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

Wyckoff positions of Space groups for Dihedral quasicrystals I : Decagonal groups

Wyckoff positions of decagonal space groups are given.

Abstract: Wyckoff positions of five-dimensional decagonal, octagonal and dodecagonal space groups are calculated to facilitate the model building and structure analysis of dihedral quasicrystals. The Wyckoff positions in the decagonal ones and their representative coordinates are given. The Wyckoff positions of octagonal and dodecagonal ones together with the orbits of Wyckoff positions are available in the web site (http://wcp2-ap.eng.hokudai.ac.jp/yamamoto/)

1 Introduction

The program Carat implemented in the program package GAP is applied to the dihedral space groups. This requires only the matrix representation of the point group and calculate the normalizer for classifying the generated space groups. All the possible space groups in these quasicrystals are known and tabulated.[4] However, the known tables are calculated under different equivalence relations, where the scaling transformations of the dihedral lattice is not taken into account and enantiomorphic pairs are regarded as independent. We employ the affine equivalence as in Carat. Then the classification in this paper is broader than that in the present paper, leading to a smaller number of non-equivalent space groups.

In the structure analysis of quasicrystals, the tables of Wyckoff positions are useful, since the number of independent parameters of an occupation domain (OD) is determined by the site symmetry of the center of the OD. The shape of ODs is also restricted by the site symmetry. The symmetry of the OD should be higher than or equal to the site symmetry. Thus the Wyckoff positions and their site symmetry are fundamental information of quasicrystallography. The method of calculating Wyckoff position is established but there are no tables of all Wyckoff positions of quasicrystals. This paper calculates those of decagonal, octagonal and dodecagonal quasicrystals, since they are found so far. In this paper, tables only for the decagonal quasicrystals are given but other more comprehensive tables are accessible in the web.(http://wcp-ap.eng.hokudai.ac.jp/yamamoto/)

2 Calculation Method

We use Carat in GAP package as in a previous paper.[2, 9] The n× n matrix representations of a point group is necessitated in Carat. The point group of axial quasicrystals is a dihedral group or its subgroup. Therefore it is given by one p-fold rotation or rote-inversion (p>2) and one 2-fold rotation or mirror. For centrosymmetric groups, we use one p-fold rotation (p>2) and one 2-fold axis and inversion. All the cases, there is only a primitive lattice, although pentagonal space groups have the primitive and one centered lattice, which is called Q-centered lattice.[3, 4]

Table 1: The generators of the decagonal space groups (excluding those for lattice translations) employed in Tables 2-7. [R₁₀≡ −u,x+y+z+u,−x,−y,v, R₂₁≡ −u,−z,−y,−x,−v, R₂₂≡ u,z,y,x,−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)/10, τ₃=(0,0,0,0,1)/5] Note that P10₁22(10⁷22) is equivalent to P10₃22(10⁷22) and P10₂22(10⁷22) to P10₄22(10⁷22) since they are related by the scaling transformation of the decagonal lattice.

