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

Wyckoff positions and their representative coordinates of icosahedral space groups and their applications are given.

Abstract: Wyckoff positions of all icosahedral space groups, which are important for model building and refinement of the icosahedral quasicrystals are given. Several applications of the Wyckoff positions are demonstrated. The orbits of Wyckoff positions are available in the web site (http://wcp2-ap.eng.hokudai.ac.jp/yamamoto/)

1 Introduction

Occupation domain (OD) positions in quasicrystals play an essential role for describing and determining quasicrystal structures as same as the role of atom positions in conventional crystals.[14] They are classified by the point symmetry into groups, which are known as Wyckoff positions. This is necessary for model building of quasicrystals, since the point symmetry restricts the shape of the OD. In the structure refinement program qcdiff written by the author therefore, the site symmetry is calculated from the OD position. Other two examples of the use of Wyckoff positions are shown.

In icosahedral quasicrystals found so far, the space groups were symmorphic. However, icosahedral quasicrystals with non-symmorphic space group may exist as in decagonal quasicrystals. There are five non-symmorphic space groups among eleven icosahedral ones.[6, 11, 9] For non-symmorphic space groups, the calculations of the Wyckoff positions are not easy in general. The table of all Wyckoff positions has not be given, although limited Wyckoff positions of symmorphic space groups in the reciprocal lattice are calculated for electronic structures of quasicrystals.[10] There exists a computer program, Carat, for calculating Wyckoff positions of space groups which is implemented in the computer algebra system GAP.[1] (http://www.gap-system.org/Packages/Cryst) This enables us to derive all non-equivalent icosahedral space groups and their Wyckoff positions.

In ths paper, Wyckoff positions together with their symbols and site symmety of all (eleven) icosahedral space groups in the six dimensional (6D) space are shown and their applications are demonstrated.

2 Calculation Method

The program Carat requires n× n matrix representations of the generators of a point group for nD space groups. The matrices are (3+d) reducible (d=n−3) for nD (n=5 or 6) point groups of quasicrystals. The point group is a semi-direct product of the 3D and dD point groups, so that they are isomorphic to a 3D point group. The latter is generated by at most three generators. As a result, the nD point group is generated by the same number of generators. When the point group G is centrosymmetric, we choose generators of its non-centrosymmetric factor group G/I and inversion as the generators of the point group. Then number of generators is at most four. As the generators of the 3D point groups, we can choose one 2-fold rotation and one or two p-fold rotation with p>2 point group and inversion for centrosymmetric groups, while for non-centrosymmetric one, one merror or 2-fold rotation and one or two p-fold rotation or rote-inversion.

The icosahedral point group m35(m35²) is a point group with the highest symmetry, the order of which is 120. It is generated by 2-fold, 3-fold and 5-fold rotation and inversion. They are given by −x,−y,−w,−v,−u,−z; y,z,x,w,−u,−v; x,w,y,z,u,v and −x,−y,−z,−u,−v,−w in the international table (IT) format.[4] They are written as {R₂₁|τ₀}, {R₃₁|τ₀}, {R₅₁|τ₀} and {I|τ₀}, where τ₀ is the 6D zero vector. For the face- and body-centered lattices, the basis vectors of their reciprocal lattice d_i^{′ *} are given by d_i^{′ *}=∑_j=1⁶d_j^*(S_k)_ji (k=1,2) in terms of those in the primitive lattice d_j^*, where S_k is given by[6]

and

The determinants of S₁ and S₂ are 32 and 2 respectively. Their inverse matrices are given by

and

Then the coordinates, basis vectors in the (direct) lattice and matrix representation of the symmetry operations are given by x_i^′=∑_j=1⁶ x_j(S_k)_ji, d_i^′=∑_j=1⁶ (S_k⁻¹)_ijd_j and R_ij^′=∑_l=1⁶ (S_k⁻¹)_ilR_lj(S_k)_kj.[7] Using the matrices R in the primitive lattice, the matrices R^′ for generators, which are required in Carat, are calculated and all the symmetry operators in the face-centered and body-centered icosahedral lattices are derived. The obtained Wyckoff position coordinates with respect to the basis d_i^′ are converted into those in the primitive lattice system d_i by x_i=∑_j=1⁶ x_j^′(S_k⁻¹)_ji. In non-symmorphic space groups, the non-primitive translation vector coordinates with respect to d_j^′, which are given by Carat, should also be transformed by this. Thus the symmetry operators and the Wyckoff positions in a centered lattice are calculated.

