiment. ISQCS, June 23-27, 2019, Sendai, Tohoku University
Principle of the density modification method
Diffraction experiment
nD indexed reflections
: random FT-1
? -1 FT Density modification
Low density elimination, Stop FT Charge flipping Structure solution ? FT: Fourier transform ISQCS, June 23-27, 2019, Sendai, Tohoku University
Structure solution of P-type i-Zn-Mg-Ho QC
(a) r// 5f r⊥ space group Pm35 5f E V: (0,0,0,0,0,0) B B: (1,1,1,1,1,1)/2 V E: (1,0,0,0,0,0)/2 a = 0.5137 nm
(b) 3f (c) 2f r⊥ r⊥
B B
V r// r// 2f 3f ISQCS, June 23-27, 2019, Sendai, Tohoku University
Structure solution of i-Yb-Cd QC
space group Pm35 V: (0,0,0,0,0,0) B: (1,1,1,1,1,1)/2 E: (1,0,0,0,0,0)/2 a = 0.5689 nm ISQCS, June 23-27, 2019, Sendai, Tohoku University Occupation domains for i-Yb-Cd QC ISQCS, June 23-27, 2019, Sendai, Tohoku University
Constitution of the ODs for i-YbCd QC ISQCS, June 23-27, 2019, Sendai, Tohoku University
H.Takakura, C.P.Gomez, A.Yamamoto, M.deBoissue, A.P.Tsai, Nat. Mater. 2007, 6, 58. ISQCS, June 23-27, 2019, Sendai, Tohoku University Summary
• The 6D structure of iQCs can be known by the 6D electron densities obtained through a phase retrieval of diffraction data. • Respective iQCs have different 6D crystal structures. • Result of phase retrieval is a starting point of structure analysis of iQCs. • The purpose of the structure analysis of QCs is to determine the 3D atomic structure, which means that to determine the detailed shape of occupation, to specify its location and to reveal the distribution of constituent elements in it.