Space group generators

P10/mmm(10⁷1mm) {R₁₀|τ₀},{R₂₁|τ₀},{I|τ₀}

P10/mcc(10⁷1mm) {R₁₀|τ₀},{R₂₁|τ₁},{I|τ₀}

P10₅/mcm(10⁷1mm) {R₁₀|τ₁},{R₂₁|τ₀},{I|τ₀}

P10₅/mmc(10⁷1mm) {R₁₀|τ₁},{R₂₁|τ₁},{I|τ₀}

P10mm(10⁷mm) {R₁₀|τ₀},{R₂₁|τ₀}

P10cc(10⁷mm) {R₁₀|τ₀},{R₂₁|τ₁}

P10₅cm(10⁷mm) {R₁₀|τ₁},{R₂₁|τ₀}

P10₅mc(10⁷mm) {R₁₀|τ₁},{R₂₁|τ₁}

P10 22(10⁷mm) {R₁₀|τ₀},{R₂₁|τ₀}

P10₁22(10⁷mm) {R₁₀|τ₂},{R₂₁|τ₀}

P10₂22(10⁷mm) {R₁₀|τ₃},{R₂₁|τ₀}

P10₅22(10⁷mm) {R₁₀|τ₁},{R₂₁|τ₀}

P10/m(10⁷1) {R₁₀|τ₀},{σ_z|τ₀}

P10₅/m(10⁷1) {R₁₀|τ₁},{σ_z|τ₀}

P10(10⁷) {R₁₀|τ₀}

P10₁(10⁷) {R₁₀|τ₂}

P10₂(10⁷) {R₁₀|τ₃}

P10₅(10⁷) {R₁₀|τ₁}

P102m(5mm) {IR₁₀|τ₀},{R₂₁|τ₀}

P102c(5mm) {IR₁₀|τ₀},{R₂₁|τ₁}

P10m2(5mm) {IR₁₀|τ₀},{R₂₂|τ₀}

P10c2(5mm) {IR₁₀|τ₀},{R₂₂|τ₁}

P10(5) {IR₁₀|τ₀}

3 Discussion

The decagonal quasicrystals have several different space groups. The space groups used in the structure analysis so fat are P10₅/mmc(10⁷1mm), P10₅mc(10⁷mm), 10(5), P10m2(5mm) and P102c(5mm) [8, 7, 1, 5, 6] However, the decagonal quasicrystals analized so far are limitted to the cases with the period ≃ 4, 8, 12 Å along the 10-fold axis. Some decagonal quasicrystals have 16Å period. In d-Al-Ni-Co, there are at least 5 quasicrystal phases. Among them, three phases are determined. The unsolved structures may necessitate other space groups.

Table 2: The Wyckoff positions of the space groups P10/mmm(10⁷1mm). The first column represents Wyckoff symbol (W.S.). The second and third columns denote the site symmetry and the representative coordinates, respectively.

W.S. site symmetry coordinates

1a 10/mmm(10⁷1mm) (0,0,0,0,0)

1b 10/mmm(10⁷1mm) (0,0,0,0,1/2)

2a 10m(5m) (1/5,1/5,1/5,1/5,0)

2b 10m(5m) (1/5,1/5,1/5,1/5,1/2)

2c 10m(5m) (2/5,2/5,2/5,2/5,0)

2d 10m(5m) (2/5,2/5,2/5,2/5,1/2)

5a mmm(mm1) (1/2,0,0,1/2,0)

5b mmm(mm1) (1/2,0,0,1/2,1/2)

5c mmm(mm1) (0,1/2,1/2,0,0)

5d mmm(mm1) (0,1/2,1/2,0,1/2)

5e mmm(mm1) (1/2,1/2,1/2,1/2,0)

5e mmm(mm1) (1/2,1/2,1/2,1/2,1/2)

2e 10mm(10⁷mm) (0,0,0,0,v)

4e 5m(5²m) (1/5,1/5,1/5,1/5,v)

4f 5m(5²m) (2/5,2/5,2/5,2/5,v)

10a mm2(mm1) (1/2,0,0,1/2,v)

10b mm2(mm1) (0,1/2,1/2,0,v)

10c mm2(mm1) (1/2,1/2,1/2,1/2,v)

10d 2mm(mm) (x,y,−y,−x,0)

10e 2mm(mm) (x,y,−y,−x,1/2)

10f mm2(m) (x,y,−y,−x,v)

10g mm2(m) (x,y,y,x,v)

20a m(1) (x,y,z,u,0)

20b m(1) (x,y,z,u,1/2)

40a 1(1) (x,y,z,u,v)

Table 3: The Wyckoff positions of the space groups P10/mcc(10⁷1mm).

