point substitution processes for generating icosahedral ... · point substitution processes for...
TRANSCRIPT
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Point substitution processes for generating icosahedral tilings
Nobuhisa Fujita
IMRAM, Tohoku University, Sendai 980-8577, Japan
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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
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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
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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
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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
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e1 e2
e3
e4e5
e6
( )
−−
−=
1010
1001
0110
654321
ττττ
ττeeeeee
Icosahedral basis set
2
51+=τ
the golden mean
τ+= 2|||| 1e
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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
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[ ]
+∪
+∪+∪=
+∪=
)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
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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.
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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
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R P H
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
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(1)Expansion(σ =τ 2)
R P H
Point substitution processes for decagonal tilings
N. Fujita, Acta Cryst. A 65, 342 (2009)
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(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)
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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)
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(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)
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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)
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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)
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N. Fujita, Acta Cryst. A 65, 342 (2009)
Window
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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)
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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
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|||| 1eτ
Windows
T *(2F)
F-typeT *(P)
P-type
|||| 1e
Point density
P
PW
Ω P
P
F
F WW
Ω=
Ω 2
3τ
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)111000(:3
)111111(2
1:2
)000000(:1
S
Ih-star
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Point inflation rule(viewed in the external space)
seed 1st iteration 2nd iteration
τ× τ×
Q∞∞∞∞(S,ττττ)
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)111000(:3
)111111(2
1:2
)000000(:1
S
T
Ih-star(mapped to the internal space)
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window
in the internal space
)111000(:3
)111111(2
1:2
)000000(:1
2.618031
11 2
154321 ==
−=++++++ −
−−−−− ττ
τττττ L
|||| 1e
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5-fold direction
Q∞∞∞∞(S,ττττ)∩∩∩∩MP
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Inflation rules by τ 3 scalingT. Ogawa, J. Phys. Soc. Jpn. 54, 3205 (1985).
Inflation rule of the Ammann-Kramer tiling
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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
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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
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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
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M. Mihalkovic et al., Phys. Rev. B 53, 9002-9020 (1996).
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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
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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)
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5-fold view
Point sutstitution processes for icosahedral CCTs
Ih-starthe magic star
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The Ih-star is placed on every vertex of the expanded CCT
(scaling ratio=τ 3)
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Fix the center to bethe (68)0 type node.
The Ih-star
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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)
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1 vertex0 A-cell2 B-cells2 C-cells0 D-cell
77 vertices192 A-cells76 B-cells76 C-cells14 D-cells
BC-packing(rhombohedral)
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1 vertex0 A-cells0 B-cells0 C-cells2 D-cells
(67)333 The center is missing!
D-packing(simple hexagonal)
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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)
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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
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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.