Point substitution processes for generating icosahedral tilings. Nobuhisa Fujita IMRAM, Tohoku University, Sendai , Japan

Transcription

1 Point substitution procsss for gnrating icosahdral tilings Nobuhisa Fujita IMRAM, Tohoku Univrsity, Sndai , Japan

2 Outlin Part I. 1. Basic icosahdral tilings 2. Point substitution procsss Part II. 3. Tilings constructd with PIRs 4. Canonical cll tilings 5. Toward icosahdral CCTs

3 Outlin Part I. 1. Basic icosahdral tilings 2. Point substitution procsss Part II. 3. Tilings constructd with PIRs 4. Canonical cll tilings 5. Toward icosahdral CCTs

4 Icosahdral QCs Icosahdral tilings ED pattrn along 5-fold axis of an icosahdral quasicrystal Mn icosahdra b-c packing of icosahdral clustrs (F-typ) basd on th rhombohdral tiling (Ammann-Kramr tiling). Modl of i -(Al-Mn), M. Audir and P. Guyot, Phil. Mag. B 53, L43 (1986)

5 T *(P) tiling (Ammann-Kramr tiling) Basic tils OR, AR (Ammann rhombohdra) 5-fold viw 2-fold viw M. Dunau and A. Katz, Phys. Rv. Ltt. 54, 2688 (1985). P. Kramr and R. Nri, Acta Cryst. A 40, 580 (1984).

6 ( ) = τ τ τ τ τ τ Icosahdral basis st = τ th goldn man = 2 +τ 1

7 Icosahdral moduls } ) ( { : Ζ = j P n n n n n n n M Ζ[τ ] } ) ( 2, mod 0 { : Ζ = = j j j F n n n n n n n n M Ζ[τ ] [ 3 ] Ζ τ [1] T. Janssn, Acta Cryst. A 42 (1986) 261. [2] D. S. Rokhsar t al., Phys. Rv. B 35 (1987) [3] L.S. Lvitov and J. Rhynr, J. Physiqu 49 (1988) ( ) )} ( { : Ζ Ζ = j I ν ν ν ν ν ν ν M intgr ring

8 Scal invarianc of th moduls M I : = M = M P F M P (111111) [ M + (100000) ] M + ( ) M + (111111) F F 1 2 F 1 2 V V o B B o τ-scaling τ B V τ τ B o τ V o M M M P F I τ 3 τ = = M τ = M M I F P

9 T *(2F) tiling (Kramr t al.) P. Kramr t al., in Symmtris in Scinc V: Algbraic Structurs, thir Rprsntions, Ralizations and Physical Applications, Ed. by B. Grubr t al., Plnum Prss, Nw York, 1991, pp. 395.

10 Outlin Part I. 1. Basic icosahdral tilings 2. Point substitution procsss Part II. 3. Tilings constructd with PIRs 4. Canonical cll tilings 5. Toward icosahdral CCTs

11 Point substitution procsss for dcagonal tilings R P H N. Fujita, Acta Cryst. A 65, 342 (2009)

12 Point substitution procsss for dcagonal tilings (1)Expansion (σ =τ 2 ) R P H N. Fujita, Acta Cryst. A 65, 342 (2009)

13 Point substitution procsss for dcagonal tilings (1)Expansion (σ =τ 2 ) (2)Plac S at vry vrtx R P H N. Fujita, Acta Cryst. A 65, 342 (2009)

14 Point substitution procsss for dcagonal tilings (1)Expansion (σ =τ 2 ) (2)Plac S at vry vrtx R P H N. Fujita, Acta Cryst. A 65, 342 (2009)

15 Point substitution procsss for dcagonal tilings (1)Expansion (σ =τ 2 ) (2)Plac S at vry vrtx (3)Eliminat xcssiv points R P H N. Fujita, Acta Cryst. A 65, 342 (2009)

16 Point substitution procsss for dcagonal tilings (1)Expansion (σ =τ 2 ) (2)Plac S at vry vrtx (3)Eliminat xcssiv points R P H N. Fujita, Acta Cryst. A 65, 342 (2009)

17 Point substitution procsss for dcagonal tilings (1)Expansion (σ =τ 2 ) (2)Plac S at vry vrtx (3)Eliminat xcssiv points R P H N. Fujita, Acta Cryst. A 65, 342 (2009)

18 Window N. Fujita, Acta Cryst. A 65, 342 (2009)

19 Point Substitution Procss (for constructing icosahdral quasipriodic tilings) (1)Expansiv similarity transformation: T i σ T i (T i M, σ =ρ n, ρ=τ 3 (P), τ(f), τ(i)) (2) Rplicat th I h -star at vry vrtx: T i = σ T i + S (S M) (3) Dcimation of points by local ruls: T i T i+1 ( T i ) N. Fujita, Acta Cryst. A 65, 342 (2009) Stp (3) is ndd if thr is rdundancy in th points gnratd through (1) and (2) ( Point inflation rul) K. Niizki, J.Phys.A:Math.Thor.41, (2008)