W.S. site symmetry coordinates

2a 10/m(10⁷1) (0,0,0,0,0)

2b 1022(10⁷mm) (0,0,0,0,1/4)

2c 1022(5mm) (1/5,1/5,1/5,1/5,0)

2d 1022(5mm) (2/5,2/5,2/5,2/5,0)

2e 52(10m) (1/5,1/5,1/5,1/5,1/4)

2f 52(10m) (2/5,2/5,2/5,2/5,1/4)

10a m2m(mm1) (0,0,0,1/2,0)

10b m2m(mm1) (0,0,1/2,0,0)

10c m2m(mm1) (0,1/2,1/2,1/2,0)

10d 222(mm1) (1/2,0,0,1/2,1/4)

10d 222(mm1) (0,1/2,1/2,0,1/4)

10e 222(mm1) (1/2,1/2,1/2,1/2,1/4)

4a 10(10⁷) (0,0,0,0,v)

4b 10(10⁷) (1/5,1/5,1/5,1/5,v)

4c 10(10⁷) (2/5,2/5,2/5,2/5,v)

10a mm2(mm1) (0,0,0,1/2,v)

10b mm2(mm1) (0,0,1/2,0,v)

10b mm2(mm1) (0,1/2,1/2,1/2,v)

10d 2/m(mm) (x,y,−y,−x,1/4)

10f m(m) (x,y,y,x,1/4)

20a m(1) (x,y,z,u,0)

40a 1(1) (x,y,z,u,v)

Table 4: The Wyckoff positions of the space groups P10₅/mmc(10⁷1mm).

W.S. site symmetry coordinates

2a 10m(5m) (0,0,0,0,3/4)

2c 10m(5m) (1/5,1/5,1/5,1/5,3/4)

2d 10m(5m) (1/5,1/5,1/5,1/5,1/4)

2e 10m(5m) (2/5,2/5,2/5,2/5,3/4)

2f 10m(5m) (2/5,2/5,2/5,2/5,1/4)

2b 5m(10⁹m) (0,0,0,0,0)

10a 2/m(mm) (1/2,0,0,1/2,0)

10b 2/m(mm) (0,1/2,1/2,0,0)

10c 2/m(mm) (1/2,1/2,1/2,1/2,0)

4a 5m(5²m) (0,0,0,0,v)

4b 5m(5²m) (1/5,1/5,1/5,1/5,v)

4c 5m(5²m) (2/5,2/5,2/5,2/5,v)

10f 2/m(mm) (x,y,y,x,3/4)

20f 2(m) (x,y,−y,−x,0)

20f m(m) (x,y,y,x,v)

20a m(1) (x,y,z,u,0)

40a 1(1) (x,y,z,u,v)

Table 5: The Wyckoff positions of the space groups P10₅/mcm(10⁷1mm).

W.S. site symmetry coordinates

2a 10mm(5mm) (0,0,0,0,3/4)

2b 5m(5m) (0,0,0,0,0)

2c 10(5) (1/5,1/5,1/5,1/5,3/4)

2d 10(5) (2/5,2/5,2/5,2/5,3/4)

2e 52(5²m) (1/5,1/5,1/5,1/5,0)

2f 52(5²m) (2/5,2/5,2/5,2/5,0)

10a 2/m(m1) (1/2,0,0,1/2,0)

10b 2/m(m1) (0,1/2,1/2,0,0)

10c 2/m(m1) (1/2,1/2,1/2,1/2,0)

4a 5m(5²m) (0,0,0,0,v)

8a 5(5²) (1/5,1/5,1/5,1/5,v)

8b 5(5²) (2/5,2/5,2/5,2/5,v)

10d 2/m(mm) (x,y,−y,−x,3/4)

20a 2(m) (x,y,y,x,0)

20b m(m) (x,y,−y,−x,v)

20c m(m) (x,y,z,u,3/4)

40a 1(1) (x,y,z,u,v)

The solved structures are analyzed and described without using tables of Wyckoff positions, since no tables were present. The refined parameter tables are given in the papers but neither Wyckoff symbols nor site symmetry symbols have not been given. Although we can calculate the site symmetry if the coordinates of the OD center are known and in the refinement promgam for quasicrystals, Qcdiff, written by the author, the site symmetry is calculated from the coordinates. However, it may not be easy to know the site symmetry from the coordinates, since it depends on the space group. Therefore the table of refined structural parameters together with the Wyckoff symbols is useful to know symmetrical information of quasicrystal structures, as is so in conventional crystals. The tables listed in this paper will be useful for the description of the quasicrystal structures.

