点集合置換法による正二十面体対称準周期タイリングの作成 (準周期秩序の数理)

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1 (Nobuhisa Fujita) Institute of Multidisciplinary Research for Advanced Materials, Tohoku University 1. (Dirac peak) (Z-module) $d$ (rank) $r$ r $\backslash$ (Bravais lattice) $d$ $d$ $r$ (window) (cut-and-projection method) $n$ $\varphi(n)(n$ $n$ ) $=$ $r=6$ 53 $2/m$ (Hermann-Mauguin ) 1 I ( $p$ $F$ ) [1-3] ( ) - (Ammann-Kramer tiling) [4, 5] (rhombic Penrose tiling)[6, 7] ($3D$ Penrose tiling) (Ammann rhombohedra) (1 ) - $P$ (rhombic O 2 1 Sch\"onflies 2 ( )

2 2 triacontahedron) [4] 3 $-$ $(D6)$ $T$ [8] (Danzer tiling) [9, 10] () $F$ ( ) 1: $($acute rhombohedron A $R)$ (obtuse rhombohedron OR) (3 ) $e,$ $(j=1,2, \ldots, 6)$ (section method) $($dual-grid $method)$ (substitution) [11] - ( ) ( $)$ (atomic surface) [12] ( ) ( ) ( ) (pentagonal Penrose ti $ ing$ ) $[13]$ (Stampfli tiling)[14, 15] ( ) 3

3 $\Sigma$ 3 iterated function system $[16-2I]$ 2. $)$ ( $\eta$ $Z[_{P}]$ Pisot $p$ $(\eta=p, p^{2}, \ldots)$ [20, 21] $G$ $S1$ : $\eta$ $\eta\sigma$ $S2$ : $S$ Gsymmetric ( ) 4 $\eta\sigma+s$ $\Sigma $ $S3$ : $\Sigma \subset\eta\sigma+$ S. [20, 21]. 4 $+$ $A+B\equiv\{a+b ^{\forall}a\in A$ $\forall_{b\in B\}}$,

4 $\tau^{3}$ ZP $\tau$ ZF 4 (4 ) 3. Z $P$ $F$ I [1-3] $Z_{P\text{}}Z_{F\text{}}$ $Z_{1}$ $e_{l}(i=1,2,\ldots, 6)$ $(e_{1}$ $e_{2}$ $e_{3}$ $e_{4}$ $e_{5}$ $e_{6})=\{\begin{array}{llllll}\tau \tau 0l r 0 0 l -\tau 0 1 \tau -\tau 0 1\end{array}\}$ $Z_{P}$ $Z_{F}$ $Z_{P}$ $Z_{1}$ ZP ZP $(e_{1}+e_{2}+\ldots+e_{6})/2$ $Z_{P}=\{n_{I}e_{1}+n_{2}e_{2}+n_{3}e_{3}+n_{4}e_{4}+n_{5}e_{5}+n_{6}e_{6} (n_{/})\in Z^{6}\}$, $Z_{F}=\{n_{1}e_{1}+n_{2}e_{2}+n_{3}e_{3}+n_{4}e_{4}+n_{5}e_{5}+n_{6}e_{6} \sum_{/}n_{/}=0mod 2,$ $(n_{j})\in Z^{6}\}$, $Z_{l}=\{\nu_{1}e_{1}+v_{2}e_{2}+v_{3}e_{3}+v_{4}e_{4}+v_{5}e_{5}+v_{6}e_{6} (V_{j})\in Z^{6}\cup Z^{6}+\frac{1}{2}(11$ $)\}$. ZP ZF $Z_{1}$ $Z[\tau^{3]\text{}}Z[_{T}]$ $Z[_{T}]$ ( $\tau=$(1 ;5)/2 ) [1-3] $+$ $=$ $\tau$ $\tau$3 $\tau$ ( $\tau Z_{1}=$ ZP ZF ZI) $\eta$ $=$ 4. (icosahedral quasic $\gamma$stals)

