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Converting three-space matrices to equivalent six-space matrices for Delone scalars in S6

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aRonin Institute, 9515 NE 137th Street, Kirkland, WA 98034-1820, USA, bRonin Institute, c/o NSLS-II, Brookhaven National Laboratory, Upton, NY 11973, USA, and cLawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
*Correspondence e-mail: lawrence.andrews@ronininstitute.org

Edited by A. Altomare, Institute of Crystallography - CNR, Bari, Italy (Received 2 July 2019; accepted 25 October 2019)

The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three-by-three matrices that transform three-space lattice vectors. Using those three-by-three matrices when working in the six-dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three-space representation, apply the three-by-three matrices and then back-transform to the six-space representation, but it is much simpler to have the equivalent six-by-six matrices and apply them directly. The general form of the transformation from the three-space matrix to the corresponding matrix operating on Selling scalars (expressed in space S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)

1. Introduction

The transformations from the primitive cells of the centered Bravais lattices to the centered cells and between alternative unit cells have conventionally been listed as matrices that are applied to three-space lattice vectors (Burzlaff & Zimmermann, 1985[Burzlaff, H. & Zimmermann, H. (1985). Z. Kristallogr. 170, 247-262.]; Burzlaff et al., 1992[Burzlaff, H., Zimmermann, H. & de Wolff, P. M. (1992). Crystal Lattices. International Tables for Crystallography, Vol. A, ch. 9, pp. 734-744. Dordrecht: Kluwer Academic Publishers.]). However, for both the major cell reductions [Niggli (1928[Niggli, P. (1928). Krystallographische und Strukturtheoretische Grundbegriffe. Handbuch der Experimentalphysik, Vol. 7, Part 1. Leipzig: Akademische Verlagsgesellschaft.]) and Delone (1933[Delone, B. N. (1933). Z. Krystallogr. 84, 109-149.])], it is convenient to work in a higher-dimension space than E3, as reported by Andrews & Bernstein (1988[Andrews, L. C. & Bernstein, H. J. (1988). Acta Cryst. A44, 1009-1018.]) for G6 reduction and Andrews et al. (2019[Andrews, L. C., Bernstein, H. J. & Sauter, N. K. (2019). Acta Cryst. A75, 115-120.]) for S6. Therefore, as we did for G6 (Andrews & Bernstein, 1988[Andrews, L. C. & Bernstein, H. J. (1988). Acta Cryst. A44, 1009-1018.]), we need to provide the mathematically equivalent six-by-six matrices for centering in S6. This reduces the need to convert repeatedly from S6 into three-space vectors, transform the three-space vectors, and then transform back into S6. We derive the general form and list the particular matrices for converting from the 24 canon­ical Delone types to centered lattices in S6.

2. Background and notation

2.1. The space [{\bf S}^{\bf 6}]

Andrews et al. (2019[Andrews, L. C., Bernstein, H. J. & Sauter, N. K. (2019). Acta Cryst. A75, 115-120.]) introduced the space S6 as an alternative representation of crystallographic lattices. The space is defined in terms of the `Selling scalars' used in Selling reduction (Selling, 1874[Selling, E. (1874). J. Reine Angew. Math. (Crelle's J.), 1874(77), 143-229.]) and by Delone (1933[Delone, B. N. (1933). Z. Krystallogr. 84, 109-149.]) for the classification of lattices. A point s in S6 is defined by

[s = [{\bf b} \cdot {\bf c}, {\bf a} \cdot {\bf c}, {\bf a} \cdot {\bf b}, {\bf a} \cdot {\bf d}, {\bf b} \cdot {\bf d}, {\bf c} \cdot {\bf d}] , \eqno(1)]

where d = −abc.