Table 6: The Wyckoff positions of the space groups P10mm(10⁷mm).

P10mm(10⁷mm)

W.S. site symmetry coordinates

1a 10mm(10⁷mm) (0,0,0,0,v)

1b 10mm(10⁷mm) (1/5,1/5,1/5,1/5,v)

1c 10mm(10⁷1mm) (2/5,2/5,2/5,2/5,v)

5a mm2(mm1) (1/2,0,0,1/2,v)

5b mm2(mm1) (1/2,0,0,1/2,v)

5c mm2(mm1) (0,1/2,1/2,0,v)

5e mm2(mm1) (1/2,1/2,1/2,1/2,v)

10a m(m) (x,y,−y,−x,v)

10b m(m) (x,x,0,y,z)

20a 1(1) (x,y,z,u,v)

Table 7: The Wyckoff positions of the space group P1022(10⁷mm).

P1022(10⁷mm)

W.S. site symmetry coordinates

1a 1022(10⁷mm) (0,0,0,0,0)

1a 1022(10⁷mm) (0,0,0,0,1/2)

1b 1022(10⁷mm) (1/5,1/5,1/5,1/5,0)

1c 1022(10⁷mm) (1/5,1/5,1/5,1/5,1/2)

1c 1022(10⁷1mm) (2/5,2/5,2/5,2/5,0)

1d 1022(10⁷1mm) (2/5,2/5,2/5,2/5,1/2)

5a 222(mm1) (1/2,0,0,1/2,0)

5a 222(mm1) (1/2,0,0,1/2,1/2)

5b 222(mm1) (1/2,0,0,1/2,v)

5c 222(mm1) (0,1/2,1/2,0,0)

5c 222(mm1) (0,1/2,1/2,0,1/2)

5e 222(mm1) (1/2,1/2,1/2,1/2,0)

5e 222(mm1) (1/2,1/2,1/2,1/2,1/2)

2a 10(10⁷) (0,0,0,0,v)

2e 10(10⁷) (1/5,1/5,1/5,1/5,v)

2e 10(10⁷) (2/5,2/5,2/5,2/5,v)

10a 2(1) (0,0,0,1/2,v)

10b 2(1) (0,0,1/2,1/2,v)

10c 2(1) (0,1/2,0,1/2,v)

10d 2(m) (x,y,−y,−x,0)

10e 2(m) (x,y,−y,−x,1/2)

10f 2(m) (x,x+2y,2x+2y,x+y,0)

10g 2(m) (x,x+2y,2x+2y,x+y,1/2)

20a 1(1) (x,y,z,u,v)

Table 8: The Wyckoff positions of the space groups P102m(5mm).

P102m(5mm)

W.S. site symmetry coordinates

1a 102m(5mm) (0,0,0,0,0)

1b 102m(5mm) (0,0,0,0,1/2)

1c 102m(5mm) (1/5,1/5,1/5,1/5,0)

1d 102m(5mm) (1/5,1/5,1/5,1/5,1/2)

1e 102m(5mm) (2/5,2/5,2/5,2/5,0)

1f 102m(5mm) (2/5,2/5,2/5,2/5,1/2)

1g 102m(5mm) (3/5,3/5,3/5,3/5,0)

1h 102m(5mm) (3/5,3/5,3/5,3/5,1/2)

1i 102m(5mm) (4/5,4/5,4/5,4/5,0)

1j 102m(5mm) (4/5,4/5,4/5,4/5,1/2)

5a mm2(mm1) (0,0,0,0,v)

5a mm2(mm1) (1/5,1/5,1/5,1/5,v)

5a mm2(mm1) (2/5,2/5,2/5,2/5,v)

5a mm2(mm1) (3/5,3/5,3/5,3/5,v)

5a mm2(mm1) (4/5,4/5,4/5,4/5,v)