3 Wyckoff Positions

There are 11 non-equivalent space groups in the icosahedral lattice in 6D space. We write them as Fm35(m35²), Fm_xy35(m35²), Im35(m35²), Pm35(m35²), Pm_xy35(m35²), F235(235²), F235₁(235²), I235(235²), I235₁(235²), P235(235²), and P235₁(235²). They are quite similar to the symbols recommended by the IUCr[7] except that the order of the generators are inverted, so as to be consistent with the cubic space group symbols in 3D space. (Note that in the latter, Pm3 represents a cubic space group, while P3m stands for trigonal one.) In alternative symbols, the first one is also written as F53m, I^*53m, or F53m(5²3m).[6, 11, 7]) The orders of the centrosymmetric and non-centrosymmetric point groups are 120 and 60, respectively. The corresponding space groups are generated by four or three generators together with six generators for the lattice translations. (See Table 1.) The Wyckoff potions, their site symmetry and representative coordinates of the Wyckoff position are given in Tables 2-7.

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Table 1: The generators of the icosahedral space groups (excluding those for lattice translations) employed in Tables 2-7. [R51≡ x,w,y,z,u,v, R31≡ y,z,x,w,−u,−v, R21≡ −x,−y,−w,−v,−u,−z, I≡ −x,−y,−z,−u,−v,−w, τ0=(0,0,0,0,0,0),τ1=(1,1,1,1,1,1)/2, τ2=(3,1,1,1,1,3)/4, τ3=(1,0,0,0,0,0)/5, τ4=(0,1,4,2,3,1)/5, τ5=(1,2,0,1,0,2)/10, τ6=(7,6,2,2,2,5)/10, τ7=(9,1,1,1,1,1,1)/10, τ8=(1,9,0,3,5,2)/10,τ9=(0,1,0,0,0,3)/4,τ10=(3,0,1,1,3,2)/4. τ11=(1,3,2,3,3,2)/4,τ12=(0,0,0,0,1,1)/2] Note that the origin is not at the inversion center in Fmxy35 and Pmxy35. The glide planes {σ21|τ} in Fmxy35 and Pmxy35 are given by {I|τ2}{R21|τ0} and {I|τ1}{R21|τ0}, respectively. (σ21≡ x,y,w,v,u,z.) When the inversion is not on the origin in centrosymmetric space groups, the second setting is given where the origin is at the inversion.

Space group	generators
Fm35(m352)	{R51|τ0},{R31|τ0},{R21|τ0},{I|τ0}
Fmxy35(m352)	{R51|τ0},{R31|τ0},{R21|τ0},{I|τ2}
(2nd setting)	{R51|τ9},{R31|τ10},{R21|τ11},{I|τ0}
Im35(m352)	{R51|τ0},{R31|τ0},{R21|τ0},{I|τ0}
Pm35(m352)	{R51|τ0},{R31|τ0},{R21|τ0},{I|τ0}
Pmxy35(m352)	{R51|τ0},{R31|τ0},{R21|τ0},{I|τ1}
(2nd setting)	{R51|τ0},{R31|τ12},{R21|τ1},{I|τ0}
F235(2352)	{R51|τ0},{R31|τ0},{R21|τ0}
F2351(2352)	{R51|τ5},{R31|τ6},{R21|τ0}
I235(2352)	{R51|τ0},{R31|τ0},{R21|τ0}
I2351(2352)	{R51|τ7},{R31|τ8},{R21|τ0}
P235(2352)	{R51|τ0},{R31|τ0},{R21|τ0}
P2351(2352)	{R51|τ3},{R31|τ4},{R21|τ0}

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Table 2: The Wyckoff positions of the space group Fm35. The first column represents Wyckoff symbol (W.S.). The second and third columns denote the site symmetry and the representative coordinates, respectively. In the Wyckoff positions in the second setting in Fmxy35(m35), −τ2/2 shown in Table 1 should be added to the coordinates.

