AkijiYamamotoYAMAMOTO.Akiji@nims.go.jp National Institute for Materials Science, Tsukuba, Ibaraki, 305, Japan

Wyckoff positions of Space groups for Dihedral quasicrystals II Dodecagonal groups

Wyckoff positions of dodecagonal space groups are given.

Abstract: Wyckoff positions of five-dimensional dodecagonal space groups are calculated to facilitate the model building and structure analysis of quasicrystals. There are 23 (6 centrosymmetric and 17 non-centrosymmetric) space groups. The orbit of Wyckoff positions are available in the web site (http://wcp-ap.eng.hokudai.ac.jp/yamamoto/)

1 Introduction

Dodecagonal quasicrystals (ddQCs) are found in alloys, tellurides and soft matters with differenct scale.[7, 2, 3, 5, 8, 6] They are related with the dodecagonal tiling.[4] In particular triangle-square tiling appear frequently. This has a fractal occupation domains.[1, 10] This simple structure is generated by a fractal OD located at the origin. However, in more complicated structures, ODs may be located at several different Wyckoff positions. In general, the quasi-peiodic arrangement of atom clusters leads to a complecated model consisting of several ODs at different Wyckoff positions. For example, in tantalum tellurides, atom positions are generated by occupation domains which are located at several different Wyckoff positions.[10] All dodecagonal spacegroups are known[9] but their Wyckoff positions are not shown. Therefore, a table of Wyckoff positions will facilitate the modeling of such quasicrystals. In this paper, Wyckoff positions of 5D dodecagonal space groups are shown.

2 Calculation Method

The program Carat implemented in the program package GAP is applied to the dodecagonal space groups. This requires only the matrix representation of the point group and calculates the normalizer for classifies the generated space groups. All the possible space groups in dodecagonal quasicrystals are known and tabulated.[9] However, the known tables are calculated in a different equivalence conditions, where the scaling transformations of the dodecagonal lattice is not taken into account. As a result, the classification is finer than that in the present paper. (The known tables includes more space groups than those in the present paper as discussed later.)

3 Discussion

Table 1: The generators of the dodecagonal space groups (excluding those for lattice translations) employed in Tables 2-7. [R₁₂≡ y,z,u,−x+z,v, R₂₁≡ σ_y≡ x,y−u,x−z,−u,−v,σ_z=x,y,z,u,−v, I≡ −x,−y,−z,−u,−v, τ₀=(0,0,0,0,0),τ₁=(0,0,0,0,1)/2, τ₂=(0,0,0,0,1)/12, τ₃=(0,0,0,0,1)/6, τ₄=(0,0,0,0,1)/4, τ₅=(0,0,0,0,1)/3] Note that P12₁22(12⁵22) is equivalent to P12₃22(12⁵22) and P12₂22(12⁵22) to P12₄22(12⁵22) since they are related by the scaling transformation of the decagonal lattice.