5a mm2(mm1) (1/2,0,0,1/2,1/2)

5b mm2(mm1) (1/2,0,0,1/2,v)

5c mm2(mm1) (0,1/2,1/2,0,0)

5c mm2(mm1) (0,1/2,1/2,0,1/2)

5e mm2(mm1) (1/2,1/2,1/2,1/2,0)

5e mm2(mm1) (1/2,1/2,1/2,1/2,1/2)

10a 2(m) (x,y,−y,−x,0)

10b 2(m) (x,y,−y,−x,1/2)

10c 2(m) (x,y,z,u,0)

10d 2(m) (x,y,z,u,1/2)

20a 1(1) (x,y,z,u,v)

Table 9: The Wyckoff positions of the space group P10m2(5mm).

P10m2(5mm)

W.S. site symmetry coordinates

1a 10m2(m5mm) (0,0,0,0,0)

1a 10m2(5mm) (0,0,0,0,1/2)

1b 10m2(5mm) (1/5,1/5,1/5,1/5,0)

1b 10m2(5mm) (1/5,1/5,1/5,1/5,1/2)

1c 10m2(5mm) (2/5,2/5,2/5,2/5,0)

1c 10m2(5mm) (2/5,2/5,2/5,2/5,1/2)

5a mm2(mm1) (0,0,0,0,v)

5a mm2(mm1) (1/5,1/5,1/5,1/5,v)

5a mm2(mm1) (2/5,2/5,2/5,2/5,v)

10a 2/m(m1) (x,y,−y,−x,0)

10b 2/m(m1) (x,y,−y,−x,1/2)

10a 2(m) (x,y,−y,−x,z)

10c m(1) (x,y,z,u,0)

10d m(1) (x,y,z,u,1/2)

20a 1(1) (x,y,z,u,v)

Table 10: The Wyckoff positions of the space groups P10(5).

P10(5)

W.S. site symmetry coordinates

1a 10(5) (0,0,0,0,0)

1a 10(5) (0,0,0,0,1/2)

1b 10(5) (1/5,1/5,1/5,1/5,0)

1b 10(5) (1/5,1/5,1/5,1/5,1/2)

1c 10(5) (2/5,2/5,2/5,2/5,0)

1c 10(5) (2/5,2/5,2/5,2/5,1/2)

1c 10(5) (3/5,3/5,3/5,3/5,0)

1c 10(5) (3/5,3/5,3/5,3/5,1/2)

1c 10(5) (4/5,4/5,4/5,4/5,0)

1c 10(5) (4/5,4/5,4/5,4/5,1/2)

1a 5(5²) (0,0,0,0,v)

1b 5(5²) (1/5,1/5,1/5,1/5,v)

1c 5(5²) (2/5,2/5,2/5,2/5,v)

1c 5(5²) (3/5,3/5,3/5,3/5,v)

1c 5(5²) (4/5,4/5,4/5,4/5,v)

10c m(1) (x,y,z,u,0)

10d m(1) (x,y,z,u,1/2)

20a 1(1) (x,y,z,u,v)

Table 11: The Wyckoff positions of the space groups P10(10⁷).

P10(10⁷)

W.S. site symmetry coordinates

1a 5(5²) (0,0,0,0,v)

1b 5(5²) (1/5,1/5,1/5,1/5,v)

1c 5(5²) (2/5,2/5,2/5,2/5,v)

1c 5(5²) (3/5,3/5,3/5,3/5,v)

1c 5(5²) (4/5,4/5,4/5,4/5,v)

10c m(1) (x,y,z,u,0)

10d m(1) (x,y,z,u,1/2)

20a 1(1) (x,y,z,u,v)

4 Summary

The Wyckoff positions of decagonal, octagonal and dodecagonal space groups for quasicrystals were calculated by the program Carat implemented in the package GAP and listed for the decagonal space groups. The Wyckof symbols and their site symmetries are given. Tables for the octagonal and dodecagonal space groups and all equivalent positions for each Wyckoff positions are obtainable from the web.