2.2. The space [{\bf E}^{{\bf 3}\times{\bf 3}}]

A crystallographic unit cell is commonly represented as three cell edge lengths and three angles, [a, b, c, α, β, γ], but, when presenting common operations on unit cells, it is convenient to express each of the cell edges as a vector in the three-dimensional space of real numbers E3 (also written as R3) with each cell expressed as a 3 × 3 matrix of real numbers, i.e. as an element of E3 × E3 (see Burzlaff et al., 1992[Burzlaff, H., Zimmermann, H. & de Wolff, P. M. (1992). Crystal Lattices. International Tables for Crystallography, Vol. A, ch. 9, pp. 734-744. Dordrecht: Kluwer Academic Publishers.]). An issue with this approach is that we should get the same crystallographic unit cell after any proper rotation of E3, i.e. by any unitary matrix of determinant +1. Such proper rotation matrices form the Lie group SO(3). Therefore, formally we should treat any matrix representation of a cell cE3 × E3 as equivalent to rc, for all rSO(3), and work in the space of (E3 × E3)/SO(3). We call this space of equivalence classes E3×3.

Because the matrices that multiply cells represented in E3×3 are indistinguishable from ordinary 3 × 3 matrices, we will designate them as ME3, with the understanding that they may be applied in either space E3 or space E3×3.

The convention in E3 is to use the cell edges as the basis vectors of the space. There are infinitely many choices of the orientation to form the basis. Currently, it is the common convention to orient one edge vector along the x axis etc. in a right-handed setting. The convention in S6 is to use unit vectors [100000], [010000], ….

2.3. The method for deriving a transformation in one space from a transformation in another

Consider two spaces X and Y with one invertible conversion MYX mapping

[M_{YX}:Y\rightarrow X , \eqno(2)]

and a not necessarily invertible mapping

[M_{XY} = M_{YX}^{-1}:X\rightarrow Y , \eqno(3)]

and a transformation of X into X

[T:X\rightarrow X . \eqno(4)]

We compose the mappings and the transformation to define a new transformation U of Y into Y

[U:Y\rightarrow Y = M_{XY} \, T \, M_{YX} . \eqno(5)]

If Y is a finite-dimensional linear vector space and U is linear, then we can represent U as a matrix [https://en.wikipedia.org/wiki/Linear_map] by choosing an appropriate basis. Section A in the supporting information considers the linearity in more detail.

If X is in E3×3 and Y is in S6, we can map E3×3 to S6,

[\eqalignno{M_{XY} ({\bf a}, {\bf b}, {\bf c}) = & \, [{\bf b} \cdot {\bf c}, {\bf a} \cdot {\bf c}, {\bf a} \cdot {\bf b}, {\bf a} \cdot (-{\bf a} - {\bf b} - {\bf c}), \cr & \, {\bf b} \cdot (-{\bf a} - {\bf b} - {\bf c}), {\bf c} \cdot (-{\bf a} - {\bf b} - {\bf c})] . &(6)}]

Because MXY is invariant under rotation, there are infinitely many choices for the inverse. We can choose, for example,

[M_{YX} (y) = [{\bf a} (y), {\bf b} (y), {\bf c} (y)] , \eqno(7)]

where

[{\bf a} (y) = \left [ \left ( -y_{2} - y_{3} - y_{4} \right )^{1/2}, 0, 0 \right ] , \eqno(8)]

[{\bf b} (y) = \left [ {{y_{3}} \over {{\bf a} (y)_{1}}}, \left ( -y_{1} - y_{3} - y_{5} + {{y_{3}^{2}} \over {y_{2} + y_{3} + y_{4}}} \right )^{1/2}, 0 \right ] , \eqno(9)]

[{\bf c_{12}} (y) = \left [ {{y_{2}} \over {{\bf a} (y)_{1}}}, y_{1} - {\bf b} (y)_{1} {{y_{2}} \over {{\bf a} (y)_{1} {\bf b} (y)_{2}}}, 0 \right ] , \eqno(10)]

[\eqalignno{{\bf c} (y) = & \, \Big [ {\bf c_{12}} (y)_{1}, {\bf c_{12}} (y)_{2}, \cr & \, \left ( -s[6] - s[2] - s[1] - {\bf c_{12}} (y)_{1}^{2} - {\bf c_{12}} (y)_{2}^{2} \right )^{1/2} \Big ] , &(11)}]

which is applicable for a reasonable set of valid S6 cells. This would then allow a similar demonstration to that given in Section A of the supporting information with the mapping from Y to X being the simple square root that a linear T generates a linear U in this more complex but similar case. The details are left as an exercise for the reader.