Fm35(m352)
W.S.	site symmetry	coordinates
32· 1a	m35(m352)	(0,0,0,0,0,0)
32· 1b	m35(m352)	(1,0,0,0,0,0)/2
32· 1c	m35(m352)	(1,1,1,1,1,1)/4
32· 1d	m35(m352)	(3,1,1,1,1,1)/4
32· 15a	mmm(mmm)	(1,1,1,0,0,1)/4
32· 15b	mmm(mmm)	(2,0,1,1,1,1)/4
32· 15c	mmm(mmm)	(2,0,1,0,0,1)/4
32· 15d	mmm(mmm)	(1,1,0,0,0,2)/4
32· 12a	5m(52m)	(x,y,y,y,y,y)
32· 20a	3m(3m)	(x,x,y,−y,y,x)
32· 30a	2mm(2mm)	(x,x,y,0,0,y)
32· 30b	2mm(2mm)	(1/4+x,1/4+x,y,1/4,1/4,1/2+y)
32· 30c	2mm(2mm)	(1/4+x,1/4+x,y,0,0,1/2+y)
32· 30d	2mm(2mm)	(1/2+x,x,y,1/4,1/4,y)
32· 60a	m(m)	(x+y+z+u,x,y,u,0,z)
32· 120a	1(1)	(x,y,z,u,v,w)

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In the tables, we use the conventional lattice basis with the centering translations for the face-centered (F) and body-centered (I) lattices, although the the Wyckoff positions were calculated in different (primitive) basis. In the I and F lattices, 2 and 32 centering translations exist. The former is given by (0,0,0,0,0,0) and (1/2,1/2,1/2,1/2,1/2,1/2), while the latter includes the permutations of (1/2,1/2,0,0,0,0) and (0,0,1/2,1/2,1/2,1/2) in additions to the above two. The the conventional basis vectors are in parallel to 5-fold axes.[6, 2] (See Appendix.)

The symbols of the Wyckoff position are quite similar to those in 3D space groups but the convention of the latter is slightly modified owing to several reasons. The number of equivalent positions is separated into two parts for the centered lattice, the number of centering translations and the number of equivalent positions in the asymmetric unit. They are separated by the center dot. In addition, the number of equivalent position is followed by the alphabetical character which begins with a in a class with the same number of equivalent positions. [The number of alphabets (26) is too small to distinguish Wyckoff positions in some 6D space groups if the convention of the 3D space groups in the International Table is employed.[4]] For example, in the 6D space group Pm3(m3), not discussed here, there are 47 Wyckoff positions.)

4 Crystalline Approximants

As the first application of the Wyckoff position, we consider the space group of the crystalline approximants of icosahedral quasicrystals, which are obtained from a six-dimensional (6D) model. As is well known, the period of the approximants are determined by the linear phason strain.[5, 8, 13] The space group of the approximants depends on the 3D hyperplane used. In order to obtain a given space group symmetry, the hyperplane passing through a point in 6D lattice with the point symmetry equal to or higher than the highest site symmetry in the space group has to be used. For example, cubic approximants with Pm3, Im3 Pa3, Ia3 are known. The highest site symmetry of the former two is m3 while that of the latter two is 3. The Wyckoff positions of the space group Pm35(m35²), which have a point group equal to or higher than these, are 1a, 1b, 10a and 10b. (See Table4.) The 3D hyperplane passing through the former two leads to the cubic approximants with the space group Pm3 or Im3, while those through the latter two gives Pa3 or Ia3 depending on the order of the approximant. When the order of the approximant i is defined by the Fibonacci number of the approximant F_i/F_i+1, (F₀=1, F₁=1, F_i+1=F_i−1+F_i) the approximant with i=0 mod 3 has the body centered lattice, while that with i=1,2 mod 3 has the primitive lattice. Thus Im3 and Pa3 are possible for the 1/1 and 1/2 approximants. A 3/2 approximant is recently found in i-Al-Pd-Cr-Fe system. This has Pa3, consistent with the above results.[3].