References

[1]: A. ‍Cervellino, T. ‍Haibach, and W. ‍Steurer. Structure solution of the basic decagonal Al-Co-Ni phase by the atomic surfaces modelling method. Acta Crystallogr., B58:8–33, 2002.
[2]: B. ‍Eick, F. ‍Gähler, and W. ‍Nickel. Computing Maximal Subrgoups and Wyckoff Positions of Space Groups. Acta Crystallogr., A53:467, 1997.
[3]: T. ‍Janssen, J. ‍L. Birman, F. ‍Denoyer, V. ‍A. Koptsik, J. ‍L. Verger-Gaugry, D. ‍Weigel, A. ‍Yamamoto, and S. ‍C. Abrahams. Report of a Subcommittee on the Nomenclature of n-Dimensional Crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups. Acta Crystallogr., A58:605–621, 2002.
[4]: D. ‍A. Rabson, N. ‍D. Mermin, D. ‍S. Rokhsar, and D. ‍C. Wright. The space groups of axial crystals and quasicrystals. Rev. Mod Phys., 63:699–733, 1991.
[5]: A. ‍Strutz, A. ‍Yamamoto, and W. ‍Steurer. Basic Co-rich decagonal Al-Co-Ni: Average structure. Phys. Rev., B80:184102, 2009.
[6]: A. ‍Strutz, A. ‍Yamamoto, and W. ‍Steurer. Basic Co-rich decagonal Al-Co-Ni: Superstructure. Phys. Rev., B82:064107, 2010.
[7]: H. ‍Takakura, A. ‍Yamamoto, and A. ‍P. Tsai. The structure of a decagonal Al₇₂Ni₂₀Co₈ quasicrystal. Acta Crystallogr., A57:576, 2001.
[8]: S. ‍Weber and A. ‍Yamamoto. Application of the five-dimensional maximum entropy method to the structure refinement of decagonal Al₇₀Mn₁₇Pd₁₃. Phil. Mag., A 76:85–106, 1997.
[9]: A. ‍Yamamoto. Wyckoff positions of icosahedral space groups and their applications. Acta Crystallogr., page to be submitted, 2013.

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

Space group	generators
P10/mmm(10⁷1mm)	{R₁₀\|τ₀},{R₂₁\|τ₀},{I\|τ₀}
P10/mcc(10⁷1mm)	{R₁₀\|τ₀},{R₂₁\|τ₁},{I\|τ₀}
P10₅/mcm(10⁷1mm)	{R₁₀\|τ₁},{R₂₁\|τ₀},{I\|τ₀}
P10₅/mmc(10⁷1mm)	{R₁₀\|τ₁},{R₂₁\|τ₁},{I\|τ₀}
P10mm(10⁷mm)	{R₁₀\|τ₀},{R₂₁\|τ₀}
P10cc(10⁷mm)	{R₁₀\|τ₀},{R₂₁\|τ₁}
P10₅cm(10⁷mm)	{R₁₀\|τ₁},{R₂₁\|τ₀}
P10₅mc(10⁷mm)	{R₁₀\|τ₁},{R₂₁\|τ₁}
P10 22(10⁷mm)	{R₁₀\|τ₀},{R₂₁\|τ₀}
P10₁22(10⁷mm)	{R₁₀\|τ₂},{R₂₁\|τ₀}
P10₂22(10⁷mm)	{R₁₀\|τ₃},{R₂₁\|τ₀}
P10₅22(10⁷mm)	{R₁₀\|τ₁},{R₂₁\|τ₀}
P10/m(10⁷1)	{R₁₀\|τ₀},{σ_z\|τ₀}
P10₅/m(10⁷1)	{R₁₀\|τ₁},{σ_z\|τ₀}
P10(10⁷)	{R₁₀\|τ₀}
P10₁(10⁷)	{R₁₀\|τ₂}
P10₂(10⁷)	{R₁₀\|τ₃}
P10₅(10⁷)	{R₁₀\|τ₁}
P102m(5mm)	{IR₁₀\|τ₀},{R₂₁\|τ₀}
P102c(5mm)	{IR₁₀\|τ₀},{R₂₁\|τ₁}
P10m2(5mm)	{IR₁₀\|τ₀},{R₂₂\|τ₀}
P10c2(5mm)	{IR₁₀\|τ₀},{R₂₂\|τ₁}
P10(5)	{IR₁₀\|τ₀}

