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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.