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Table 3: The Wyckoff positions of the space group Fmxy35. The first column represents Wyckoff symbol (W.S.). The second and third columns denote the site symmetry and the representative coordinates, respectively. In the Wyckoff positions in the second setting in Fmxy35(m35), −τ2/2 shown in Table 1 should be added to the coordinates.

Fmxy35(m352)
W.S.	site symmetry	coordinates
32· 2a	235(235)	(0,0,0,0,0,0)
32· 2b	235(235)	(1,1,0,0,0,1)/2
32· 12a	5(52)	(5,1,1,1,1,5)/8
32· 12b	5(52)	(3,3,3,3,3,7)/8
32· 20a	3(3)	(7,3,1,3,1,7)/8
32· 20b	3(3)	(1,5,3,1,3,1)/8
32· 30a	222(222)	(1,1,0,1,1,2)/4
32· 30b	222(222)	(1,1,0,0,0,2)/4
32· 24a	5(52)	(x,y,y,y,y,y)
32· 40a	3(3)	(x,x,y,−y,y,x)
32· 60a	2(2)	(0,0,x,−y,y,−x)
32· 60b	2(2)	(1/4,1/4,x,−y,y,−x)
32· 120a	1(1)	(x,y,z,u,v,w)

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Table 4: The Wyckoff positions of the space group Im35(m352).

Im35(m352)
W.S.	site symmetry	coordinates
2· 1a	m35(m352)	(0,0,0,0,0,0)
2· 6a	5m(52m)	(1,0,0,0,0,0)/2
2· 6b	5m(52m)	(1,1,1,1,1,1)/4
2· 6c	5m(52m)	(3,1,1,1,1,1)/4
2· 10a	3m(3m)	(1,1,1,0,0,0)/2
2· 10b	3m(3m)	(1,1,1,3,1,3)/4
2· 10c	3m(3m)	(1,1,1,1,3,1)/4
2· 12a	5m(52m)	(x,y,y,y,y,y)
2· 15a	mmm(mmm)	(1,1,0,0,0,0)/2
2· 20a	3m(3m)	(x,x,x,y,y,y)
2· 30a	222(222)	(1,3,0,3,3,2)/4
2· 30b	222(222)	(2,0,3,3,1,1)/4
2· 30c	2mm(2mm)	(x,x,y,0,0,y)
2· 60a	m(m)	(x,x,0,y,y,0)
2· 120a	1(1)	(x,y,z,u,v,w)

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5 i-Cd-Yb Structure without Positional Disorder

The i-Cd-Yb is composed of the Tsai-type cluster, the innermost shell of which is a tetrahedron. The cluster center positions are generated by the central part of the occupation domain located at the origin in the space group Pm35(m35²). The origin has the site symmetry of m35(m35) which is higher than the point symmetry of the tetrahedron, 23. As a result, when the orientation of the tetrahedron is the same everywhere, the symmetry of the QC becomes 23. In the structure refinement, therefore, the randomly orientated tetrahedra with the occupancy of 1/10 are assumed.[12]

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Table 5: The Wyckoff positions of the space group Pm35(m352). In the Wyckoff positions for the second setting in Pmxy35(m35), −τ1/2 in Table 1 should be added to the coordinates.

Pm35(m352)
W.S.	site symmetry	coordinates
1a	m35(m352)	(0,0,0,0,0,0)
1b	m35(m352)	(1,1,1,1,1,1)/2
6a	5m(52m)	(0,1,1,1,1,1)/2
6b	5m(52m)	(1,0,0,0,0,0)/2
10a	3m(3m)	(1,1,0,0,0,1)/2
10b	3m	(0,0,1,1,1,0)/2
12a	5m(52m)	(x,y,y,y,y,y)
15a	mmm(mmm)	(0,0,1,0,0,1)/2
15b	mmm(mmm)	(0,0,1,1,1,1)/2
20a	3m(3m)	(x,x,y,y,y,x)
30a	2mm(2mm)	(x,x,y,0,0,y)
30b	2mm(2mm)	(x,x,y,1/2,1/2,y)
60a	2(2)	(0,1/2,x,y,−y,−x)
60b	m(m)	(x,y,z,u,u,z)
120a	1(1)	(x,y,z,u,v,w)

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Table 6: The Wyckoff positions of the space group Pmxy35(m352). In the Wyckoff positions for the second setting in Pmxy35(m35), −τ1/2 in Table 1 should be added to the coordinates.