The point is that, because U is linear, the components of its representation as a matrix can be determined by applying it to basis vectors each with only one non-zero component, letting X be in E3×3, Y be in S6 and MXY be E3toS6 (see Section 3.1[link]).

3. Converting an [{\bf E}^{\bf 3}] matrix to an [{\bf S}^{\bf 6}] matrix

If we represent the cell as an S6 vector (Andrews et al., 2019[Andrews, L. C., Bernstein, H. J. & Sauter, N. K. (2019). Acta Cryst. A75, 115-120.]), we can define an operator E3toS6 where E3toS6(a, b, c) = [b · c, a · c, a · b, a · d, b · d, c · d], where d = −abc.

We form a matrix operating in E3×3, ME3 = [[m1,1, m1,2, m1,3], [m2,1, m2,2, m2,3], [m3,1, m3,2, m3,3]]. We need to compute a 6 × 6 matrix, MS6, to operate on S6 vectors.

The obvious basis vectors for S6, [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0] and [0, 0, 0, 0, 0, 1], do not correspond to real vectors in E3, since a dot product of 1 for real non-zero unit basis vectors would imply an angle between them of zero, i.e. that they are identical, but if two unit basis E3 vectors, say a and b, are identical, and one, say c, is perpendicular to both a and b, then the d = −abc vector cannot be perpendicular to a or b, because a · d = b · d = −a · aa · ba · c = −2a · a, which cannot be zero. Therefore we use the negatives of those S6 basis vectors.

The E3×3 basis vectors we choose are shown in Table 1[link] with the corresponding S6 vectors.

Table 1
E3×3 basis vectors matched to S6 basis vectors

E3×3 basis vector S6 basis vector
[[0, 0, 0], [1, 0, 0], [[\overline{1},0,0]]] [[\overline{1},0,0,0,0,0]]
[[1, 0, 0], [0, 0, 0], [[\overline{1},0,0]]] [[0,\overline{1},0,0,0,0]]
[[1, 0, 0], [[\overline{1},0,0]], [0, 0, 0]] [[0,0,\overline{1},0,0,0]]
[[1, 0, 0], [0, 0, 0], [0, 0, 0]] [[0,0,0,\overline{1},0,0]]
[[0, 0, 0], [1, 0, 0], [0, 0, 0]] [[0,0,0,0,\overline{1},0]]
[[0, 0, 0], [0, 0, 0], [1, 0, 0]] [[0,0,0,0,0,\overline{1}]]

3.1. Relationship to [{\bf S}^{\bf 6}]

We use the operator E3toS6 that converts a vector in E3×3 to one in S6 (see above). In our case, we are starting from reduced unit cells, which means that in S6 all six scalars are zero or negative. We choose the S6 basis vectors to have zero scalars except for a single −1 in each. In E3×3 we choose an orthogonal set (see above), where for each E3×3 vector E3toS6 produces only the corresponding S6 basis vector.

As an illustrative example, we choose the first basis vector [above, and matrix E in step (b) in Fig. 1[link]],

[[[0, 0, 0], [1, 0, 0], [-1, 0, 0]], ({\bf d} = [0, 0, 0]). ]

We apply ME3 to that vector [step (a) in Fig. 1[link]] and the corresponding MS6 to the corresponding S6 vector. Because only one element of the S6 basis vector is non-zero, the result of multiplying by MS6 produces only the elements of the corresponding column of MS6, with the other elements being zero [step (c) in Fig. 1[link]]. When we multiply that E3×3 [step (b) in Fig. 1[link]] basis vector by ME3 and then convert to S6 using E3toS6 [step (d) in Fig. 1[link]], the resulting elements of S6 are the same S6 column elements expressed in terms of the elements of ME3.