W.S.	site symmetry	coordinates
1a	10/mmm(10⁷1mm)	(0,0,0,0,0)
1b	10/mmm(10⁷1mm)	(0,0,0,0,1/2)
2a	10m(5m)	(1/5,1/5,1/5,1/5,0)
2b	10m(5m)	(1/5,1/5,1/5,1/5,1/2)
2c	10m(5m)	(2/5,2/5,2/5,2/5,0)
2d	10m(5m)	(2/5,2/5,2/5,2/5,1/2)
5a	mmm(mm1)	(1/2,0,0,1/2,0)
5b	mmm(mm1)	(1/2,0,0,1/2,1/2)
5c	mmm(mm1)	(0,1/2,1/2,0,0)
5d	mmm(mm1)	(0,1/2,1/2,0,1/2)
5e	mmm(mm1)	(1/2,1/2,1/2,1/2,0)
5e	mmm(mm1)	(1/2,1/2,1/2,1/2,1/2)
2e	10mm(10⁷mm)	(0,0,0,0,v)
4e	5m(5²m)	(1/5,1/5,1/5,1/5,v)
4f	5m(5²m)	(2/5,2/5,2/5,2/5,v)
10a	mm2(mm1)	(1/2,0,0,1/2,v)
10b	mm2(mm1)	(0,1/2,1/2,0,v)
10c	mm2(mm1)	(1/2,1/2,1/2,1/2,v)
10d	2mm(mm)	(x,y,−y,−x,0)
10e	2mm(mm)	(x,y,−y,−x,1/2)
10f	mm2(m)	(x,y,−y,−x,v)
10g	mm2(m)	(x,y,y,x,v)
20a	m(1)	(x,y,z,u,0)
20b	m(1)	(x,y,z,u,1/2)
40a	1(1)	(x,y,z,u,v)

P1022(10⁷mm)
W.S.	site symmetry	coordinates
1a	1022(10⁷mm)	(0,0,0,0,0)
1a	1022(10⁷mm)	(0,0,0,0,1/2)
1b	1022(10⁷mm)	(1/5,1/5,1/5,1/5,0)
1c	1022(10⁷mm)	(1/5,1/5,1/5,1/5,1/2)
1c	1022(10⁷1mm)	(2/5,2/5,2/5,2/5,0)
1d	1022(10⁷1mm)	(2/5,2/5,2/5,2/5,1/2)
5a	222(mm1)	(1/2,0,0,1/2,0)
5a	222(mm1)	(1/2,0,0,1/2,1/2)
5b	222(mm1)	(1/2,0,0,1/2,v)
5c	222(mm1)	(0,1/2,1/2,0,0)
5c	222(mm1)	(0,1/2,1/2,0,1/2)
5e	222(mm1)	(1/2,1/2,1/2,1/2,0)
5e	222(mm1)	(1/2,1/2,1/2,1/2,1/2)
2a	10(10⁷)	(0,0,0,0,v)
2e	10(10⁷)	(1/5,1/5,1/5,1/5,v)
2e	10(10⁷)	(2/5,2/5,2/5,2/5,v)
10a	2(1)	(0,0,0,1/2,v)
10b	2(1)	(0,0,1/2,1/2,v)
10c	2(1)	(0,1/2,0,1/2,v)
10d	2(m)	(x,y,−y,−x,0)
10e	2(m)	(x,y,−y,−x,1/2)
10f	2(m)	(x,x+2y,2x+2y,x+y,0)
10g	2(m)	(x,x+2y,2x+2y,x+y,1/2)
20a	1(1)	(x,y,z,u,v)