Space group generators

P12/mmm(12⁵1mm) {R₁₂|τ₀},{R₂₁|τ₀},{I|τ₀}

P12/mmc(12⁵1mm) {R₁₂|τ₀},{R₂₁|τ₁},{I|τ₀}

P12₆/mmm(12⁵1mm) {R₁₂|τ₁},{R₂₁|τ₀},{I|τ₀}

P12mm(12⁵mm) {R₁₂|τ₀},{R₂₁|τ₀},{I|τ₀}

P12mc(12⁵mm) {R₁₂|τ₀},{R₂₁|τ₁},{I|τ₀}

P12₆mm(12⁵mm) {R₁₂|τ₁},{R₂₁|τ₀},{I|τ₀}

P12 22(12⁵mm) {R₁₂|τ₀},{R₂₁|τ₀}

P12₁22(12⁵mm) {R₁₂|τ₂},{R₂₁|τ₀}

P12₂22(12⁵mm) {R₁₂|τ₃},{R₂₁|τ₀}

P12₃22(12⁵mm) {R₁₂|τ₄},{R₂₁|τ₀}

P12₄22(12⁵mm) {R₁₂|τ₅},{R₂₁|τ₀}

P12₆22(12⁵mm) {R₁₂|τ₁},{R₂₁|τ₀}

P12/m(12⁵1) {R₁₂|τ₀},{σ_z|τ₀}

P12₆/m(12⁵1) {R₁₂|τ₀},{σ_z|τ₀}

P12(12⁵) {R₁₂|τ₀}

P12₁(12⁵) {R₁₂|τ₂}

P12₂(12⁵) {R₁₂|τ₃}

P12₃(12⁵) {R₁₂|τ₄}

P12₄(12⁵) {R₁₂|τ₅}

P12₆(12⁵) {R₁₂|τ₁}

P122m(12¹¹mm) {IR₁₂|τ₀},{R₂₁|τ₀}

P122c(12¹¹mm) {IR₁₂|τ₀},{R₂₁|τ₁}

P12(12¹¹) {IR₁₂|τ₀}

Table 2: Wyckoff positions of P12/mmm(12⁵1mm)

W.S. site symmetry coordinates

1a xxx (0,0,0,0,0)

1b xxx (0,0,0,0,1/2)

3a xxx (0,1/2,1/2,0,0)

3b xxx (0,1/2,1/2,0,1/2)

4a xxx (1/3,2/3,2/3,1/3,0)

4b xxx (1/3,2/3,2/3,1/3,1/2)

4c xxx (0,2/3,0,1/3,0)

4d xxx (0,2/3,0,1/3,1/2)

6a xxx (0,1/2,0,0,0)

6b xxx (0,1/2,0,0,1/2)

6c xxx (1/2,0,0,1/2,0)

6d xxx (1/2,0,0,1/2,1/2)

2a xxx (0,0,0,0,x)

6e xxx (0,1/2,1/2,0,x)

8a xxx (0,2/3,0,1/3,x)

8b xxx (1/3,2/3,2/3,1/3,x)

12a xxx (1/2,0,0,1/2,x)

12b xxx (0,1/2,0,0,x)

12c xxx (0,x,y,0,0)

12d xxx (0,x,y,0,1/2)

12e xxx (x,y,y,x,0)

12f xxx (x,y,y,x,1/2)

24a xxx (x,y,y,x,z)

24b xxx (0,x,y,0,z)

24c xxx (x,y,z,u,0)

24d xxx (x,y,z,u,1/2)

48a xxx (x,y,z,u,v)

Table 3: Wyckoff positions of P12/mmc(12⁵1mm)

W.S. site symmetry coordinates

2a xxx (0,0,0,0,1/4)

2b xxx (0,0,0,0,0)

6a xxx (0,1/2,1/2,0,1/4)

6b xxx (1/2,0,1/2,1/2,0)

8a xxx (1/3,2/3,2/3,1/3,1/4)

8b xxx (0,2/3,0,1/3,0)

8c xxx (2/3,2/3,1/3,1/3,0)

8d xxx (0,2/3,0,1/3,1/4)

12a xxx (0,0,0,1/2,0)

12b xxx (0,0,1/2,1/2,0)

12c xxx (1/2,0,0,1/2,1/4)

12d xxx (0,1/2,0,0,1/4)

4a xxx (0,0,0,0,x)

12e xxx (1/2,0,1/2,1/2,x)

16a xxx (0,2/3,0,1/3,x)

16b xxx (2/3,2/3,1/3,1/3,x)

24a xxx (0,0,0,1/2,x)

24b xxx (0,0,1/2,1/2,x)

24c xxx (0,x,y,0,1/4)

24d xxx (x,y,y,x,1/4)

24e xxx (x,y,z,u,0)

48a xxx (x,y,z,u,v)

48b xxx (x,y,z,u,v)

48c xxx (x,y,z,u,v)

Table 4: Wyckoff positions of P12₆/mcm(12⁵1mm)

W.S. site symmetry coordinates

2a xxx (0,0,0,0,1/2)

2b xxx (0,0,0,0,1/4)

4a xxx (0,2/3,0,1/3,1/2)

4b xxx (0,2/3,0,1/3,0)

6a xxx (0,1/2,0,0,1/2)

6b xxx (0,1/2,0,0,0)

6c xxx (0,1/2,1/2,0,1/2)

