Point substitution processes for generating icosahedral tilings
Nobuhisa Fujita
IMRAM, Tohoku University, Sendai 980-8577, Japan
Outline
Part I.
1. Basic icosahedral tilings
2. Point substitution processes
Part II.
3. Tilings constructed with PIRs
4. Canonical cell tilings
5. Toward icosahedral CCTs
Outline
Part I.
1. Basic icosahedral tilings2. Point substitution processes
Part II.
3. Tilings constructed with PIRs
4. Canonical cell tilings
5. Toward icosahedral CCTs
ED pattern along 5-foldaxis of an icosahedral quasicrystal
Model of i -(Al-Mn),M. Audier and P. Guyot, Phil. Mag. B 53, L43 (1986)
Icosahedral QCs
b-c packing of icosahedral clusters (F-type) based on the rhombohedral tiling (Ammann-Kramer tiling).
Mn icosahedra
Icosahedral tilings
T *(P) tiling
(Ammann-Kramer tiling)
M. Duneau and A. Katz, Phys. Rev. Lett. 54, 2688 (1985).P. Kramer and R. Neri, Acta Cryst. A 40, 580 (1984).
Basic tiles OR, AR(Ammann rhombohedra)
5-fold view 2-fold view
e1 e2
e3
e4e5
e6
( )
−−
−=
1010
1001
0110
654321
ττττ
ττeeeeee
Icosahedral basis set
2
51+=τ
the golden mean
τ+= 2|||| 1e
Icosahedral modules
)(|: 6665544332211 Ζ∈+++++= jP nnnnnnn eeeeeeM
][τΖ)(,2mod0
|:6
665544332211
Ζ∈=
+++++=
∑ jj
j
F
nn
nnnnnn eeeeeeM
][τΖ
][ 3τΖ
[1] T. Janssen, Acta Cryst. A 42 (1986) 261.
[2] D. S. Rokhsar et al., Phys. Rev. B 35 (1987) 5487.
[3] L.S. Levitov and J. Rhyner, J. Physique 49 (1988) 1835.
( ) )111111(2
1
|:
66
665544332211
+Ζ∪Ζ∈
+++++=
j
I
ν
νννννν eeeeeeM
integer ring
[ ]
+∪
+∪+∪=
+∪=
)111111(2
1)111111(
2
1)100000(
)111111(2
1:
FFFF
PPI
MMMM
MMM
Ve Vo Be Bo
Vo
Be Bo
τ-scaling×τ
×τ
×τ
×τVe II
FF
PP
MM
MM
MM
=×=×=×
τττ 3
Scale invariance of the modules
T *(2F) tiling
(Kramer et al.)
P. Kramer et al., in Symmetries in Science V: Algebraic Structures, their Representions, Realizations and Physical Applications, Ed. by B. Gruber et al., Plenum Press, New York, 1991, pp. 395.
Outline
Part I.1. Basic icosahedral tilings
2. Point substitution processes
Part II.3. Tilings constructed with PIRs
4. Canonical cell tilings
5. Toward icosahedral CCTs
R P H
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
(1)Expansion(σ =τ 2)
R P H
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
(1)Expansion(σ =τ 2)
(2)Place S at every vertex
R P H
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
R P H
(1)Expansion(σ =τ 2)
(2)Place S at every vertex
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
(3)Eliminate excessive points
R P H
(1)Expansion(σ =τ 2)
(2)Place S at every vertex
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
R P H
(3)Eliminate excessive points
(1)Expansion(σ =τ 2)
(2)Place S at every vertex
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
R P H
(3)Eliminate excessive points
(1)Expansion(σ =τ 2)
(2)Place S at every vertex
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
N. Fujita, Acta Cryst. A 65, 342 (2009)
Window
Point Substitution Process(for constructing icosahedral quasiperiodic tilings)
(1)Expansive similarity transformation: Ti σ Ti (Ti⊂M, σ =ρn, ρ=τ3(P), τ(F), τ(I))
(2) Replicate the Ih-star at every vertex: Ti’= σ Ti + S (S⊂M)
(3) Decimation of points by local rules: Ti’ Ti+1 (⊂Ti’ )
N. Fujita, Acta Cryst. A 65, 342 (2009)
Step (3) is needed if there is redundancy in the points generated through (1) and (2) (Point inflation rule)
K. Niizeki, J.Phys.A:Math.Theor.41,175208 (2008)
Outline
Part I.1. Basic icosahedral tilings
2. Point substitution processes
Part II.