$5 (2) $P$ Cd-Yb [22] - $+$ ( ) $5[23]_{0}$ $c=(\sqrt 3/2)b$ $b=2(\tau^{3}/\sqrt 5)^{1/2}a_{R}$ $b$ $a_{r}$ $c$ $+$ [23L $[24-26]_{0}$ Cd-Yb$

5 5 (2) $P$ Cd-Yb [22] - $+$ ( ) $5[23]_{0}$ $c=(\sqrt 3/2)b$ $b=2(\tau^{3}/\sqrt 5)^{1/2}a_{R}$ $b$ $a_{r}$ $c$ $+$ [23L $[24-26]_{0}$ Cd-Yb 2: Cd-Yb [22]O A $R$ ( ) A Cd ( ) $R$ $\tau$ ( ) $\tau^{-\delta}:\tau^{-3}:\tau^{-2}\circ$ Yb () ( ) OR ( ) $OR$ Cd $(RT)$ () RT Cd ( ) 5 $+$

6 6 (canonical cells) (A $B$ $C$ $D$ ) (3) ( $ $ ) [27] $b$ $c$ $c=(\sqrt 3/2)b$ [22] $RT$ A $R$ [28]. OR 3: ( ) $b$ $c$ ( ) [29] [30] 6 ( ) [29] 44,200 [31] 7 [32]

7 $\eta$ 7 $o^{}$ face-t face 32 [27] 32 $S$ $P$ $\tau^{3}$ $\eta=\tau^{3}$ $\eta$ $S$ ZP 363 (4) 10 $5\overline{3}2/m$ ( ) 1 1 $12$ ZP $20$ $30$ 60 1: S (4) 4: S $t\rceil$ $S$

8 $\tau^{3}$ 8 A-packing $(Im\overline{3})$, BC-packing $(R\overline{3}m)$, D-packing $(P\overline{3}m1)$, 2/1 cubic packing $(Pa\overline{3})$ [27] (Hermann-Mauguin ) A-packing A-packing A $(b.c.c.)$ A bcc (100) ( $b$ ) $c$ (5) A 8A-packing $b$ $c$ (68)0 ( [27] ) 5: $(b. c. c. )$ A A $\eta=\tau^{3}$ $S1$ A-packing $S2$ $S$ 6 $S2$ $S$ $3$ $G1$ : $S2$ $S$ ( 1 ) (68)0 ( (68)0 8 [27] A-packing 3 1/1 (1/1 ) $A$.packing 1/1 cubic packing

9 9 ) (Fixed) (Occupied) $G2$ : Fixed Occupied Fixed Occupied $c$ (Unoccupied) Fixed Unlabelled OccupiedUnoccupied G3: S2 Occupied Unoccupied 32 ( ) Occupied Fixed $G2$ $G4$ : Fixed Unoccupied $G3$ Fixed $G5$ : $G3$ Occupied ( ) $G1$ (68)0 $G6$ : $G3$ Occupied 9 $($ A-packing (68) $G1)$ $($ $G2)$ 6 Occupied ( ) (67)333 (6 ) Unlabelled ( ) Occupied () 9

$$\tau^{3}$ $\cross$ $\cross$ 10 Occupied (67)333 (76) (67)333 ( 6 ) $G3$ Unlabelled (6 ) $G3$ Fixed Unoccupied$

10 $\tau^{3}$ $\cross$ $\cross$ 10 Occupied (67)333 (76) (67)333 ( 6 ) $G3$ Unlabelled (6 ) $G3$ Fixed Unoccupied (6 ) $S3$ A-packing $\tau^{3}b$ $\cross$ 136 $D$ $\cross 138$ $\cross$ A 348 $B$ 136 $C$ : 6. $S1$ -S 3 ( )

11 $J^{\cdot}$ $:_{\backslash }$ $\alpha^{t}$ 11 $*.\iota$.. $\vee\sim\iota$. $\cdot$ $ \nearrow$. $\backslash -\bullet\sim-\overline{.\backslash }\backslash ---/ $ $arrow\perp.-arrow$.