Pmxy35(m352)
WS	site symmetry	coordinates
2a	235(2352)	(0,0,0,0,0,0)
12a	52(522)	(0,1,1,1,1,1)/2
12b	5(52)	(3,3,3,3,3,3)/4
12c	5(52)	(3,1,1,1,1,1)/4
20a	32(32)	(1,1,0,0,0,1)/2
20b	3(3)	(3,3,3,1,3,3)/4
20c	3(3)	(1,1,3,1,3,1)/4
30a	222(222)	(0,0,1,0,0,1)/2
24a	5(52)	(x,y,y,y,y,y)
40a	3(3)	(x,x,y,−y,y,−x)
60a	2(2)	(0,0,x,y,−y,−x)
60b	2(2)	(0,1/2,x,y,−y,−x)
120a	1(1)	(x,y,z,u,v,w)

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The ODs for the atom positions of the first shell are located at the Wyckoff position 20a in Pm35(m35²) and P235(235²). (See Tables 4 and 7.) Their site symmetry is 3m(3m) and 3(3). In order to obtain a completely ordered i-Cd-Yb structure, the site symmetry of the cluster center has to be 23(23). In the non-centrosymmetric icosahedral group P235(235²), the site symmetry of the origin is 235(235²). The OD of the cluster center can be subdivided into 5 disconnected domains with the symmetry 23(23). The ODs of the constituent atoms of the first shell is obtained from these disconnected ODs by shifting them in parallel to the external space.[16, 15] The shifted positions belong to the Wyckoff position 20a. In the latter, the site symmetry of the OD is higher than that of the Wyckoff position. Therefore, this model gives the icosahedral space group P235(235²). On the other hand, the symmetry of the disconnected OD breaks the total symmetry in the former. Thus we can simply conclude that the total symmetry of the completely ordered model is P235(235²) from the site symmetry of the Wyckoff position. The full detail of this model will be shown in a sepalate paper.

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Table 7: The Wyckoff positions of the space group F235(2352).

F235
W.S.	site symmetry	coordinates
32· 1a	235(2352)	(0,0,0,0,0,0)
32· 1b	235(2352)	(1,1,1,1,1,1)/4
32· 1c	235(2352)	(3,1,1,1,1,1)/4
32· 6a	52(522)	(1,0,0,0,0,0)/2
32· 12a	5(52)	(x,y,y,y,y,y)
32· 15a	222(222)	(0,0,0,1,1,2)/4
32· 15b	222(222)	(1,1,0,1,1,0)/4
32· 15c	222(222)	(1,1,0,0,0,0)/4
32· 15d	222(222)	(1,1,0,1,1,2)/4
32· 20a	3(3)	(x,x,y,−y,y,x)
32· 30a	2(2)	(0,0,x,−y,y,−x)
32· 30b	2(2)	(1/4,1/4,x,−y,+y,1/2−x)
32· 30c	2(2)	(1/4,1/4,x,1/4−y,1/4+y,1/2−x)
32· 30d	2(2)	(1/2,0,x,1/4−y,1/4+y,1/2−x)
32· 60a	1(1)	(x,y,z,u,v,w)

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Table 8: The Wyckoff positions of the space group F2351(2352).