6d xxx (0,1/2,1/2,0,1/4)

8a xxx (1/3,2/3,2/3,1/3,1/4)

8b xxx (2/3,2/3,1/3,1/3,1/2)

12a xxx (0,0,1/2,1/2,1/2)

12b xxx (1/2,0,0,1/2,1/4)

4c xxx (0,0,0,0,x)

8c xxx (0,2/3,0,1/3,x)

12c xxx (0,1/2,0,0,x)

12d xxx (0,1/2,1/2,0,x)

16a xxx (2/3,2/3,1/3,1/3,x)

24a xxx (0,0,1/2,1/2,x)

12e xxx (0,x,y,0,1/2)

12f xxx (0,x,y,0,0)

24b xxx (x,y,y,x,1/4)

24c xxx (0,x,y,0,z)

24d xxx (x,y,z,u,1/2)

48a xxx (x,y,z,u,v)

48b xxx (x,y,z,u,v)

Table 5: Wyckoff positions of P12mm(12⁵mm)

W.S. site symmetry coordinates

1a xxxx (0,0,0,0,x)

3a xxxx (0,1/2,1/2,0,x)

4a xxxx (1/3,2/3,2/3,1/3,x)

4b xxxx (0,2/3,0,1/3,x)

6a xxxx (0,1/2,0,0,x)

6b xxxx (1/2,0,0,1/2,x)

12a xxxx (x,y,y,x,z)

12b xxxx (0,x,y,0,z)

24a xxxx (x,y,z,u,v)

Table 6: Wyckoff positions of P12mm(12⁵mm)

W.S. site symmetry coordinates

2a xxxx (0,0,0,0,x)

6a xxxx (1/2,0,1/2,1/2,x)

8a xxxx (0,2/3,0,1/3,x)

8b xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,0,1/2,1/2,x)

24a xxxx (x,y,z,u,v)

Table 7: Wyckoff positions of P12mm(12⁵mm)

W.S. site symmetry coordinates

2a xxxx (0,0,0,0,x)

4a xxxx (0,2/3,0,1/3,x)

6a xxxx (0,1/2,0,0,x)

6b xxxx (0,1/2,1/2,0,x)

8a xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (0,0,1/2,1/2,x)

12b xxxx (0,x,y,0,z)

24a xxxx (x,y,z,u,v)

Table 8: Wyckoff positions of P1222(12⁵mm)

W.S. site symmetry coordinates

1a xxxx (0,0,0,0,0)

1b xxxx (0,0,0,0,1/2)

3a xxxx (0,1/2,1/2,0,0)

3b xxxx (0,1/2,1/2,0,1/2)

4a xxxx (2/3,2/3,1/3,1/3,0)

4b xxxx (2/3,2/3,1/3,1/3,1/2)

4c xxxx (2/3,0,1/3,0,0)

4d xxxx (2/3,0,1/3,0,1/2)

6a xxxx (0,0,1/2,0,0)

6b xxxx (0,0,1/2,0,1/2)

6c xxxx (1/2,0,0,1/2,0)

6d xxxx (1/2,0,0,1/2,1/2)

2a xxxx (0,0,0,0,x)

6e xxxx (1/2,0,1/2,1/2,x)

8a xxxx (0,2/3,0,1/3,x)

8b xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,0,1/2,1/2,x)

12c xxxx (x,y,−y,−x,0)

12d xxxx (x,y,−y,−x,1/2)

12e xxxx (0,y,x,0,0)

12f xxxx (0,y,x,0,1/2)

24a xxxx (x,y,z,u,v)

Table 9: Wyckoff positions of P1222(12⁵mm)

W.S. site symmetry coordinates

24a xxxx (x,y,−y,−x,5/8)

12a xxxx (0,y,x,0,0)

24b xxxx (x,y,z,u,v)

Table 10: Wyckoff positions of P1222(12⁵mm)

W.S. site symmetry coordinates

6a xxxx (0,0,0,0,0)

6b xxxx (0,0,1/2,0,0)

6c xxxx (0,0,1/2,0,1/2)

6d xxxx (0,1/2,1/2,0,0)