12 12 6 : A-packing $S1$ $S2$ ( ) ( ) $\sim$ $c$

13 13 ( ) (68) (Fixed) (Occupied) (Unoccupied) $c$ Occupied ( ) Occupied (67)333 $G3$ $G3$ 1. T. Janssen, Acta Cryst. A 42, 261 (1986). 2. L. S. Levitov and J. Rhyner, Journal de Physique 49, 1835 (1988). 3. D. S. Rokhsar, N. D. Mermin and D. C. Wright, Phys. Rev. $B35$, 5487 (1987). 4. M. Duneau and A. Katz, Phys. Rev. Lett. 54, 2688 (1985). 5. P. Kramer and R. Neri, Acta Clyst. A 40, 580 (1984). 6. M. Gardner, Sci. Am. 236,110 (1977). 7. N. G. de Bruijn, $Ned$. Akad. Weten. Proc. Ser 84, 39 (1981). $A$ 8. P. Kramer, Z. Papadopolos and D. Zeidler, in $AIP$ Conference Proceedings 266, ed. $by$ A. Frank, T. $H$ Seligman, $K$ B. Wolf, American Institute of Physics, New York (1992) p L. Danzer, $Disc1^{\cdot}ete$ Math. 76, 1 (1989). 10. L. Danzer, in Group theoretical methods in physics: proceedings of the XVIII Interna tional Colloquium held at Moscow, USSR, Springer, Berlin (1991) p T. Janssen, G. Chapuis and M.d. Boissieu, Aperiodic Crystals $-F_{1}o1n$ Modulated Phases to Quasiciystals, IUCr Monogiaphs on Ciystallography 20, ed. International Union of Crystallography, Oxford Science Publications (2007) chapter M. Schlottmann, Int. $J$ Mod Phys. $B7$, 1351 (1993). 13. R. Penrose, Bull. Inst. Math. Appl. 10, 216 (1974). 14. P. Stampfli, Helv. Phys. Acta 59, 1260 (1986). 15. J. Hermisson, C. Richard and M. Baake, $J$ Phys. IFrance 7, 1003 (1997). 16. E. Zobetz, Acta Ciyst. A 48, 328 (1992). 17. C. Godreche et $a1,$ $J$ Phys. IFrance 3, 1921 (1993). 18. E. Cockayne, Phys. Rev. $B51$, (1995).

14 K. Niizeki, Phil. $Mag$. $87$, 2855 (2007). 20. N. Fujita, Acta Ciyst. A 65, 342 (2009). 21. N. Fujita, $J$ Phys. : $Conf$ Ser. 226, (2010). 22. H. Takakura et $\theta I$, Nature Materials 6, 58 (2007). 23. C. L. Henley, Phys. Rev. $B34,797$ (1986). 24. Z. Olami and S. Alexander, Phys. Rev. $B37$, 3973 (1988). 25. A. P. Smith, Phys. Rev. $B42$, 1189 (1990). 26. E. Cockayne, Phys. Rev. $B49$, 5896 (1994). 27. C. L. Henley, Phys. Rev $B43,993$ (1991). 28. M. Mihalkovic et al., Phys. Rev. $B53$, 9002 (1996). 29. M. Mihalkovic and P. Mrafko, Euiophys. Lett. 21, 463 (1993). 30. M. E. J. Newman, C. L. Henley and M. Oxborrow, Phil. $Mag$. $B71,991$ (1995). 31. M. Mihalkovic, unpublished result. 32. L. Danzer, unpublished result.

The Plasma Boundary of Magnetic Fusion Devices

ASAKURA Nobuyuki, Japan Atomic Energy Research Institute, Naka, Ibaraki 311-0193, Japan e-mail: asakuran@fusion.naka.jaeri.go.jp The Plasma Boundary of Magnetic Fusion Devices Naka Fusion Research Establishment,