3. Tilings constructed with PIRs4. Canonical cell tilings
5. Toward icosahedral CCTs
|||| 1eτ
Windows
T *(2F)
F-typeT *(P)
P-type
|||| 1e
Point density
P
PW
Ω P
P
F
F WW
Ω=
Ω 2
3τ
)111000(:3
)111111(2
1:2
)000000(:1
S
Ih-star
Point inflation rule(viewed in the external space)
seed 1st iteration 2nd iteration
τ× τ×
Q∞∞∞∞(S,ττττ)
)111000(:3
)111111(2
1:2
)000000(:1
S
T
Ih-star(mapped to the internal space)
window
in the internal space
)111000(:3
)111111(2
1:2
)000000(:1
2.618031
11 2
154321 ==
−=++++++ −
−−−−− ττ
τττττ L
|||| 1e
5-fold direction
Q∞∞∞∞(S,ττττ)∩∩∩∩MP
Inflation rules by τ 3 scalingT. Ogawa, J. Phys. Soc. Jpn. 54, 3205 (1985).
Inflation rule of the Ammann-Kramer tiling
Outline
Part I.
1. Basic icosahedral tilings
2. Point substitution processes
Part II.
3. Tilings constructed with PIRs
4. Canonical cell tilings5. Toward icosahedral CCTs
A-cell B-cell C-cell D-cell
cc
cc
cc
c
c
cc
c
c
c
c
bb
b
bbb
b b
b
b
b
bb
b
b
‘Cell geometry for cluster-based quasicrystal models’,
C. L. Henley, Phys. Rev. B 43 (1991) 993.
4 polyhedra: A-, B-, C-, D-cells
There are 32 classes of nodes in a CCT
Canonical cell tiling
(67)333(68)0 (62)222222
Canonical Cell Tiling
M. Mihalkovic et al., Phys. Rev. B 53, 9002-9020 (1996).
i-Cd5.7Yb: Quasicrystal (QC)
RTH
H. Takakura &C.P. Gomez et al.,(2007).
AR
M. Mihalkovic et al., Phys. Rev. B 53, 9002-9020 (1996).
Outline
Part I.
1. Basic icosahedral tilings
2. Point substitution processes
Part II.
3. Tilings constructed with PIRs
4. Canonical cell tilings
5. Toward icosahedral CCTs
A brute force algorithm:
M.E.J. Newman, C.L. Henley, and M. Oxborrow, Phil. Mag. B 71 (1995) 991.
Is there an icosahedral CCT?
A Monte Carlo density optimization method:
M. Mihalkovic and P. Mrafko, Europhys. Lett. 21 (1993) 463.
NO PROOF is given of the existenceof an ICOSAHEDRAL CCT
Methods to construct approximant CCTs(under periodic boundary conditions)
5-fold view
Point sutstitution processes for icosahedral CCTs
Ih-starthe magic star
The Ih-star is placed on every vertex of the expanded CCT
(scaling ratio=τ 3)
Fix the center to bethe (68)0 type node.
The Ih-star
2 vertices12 A-cells0 B-cell0 C-cell0 D-cell
138 vertices348 A-cells136 B-cells136 C-cells24 D-cells
A-packing(body centered cubic)
1 vertex0 A-cell2 B-cells2 C-cells0 D-cell
77 vertices192 A-cells76 B-cells76 C-cells14 D-cells
BC-packing(rhombohedral)
1 vertex0 A-cells0 B-cells0 C-cells2 D-cells
(67)333 The center is missing!
D-packing(simple hexagonal)
1 vertex0 A-cells0 B-cells0 C-cells2 D-cells
103 vertices252 A-cells102 B-cells102 C-cells20 D-cells
(67)333 The center is missing!
D-packing(simple hexagonal)
77 vertices192 A-cells76 B-cells76 C-cells14 D-cells
138 vertices348 A-cells136 B-cells136 C-cells24 D-cells
103 vertices252 A-cells102 B-cells102 C-cells20 D-cells
584 vertices1464 A-cells576 B-cells576 C-cells104 D-cells
ττττ3××××A-packing ττττ3××××D-packing
ττττ3××××BC-packing ττττ3××××2/1 cubic-packing
Conclusion
The present scheme has turned out to be useful for constructing icosahedral tilings.
The magic star can generate all the vertices of an inflated CCT except a point in the center of each expanded D-cell.
It is likely that there exist τ3-inflation rules for generating an icosahedral CCT, the proof of which still needs to be worked out.