6e xxxx (0,0,0,0,1/4)

6f xxxx (1/2,0,0,1/2,1/4)

6g xxxx (1/2,0,0,1/2,3/4)

6h xxxx (0,1/2,1/2,0,1/4)

12a xxxx (0,0,0,0,x)

12b xxxx (0,0,0,1/2,x)

12c xxxx (0,0,1/2,1/2,x)

12d xxxx (0,1/2,1/2,0,x)

12e xxxx (x,y,−y,−x,1/4)

12f xxxx (x,y,−y,−x,3/4)

12g xxxx (0,y,x,0,0)

12h xxxx (0,y,x,0,1/2)

24a xxxx (x,y,z,u,v)

Table 11: Wyckoff positions of P1222(12⁵mm)

W.S. site symmetry coordinates

4a xxxx (0,0,0,0,7/8)

4b xxxx (2/3,2/3,1/3,1/3,7/8)

4c xxxx (2/3,2/3,1/3,1/3,3/8)

4d xxxx (0,0,0,0,0)

4e xxxx (2/3,0,1/3,0,0)

4f xxxx (2/3,0,1/3,0,1/2)

8a xxxx (0,0,0,0,x)

8b xxxx (0,2/3,0,1/3,x)

8c xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (x,y,−y,−x,7/8)

12b xxxx (0,y,x,0,0)

24a xxxx (x,y,z,u,v)

Table 12: Wyckoff positions of P1222(12⁵mm)

W.S. site symmetry coordinates

3a xxxx (0,0,0,0,0)

3b xxxx (0,0,0,0,1/2)

3c xxxx (0,1/2,1/2,0,0)

3d xxxx (0,1/2,1/2,0,1/2)

6a xxxx (0,0,1/2,0,0)

6b xxxx (0,0,1/2,0,1/2)

6c xxxx (1/2,0,0,1/2,1/2)

6d xxxx (1/2,0,0,1/2,0)

6e xxxx (0,0,0,0,x)

6f xxxx (1/2,0,1/2,1/2,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,0,1/2,1/2,x)

12c xxxx (x,y,−y,−x,1/2)

12d xxxx (x,y,−y,−x,0)

12e xxxx (0,y,x,0,0)

12f xxxx (0,y,x,0,1/2)

24a xxxx (x,y,z,u,v)

Table 13: Wyckoff positions of P1222(12⁵mm)

W.S. site symmetry coordinates

2a xxxx (0,0,0,0,0)

2b xxxx (0,0,0,0,3/4)

4a xxxx (2/3,2/3,1/3,1/3,3/4)

4b xxxx (2/3,2/3,1/3,1/3,1/4)

4c xxxx (2/3,0,1/3,0,0)

4d xxxx (2/3,0,1/3,0,1/2)

6a xxxx (0,0,1/2,0,0)

6b xxxx (0,0,1/2,0,1/2)

6c xxxx (0,1/2,1/2,0,0)

6d xxxx (1/2,0,0,1/2,3/4)

6e xxxx (1/2,0,0,1/2,1/4)

6f xxxx (0,1/2,1/2,0,3/4)

4e xxxx (0,0,0,0,x)

8a xxxx (0,2/3,0,1/3,x)

8b xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,0,1/2,1/2,x)

12c xxxx (0,1/2,1/2,0,x)

12d xxxx (x,y,−y,−x,3/4)

12e xxxx (x,y,−y,−x,1/4)

12f xxxx (0,y,x,0,0)

12g xxxx (0,y,x,0,1/2)

24a xxxx (x,y,z,u,v)

Table 14: Wyckoff positions of P12/m(12⁵1)

W.S. site symmetry coordinates

1a xxxx (0,0,0,0,0)

1b xxxx (0,0,0,0,1/2)

3a xxxx (1/2,0,1/2,1/2,0)

3b xxxx (1/2,0,1/2,1/2,1/2)

4a xxxx (0,2/3,0,1/3,0)

4b xxxx (0,2/3,0,1/3,1/2)

4c xxxx (2/3,2/3,1/3,1/3,0)

4d xxxx (2/3,2/3,1/3,1/3,1/2)