F2351(2352)
W.S.	site symmetry	coordinates
32· 5a	23(23)	(15,5,5,3,7,15)/20
32· 5b	23(23)	(5,5,0,4,1,5)/10
32· 5c	23(23)	(0,5,0,4,1,5)/10
32· 5d	23(23)	(15,5,5,3,7,15)/20
32· 10a	32(32)	(8,2,0,3,0,5)/10
32· 10b	32(32)	(11,9,5,1,5,15)/20
32· 10c	32(32)	(1,9,5,1,5,15)/20
32· 10d	32(32)	(8,2,0,3,0,5)/10
32· 15a	222(222)	(11,9,5,1,5,15)/20
32· 15b	222(222)	(1,9,5,1,5,15)/20
32· 15c	222(222)	(5,15,5,8,2,15)/20
32· 15d	222(222)	(0,10,5,8,2,15)/20
32· 20a	3(3)	(3/5+x,x,y,3/10−y,y,3/10+x)
32· 30a	2(2)	(0,0,x,−y,y,−x)
32· 30b	2(2)	(1/4,1/4,x,1/4−y,1/4+y,1/2−x)
32· 30c	2(2)	(1/4,1/4,x,−y,+y,1/2−x)
32· 60a	1(1)	(x,y,z,u,v,w)

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6 Discussion

In the structure refinement program of quasicrystals, qcdiff, the OD positions are specified by two 6D vectors, the center of large OD x₀ and an internal space component of a vector from it, x₁ⁱ. The site symmetry of x₀+x₁ⁱ is determined by x_0i+y_1i (1≤ i≤ 6) or the intersection of the site symmetries of x₀ and x₁ⁱ.

The 6D coordinates of x_i (i=1,2) and x₁ⁱ are denoted as (x_i1,x_i2,x_i3,x_i4,x_i5,x_i6) and (y₁₁,y₁₂,y₁₃,y₁₄,y₁₅,y₁₆). Then (y₁₁,y₁₂,y₁₃,y₁₄,y₁₅,y₁₆), are given by y_i=∑_j=1⁶ P_ikⁱx_k, where Pⁱ is a matrix composed of irrational elements shown in Appendix. As a result, the internal space component of a non-zero lattice vector includes irrational elements. Such vectors therefore belong to a low symmetric Wyckoff position. Therefore, the symmetry of the OD center x₀ is higher than that of xⁱ in most cases, so that the site symmetry of the OD is given by that of x₁ⁱ=(y₁₁,y₁₂,y₁₃,y₁₄,y₁₅,y₁₆). If x₀ and x₁ is known, the site symmetry of the OD is easily obtained from the table.

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Table 9: The Wyckoff positions of the space group I235(2352). (In the table the symbols ri=i/20, (i=1,5,18,19) are used.)

I235(2352)
W.S.	site symmetry	coordinates
2· 1a	235(2352)	(0,0,0,0,0,0)
2· 6a	52(522)	(3,1,1,1,1,1)/4
2· 6b	52(522)	(1,0,0,0,0,0)/2
2· 6c	52(522)	(1,1,1,1,1,1)/4
2· 10a	32(32)	(1,1,3,1,3,1)/4
2· 10b	32(32)	(0,0,1,1,1,0)/2
2· 10c	32(32)	(1,1,1,3,1,1)/4
2· 12a	5(52)	(x,y,y,y,y,y)
2· 15a	222(222)	(1,3,0,3,3,2)/4
2· 15b	222(222)	(0,0,1,0,0,1)/2
2· 15c	222(222)	(3,1,0,1,1,2)/4
2· 15d	222(222)	(2,0,3,3,1,1)/4
2· 15e	222(222)	(3,1,0,3,3,2)/4
2· 20a	3(3)	(x,x,x,y,y,y)
2· 30a	2(2)	(0,0,x,y,−y,−x)
2· 30b	2(2)	(0,1/2,x,y,−y,−x)
2· 30c	2(2)	(3/4,1/4,1/4+x,1/4+y,1/4−y,1/4−x)
2· 30d	2(2)	(1/4,1/4,3/4+x,3/4+y,3/4−y,3/4−x)
2· 60a	1(1)	(x,y,z,u,v,w)

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Table 10: The Wyckoff positions of the space group I2351(2352). (In the table the symbols ri=i/20, (i=1,5,18,19) are used.)