20 Outlin Part I. 1. Basic icosahdral tilings 2. Point substitution procsss Part II. 3. Tilings constructd with PIRs 4. Canonical cll tilings 5. Toward icosahdral CCTs

21 Windows T *(P) P-typ T *(2F) F-typ 1 τ 1 Point dnsity W P Ω P 3 W F τ W = Ω 2 Ω F P P

22 I h -star S 1: (000000) 1 2 : ( : (111000) 1)

23 Point inflation rul (viwd in th xtrnal spac) sd 1 st itration 2 nd itration τ τ Q (S,τ)

24 I h -star (mappd to th intrnal spac) S T 1: (000000) 1 2 : ( : (111000) 1)

25 in th intrnal spac 1 window 1: (000000) L = 1 τ τ + τ + τ + τ + τ 1 2 = τ = : ( : (111000) 1)

26 5-fold dirction Q (S,τ) M P

27 Inflation rul of th Ammann- Kramr tiling Inflation ruls by τ 3 scaling T. Ogawa, J. Phys. Soc. Jpn. 54, 3205 (1985).

28 Outlin Part I. 1. Basic icosahdral tilings 2. Point substitution procsss Part II. 3. Tilings constructd with PIRs 4. Canonical cll tilings 5. Toward icosahdral CCTs

29 Canonical cll tiling 4 polyhdra: A-, B-, C-, D-clls Cll gomtry for clustr-basd quasicrystal modls, C. L. Hnly, Phys. Rv. B 43 (1991) 993. A-cll B-cll C-cll D-cll c c c c b c c b c b c b b b b b c c b c b b c b c c b b b Thr ar 32 classs of nods in a CCT (68) 0 (62) (67) 333

30 i-cdcanonical 5.7 Yb: Quasicrystal Cll Tiling (QC) RTH AR H. Takakura & C.P. Gomz t al., (2007). M. Mihalkovic t al., Phys. Rv. B 53, (1996).

31 M. Mihalkovic t al., Phys. Rv. B 53, (1996).

32 Outlin Part I. 1. Basic icosahdral tilings 2. Point substitution procsss Part II. 3. Tilings constructd with PIRs 4. Canonical cll tilings 5. Toward icosahdral CCTs

33 Is thr an icosahdral CCT? NO PROOF is givn of th xistnc of an ICOSAHEDRAL CCT Mthods to construct approximant CCTs (undr priodic boundary conditions) A Mont Carlo dnsity optimization mthod: M. Mihalkovic and P. Mrafko, Europhys. Ltt. 21 (1993) 463. A brut forc algorithm: M.E.J. Nwman, C.L. Hnly, and M. Oxborrow, Phil. Mag. B 71 (1995) 991.

34 Point sutstitution procsss for icosahdral CCTs I h -star th magic star 5-fold viw

35 Th I h -star is placd on vry vrtx of th xpandd CCT (scaling ratio=τ 3 )

36 Fix th cntr to b th (68) 0 typ nod. Th I h -star

37 A-packing (body cntrd cubic) 2 vrtics 12 A-clls 0 B-cll 0 C-cll 0 D-cll 138 vrtics 348 A-clls 136 B-clls 136 C-clls 24 D-clls

38 BC-packing (rhombohdral) 1 vrtx 0 A-cll 2 B-clls 2 C-clls 0 D-cll 77 vrtics 192 A-clls 76 B-clls 76 C-clls 14 D-clls

39 D-packing (simpl hxagonal) (67) 333 Th cntr is missing! 1 vrtx 0 A-clls 0 B-clls 0 C-clls 2 D-clls

40 D-packing (simpl hxagonal) (67) 333 Th cntr is missing! 1 vrtx 0 A-clls 0 B-clls 0 C-clls 2 D-clls 103 vrtics 252 A-clls 102 B-clls 102 C-clls 20 D-clls

41 τ 3 A-packing τ 3 D-packing τ 3 BC-packing 138 vrtics 348 A-clls 136 B-clls 136 C-clls 24 D-clls τ 3 2/1 cubic-packing 103 vrtics 252 A-clls 102 B-clls 102 C-clls 20 D-clls 77 vrtics 192 A-clls 76 B-clls 76 C-clls 14 D-clls 584 vrtics 1464 A-clls 576 B-clls 576 C-clls 104 D-clls

42 Conclusion Th prsnt schm has turnd out to b usful for constructing icosahdral tilings. Th magic star can gnrat all th vrtics of an inflatd CCT xcpt a point in th cntr of ach xpandd D-cll. It is likly that thr xist τ 3 -inflation ruls for gnrating an icosahdral CCT, th proof of which still nds to b workd out.