6a xxxx (0,0,0,1/2,0)

6b xxxx (0,0,0,1/2,1/2)

6c xxxx (0,0,1/2,1/2,0)

6d xxxx (0,0,1/2,1/2,1/2)

2a xxxx (0,0,0,0,x)

6e xxxx (1/2,0,1/2,1/2,x)

8a xxxx (0,2/3,0,1/3,x)

8b xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,0,1/2,1/2,x)

12c xxxx (x,y,z,u,0)

12d xxxx (x,y,z,u,1/2)

24a xxxx (x,y,z,u,v)

Table 15: Wyckoff positions of P12/m(12⁵1)

W.S. site symmetry coordinates

2a xxxx (0,0,0,0,0)

2b xxxx (0,0,0,0,3/4)

4a xxxx (0,2/3,0,1/3,0)

4b xxxx (0,2/3,0,1/3,1/2)

4c xxxx (2/3,2/3,1/3,1/3,0)

4d xxxx (2/3,2/3,1/3,1/3,1/2)

6a xxxx (0,0,0,1/2,0)

6b xxxx (0,0,0,1/2,1/2)

6c xxxx (0,0,1/2,1/2,0)

6d xxxx (0,0,1/2,1/2,1/2)

6e xxxx (0,1/2,1/2,0,0)

6f xxxx (1/2,0,1/2,1/2,3/4)

4e xxxx (0,0,0,0,x)

8a xxxx (0,2/3,0,1/3,x)

8b xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,0,1/2,1/2,x)

12c xxxx (0,1/2,1/2,0,x)

12d xxxx (x,y,z,u,0)

24a xxxx (x,y,z,u,v)

Table 16: Wyckoff positions of P12(12⁵)

W.S. site symmetry coordinates

1a xxxx (0,0,0,0,x)

3a xxxx (1/2,0,1/2,1/2,x)

4a xxxx (0,2/3,0,1/3,x)

4b xxxx (2/3,2/3,1/3,1/3,x)

6a xxxx (0,0,0,1/2,x)

6b xxxx (0,0,1/2,1/2,x)

12a xxxx (x,y,z,u,v)

Table 17: Wyckoff positions of P12(12⁵)

W.S. site symmetry coordinates

12a xxxx (x,y,z,u,v)

Table 18: Wyckoff positions of P12(12⁵)

W.S. site symmetry coordinates

6a xxxx (0,0,0,0,x)

6b xxxx (0,0,0,1/2,x)

6c xxxx (0,0,1/2,1/2,x)

6d xxxx (0,1/2,1/2,0,x)

12a xxxx (x,y,z,u,v)

Table 19: Wyckoff positions of P12(12⁵)

W.S. site symmetry coordinates

4a xxxx (0,0,0,0,x)

4b xxxx (0,2/3,0,1/3,x)

4c xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (x,y,z,u,v)

Table 20: Wyckoff positions of P12(12⁵)

W.S. site symmetry coordinates

3a xxxx (0,0,0,0,x)

3b xxxx (1/2,0,1/2,1/2,x)

6a xxxx (0,0,0,1/2,x)

6b xxxx (0,0,1/2,1/2,x)

12a xxxx (x,y,z,u,v)

Table 21: Wyckoff positions of P12(12⁵)

W.S. site symmetry coordinates

2a xxxx (0,0,0,0,x)

4a xxxx (0,2/3,0,1/3,x)

4b xxxx (2/3,2/3,1/3,1/3,x)

6a xxxx (0,0,0,1/2,x)

6b xxxx (0,0,1/2,1/2,x)

6c xxxx (0,1/2,1/2,0,x)

12a xxxx (x,y,z,u,v)

Table 22: Wyckoff positions of P122m(12mm)

W.S. site symmetry coordinates

1a xxxx (0,0,0,0,0)

1b xxxx (0,0,0,0,1/2)

3a xxxx (0,1/2,1/2,0,0)

3b xxxx (0,1/2,1/2,0,1/2)

4a xxxx (2/3,0,1/3,0,0)

4b xxxx (2/3,0,1/3,0,1/2)

6a xxxx (0,0,1/2,0,0)