I2351(2352)
W.S.	site symmetry	coordinates
2· 5a	23(23)	(5,5,19,19,11,11)/20
2· 10a	32(32)	(19,1,19,11,3,1)/20
2· 10b	32(32)	(2,3,7,8,9,3)/10
2· 10c	32(32)	(19,1,9,1,13,1)/20
2· 10d	32(32)	(2,3,2,3,4,3)/10
2· 15a	222(222)	(0,0,19,4,16,1)/20
2· 15b	222(222)	(0,0,9,4,16,11)/20
2· 15c	222(222)	(15,5,19,9,1,11)/20
2· 15d	222(222)	(5,5,4,4,6,6)/20
2· 15e	222(222)	(0,0,19,14,6,1)/20
2· 20a	3(3)	(r19+x,r1+x,r1+y,r9−y,r5+y,r1+x)
2· 30a	2(2)	(0,0,x,y,−y,−x)
2· 30b	2(2)	(0,1/2,x,y,−y,−x)
2· 30c	2(2)	(3/4,1/4,1/4+x,1/4+y,1/4−y,1/4−x)
2· 30d	2(2)	(1/4,1/4,3/4+x,3/4+y,3/4−y,3/4−x)
2· 60a	1(1)	(x,y,z,u,v,w)

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Table 11: The Wyckoff positions of the space group P235(2352).

P235(2352)
W.S.	site symmetry	coordinates
1a	235(2352)	(0,0,0,0,0,0)
1b	235(2352)	(1,1,1,1,1,1)/2
6a	52(522)	(0,1,1,1,1,1)/2
6b	52(522)	(1,0,0,0,0,0)/2
10a	32(32)	(1,1,0,0,0,1)/2
10b	32(32)	(0,0,1,1,1,0)/2
12a	5(52)	(x,y,y,y,y,y)
15a	222(222)	(0,0,1,0,0,1)/2
15b	222(222)	(0,0,1,1,1,1)/2
20a	3(3)	(x,x,y,−y,y,x)
30a	2(2)	(0,0,x,y,−y,−x)
30b	2(2)	(0,1/2,x,y,−y,−x)
30c	2(2)	(1/2,1/2,x,y,−y,−x)
60a	1(1)	(x,y,z,u,v,w)

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Table 12: The Wyckoff positions of the space group P2351(2352).

P2351(2352)
WS	site symmetry	coordinates
5a	23(23)	(5,5,9,7,3,1)/10
5b	23(23)	(0,0,2,1,4,3)/5
10a	32(32)	(6,4,9,5,1,6)/10
10b	32(32)	(1,9,9,5,1,1)/10
10c	32(32)	(3,2,2,0,3,3)/5
10d	32(32)	(1,9,4,0,6,1)/10
15a	222(222)	(0,0,9,2,8,1)/10
15b	222(222)	(0,0,9,7,3,1)/10
20a	3(3)	(x,4/5+x,y,2/5−y,1/5+y,x)
30a	2(2)	(0,0,x,y,−y,−x)
30b	2(2)	(0,1/2,x,y,−y,−x)
30c	2(2)	(1/2,1/2,x,y,−y,−x)
60a	1(1)	(x,y,z,u,v,w)

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7 Appendix

The unit vectors d_i, (i=1,2,..,6) of the primitive icosahedral lattice are taken along the 5-fold axes. Their external and internal space components are given by d_i=∑Q_ij a_j, where a_i i=1,2,3 and i=4,5,6 are the unit vectors of the external and internal spaces. The projection operators which project a 6D vector into its external and internal space components are given by 6× 6 matrices P^e and Pⁱ, which are defined by P_ij^e=∑_k=1³Q_ik⁻¹Q_kj and P_ijⁱ=∑_k=4⁶Q_ik⁻¹Q_kj. (The tilde means the transposition.) When a_i are taken in parallel to 2-fold axes, Q is given by[14]

where τ=(1+√5)/2 and a=1/a^* is the lattice constant of the icosahedral lattice. The coordinates of the external and internal component of (x₁,x₂,x₃,x₄,x₅,x₆) are given by y_i=∑_j P_ij^e/ix_j.

The explicit forms of P^e amd Pⁱ are

and

where A=2+τ. (Note that P^e+Pⁱ=I.) This indicates that the external and internal space componens of a lattice vector or a vector with rational coordinates have always irrational compoments except for the zero vector. Therefore they belong to low symmetric Wyckoff position, which includes arbitrary parameter denoted as x, y etc in the tables.

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