6b xxxx (0,0,1/2,0,1/2)

2a xxxx (0,0,0,0,x)

4c xxxx (1/3,2/3,2/3,1/3,x)

6c xxxx (1/2,0,0,1/2,x)

6d xxxx (0,1/2,1/2,0,x)

8a xxxx (0,2/3,0,1/3,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,y,x,0,0)

12c xxxx (0,y,x,0,1/2)

12d xxxx (x,y,y,x,z)

24a xxxx (x,y,z,u,v)

Table 23: Wyckoff positions of P122m(12mm)

W.S. site symmetry coordinates

2a xxxx (0,0,0,0,3/4)

2b xxxx (0,0,0,0,0)

4a xxxx (2/3,0,1/3,0,3/4)

4b xxxx (2/3,0,1/3,0,1/4)

6a xxxx (0,0,1/2,0,3/4)

6b xxxx (0,0,1/2,0,1/4)

6c xxxx (0,1/2,1/2,0,3/4)

6d xxxx (1/2,0,1/2,1/2,0)

4c xxxx (0,0,0,0,x)

8a xxxx (0,2/3,0,1/3,x)

8b xxxx (2/3,2/3,1/3,1/3,x)

12a xxxx (0,0,0,1/2,x)

12b xxxx (0,0,1/2,1/2,x)

12c xxxx (0,1/2,1/2,0,x)

12d xxxx (0,y,x,0,3/4)

12e xxxx (0,y,x,0,1/4)

24a xxxx (x,y,z,u,v)

Table 24: Wyckoff positions of P12(12)

W.S. site symmetry coordinates

1a xxxx (0,0,0,0,0)

1b xxxx (0,0,0,0,1/2)

3a xxxx (1/2,0,1/2,1/2,0)

3b xxxx (1/2,0,1/2,1/2,1/2)

2a xxxx (0,0,0,0,x)

4a xxxx (0,2/3,0,1/3,x)

4b xxxx (2/3,2/3,1/3,1/3,x)

6a xxxx (0,0,0,1/2,x)

6b xxxx (0,0,1/2,1/2,x)

6c xxxx (0,1/2,1/2,0,x)

12a xxxx (x,y,z,u,v)

References

[1]: M. ‍Baake, R. ‍Klitzing, and M. ‍Schlottman. Fractally shaped acceptance domains of quasiperiodic square-triangle tilings with dodecagonal symmetry . Physica, A191:554–558, 1992.
[2]: M. ‍Conrad and B. ‍Harbrecht. Structure properties of Ta₉₇Te₆₀ and Ta₁₈₁Te₁₁₂, two approximants of a dodecagonal tantalum telluride. In M. ‍Boissieu, J. ‍L. Verger-Gaugry, and R. ‍Currat, editors, Aperiodic ’97, pages 205–209, Singapore, 1998. World Scientific.
[3]: M. ‍Conrad, B. ‍Harbrecht, and F. ‍Kurumeich. Structure Properties of a Dodecagonal Titanium Telluride quasicrystal. In Aperiodic ’97, pages 199–203, Singapore, 1998. World Scientific.
[4]: F. ‍Gähler and J. ‍Rhyner. Equivalence of the generalized grid and projection method for the construction of quasiperiodic tilings. J. Phys., A19:267–277, 1986.
[5]: K. ‍Hayashida, T. ‍Dotera, A. ‍Takano, and Y. ‍Matsushita. Polymeric quasicrystal: mesoscopic quasicrystalline tiling in abc star polymers. Phys. Rev. Lett., 98:195502, 2007.
[6]: C. ‍R. Iacovella, A. ‍K. Keys, and S. ‍C. Glotzer. Self-assembly of soft-matter quasicrystals and their approximants. PNAS, 108:20935–20940, 2011.
[7]: T. ‍Ishimasa, H. ‍U. Nissen, and Y. ‍Fukano. New ordered state between crystalline and amorphous in Ni-Cr particles. Phys. Rev. Lett., 55:511–513, 1985.
[8]: S. ‍Iwami and T. ‍Ishimasa. Dodecagonal quasicrystal in Mn-based quaternary alloys containing Cr, Ni, and Si. Phil. Mag. Lett., 95(4):229–236, 2015.
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[10]: A. ‍Yamamoto. Dodecagonal Ta-Te quasicrystal and a 5-dimensional model with fractal occupation domains . Acta Crystallogr., A60:142–145, 2004.

This document was translated from L^AT_EX by H^EV^EA.

Space group	generators
P12/mmm(12⁵1mm)	{R₁₂\|τ₀},{R₂₁\|τ₀},{I\|τ₀}
P12/mmc(12⁵1mm)	{R₁₂\|τ₀},{R₂₁\|τ₁},{I\|τ₀}
P12₆/mmm(12⁵1mm)	{R₁₂\|τ₁},{R₂₁\|τ₀},{I\|τ₀}
P12mm(12⁵mm)	{R₁₂\|τ₀},{R₂₁\|τ₀},{I\|τ₀}
P12mc(12⁵mm)	{R₁₂\|τ₀},{R₂₁\|τ₁},{I\|τ₀}
P12₆mm(12⁵mm)	{R₁₂\|τ₁},{R₂₁\|τ₀},{I\|τ₀}
P12 22(12⁵mm)	{R₁₂\|τ₀},{R₂₁\|τ₀}
P12₁22(12⁵mm)	{R₁₂\|τ₂},{R₂₁\|τ₀}
P12₂22(12⁵mm)	{R₁₂\|τ₃},{R₂₁\|τ₀}
P12₃22(12⁵mm)	{R₁₂\|τ₄},{R₂₁\|τ₀}
P12₄22(12⁵mm)	{R₁₂\|τ₅},{R₂₁\|τ₀}
P12₆22(12⁵mm)	{R₁₂\|τ₁},{R₂₁\|τ₀}
P12/m(12⁵1)	{R₁₂\|τ₀},{σ_z\|τ₀}
P12₆/m(12⁵1)	{R₁₂\|τ₀},{σ_z\|τ₀}
P12(12⁵)	{R₁₂\|τ₀}
P12₁(12⁵)	{R₁₂\|τ₂}
P12₂(12⁵)	{R₁₂\|τ₃}
P12₃(12⁵)	{R₁₂\|τ₄}
P12₄(12⁵)	{R₁₂\|τ₅}
P12₆(12⁵)	{R₁₂\|τ₁}
P122m(12¹¹mm)	{IR₁₂\|τ₀},{R₂₁\|τ₀}
P122c(12¹¹mm)	{IR₁₂\|τ₀},{R₂₁\|τ₁}
P12(12¹¹)	{IR₁₂\|τ₀}

W.S.	site symmetry	coordinates
1a	xxx	(0,0,0,0,0)
1b	xxx	(0,0,0,0,1/2)
3a	xxx	(0,1/2,1/2,0,0)
3b	xxx	(0,1/2,1/2,0,1/2)
4a	xxx	(1/3,2/3,2/3,1/3,0)
4b	xxx	(1/3,2/3,2/3,1/3,1/2)
4c	xxx	(0,2/3,0,1/3,0)
4d	xxx	(0,2/3,0,1/3,1/2)
6a	xxx	(0,1/2,0,0,0)
6b	xxx	(0,1/2,0,0,1/2)
6c	xxx	(1/2,0,0,1/2,0)
6d	xxx	(1/2,0,0,1/2,1/2)
2a	xxx	(0,0,0,0,x)
6e	xxx	(0,1/2,1/2,0,x)
8a	xxx	(0,2/3,0,1/3,x)
8b	xxx	(1/3,2/3,2/3,1/3,x)
12a	xxx	(1/2,0,0,1/2,x)
12b	xxx	(0,1/2,0,0,x)
12c	xxx	(0,x,y,0,0)
12d	xxx	(0,x,y,0,1/2)
12e	xxx	(x,y,y,x,0)
12f	xxx	(x,y,y,x,1/2)
24a	xxx	(x,y,y,x,z)
24b	xxx	(0,x,y,0,z)
24c	xxx	(x,y,z,u,0)
24d	xxx	(x,y,z,u,1/2)
48a	xxx	(x,y,z,u,v)