[Figure 1]
Figure 1
The logic of determining MS6. (a) E3toS6 is an operator that will generate a vector s in S6 from a vector e in E3×3. (b) E is a matrix operating on E3×3 and S is a matrix operating on S6. Correspondingly, we can rewrite (a) in this more general form. (c) Choosing as an example the first basis vector ([1, 0, 0, 0, 0, 0]) in the list of basis vectors, we can then multiply by S. The first column of elements of S can then be placed into the matrix as indicated. (d) In like manner, we can multiply the first basis vector expressed in E3×3 by the matrix E in E3 that corresponds to the matrix S. However, in this case, the elements of MS6 can be computed from the list of calculations above for the first basis vector and the values of matrix E. Repeating this process for each of the six basis vectors completes S.

In each case, only one of the Selling scalars will be −1 and the others will be 0 [step (c) in Fig. 1[link]]. Because S6 is invariant under rotations of E3, we could have used any unit vector on E3 in place of [1, 0, 0], and we would have obtained the same set of S6 basis vectors.

The computer algebra system Maxima (Version 5.36.1; Chou & Schelter, 1986[Chou, S.-C. & Schelter, W. F. (1986). J. Autom. Reasoning, 2, 253-273.]; https://maxima.sourceforge.net) was used to generate the following equations.

For simplicity, we show the definition of the first row of the S6 matrix (the complete matrix is listed in Section B of the supporting information). For any three-by-three matrix, ME3, the equation below computes the first column of the negative of the complete matrix (the supporting information has all the columns):

[\eqalignno{ & E^{3}toS^{6} (M_{E^{3}} [[0, 0, 0], [1, 0, 0], [{\overline 1}, 0, 0]]) = \cr & \quad [m_{2,3}m_{3,3}-m_{2,2}m_{3,3}-m_{2,3}m_{3,2}+m_{2,2}m_{3,2}, \cr & \quad m_{1,3}m_{3,3}-m_{1,2}m_{3,3}-m_{1,3}m_{3,2}+m_{1,2}m_{3,2}, \cr & \quad m_{1,3}m_{2,3}-m_{1,2}m_{2,3}-m_{1,3}m_{2,2}+m_{1,2}m_{2,2}, \cr & \quad -m_{1,3}m_{3,3}+m_{1,2}m_{3,3}+m_{1,3}m_{3,2}-m_{1,2}m_{3,2}-m_{1,3}m_{2,3}\cr &\quad\quad +m_{1,2}m_{2,3}+m_{1,3}m_{2,2}-m_{1,2}m_{2,2}-m_{1,3}^2+2m_{1,2}m_{1,3}&\cr&\quad\quad-m_{1,2}^2, \cr & \quad -m_{2,3}m_{3,3}+m_{2,2}m_{3,3}+m_{2,3}m_{3,2}-m_{2,2}m_{3,2}-m_{2,3}^2&\cr &\quad\quad+2m_{2,2}m_{2,3}-m_{1,3}m_{2,3}+m_{1,2}m_{2,3} -m_{2,2}^2+m_{1,3}m_{2,2}&\cr&\quad\quad-m_{1,2}m_{2,2}, \cr & \quad -m_{3,3}^2+2m_{3,2}m_{3,3}-m_{2,3}m_{3,3}+m_{2,2}m_{3,3}-m_{1,3}m_{3,3}&\cr &\quad\quad+m_{1,2}m_{3,3} -m_{3,2}^2+m_{2,3}m_{3,2}-m_{2,2}m_{3,2}+m_{1,3}m_{3,2}&\cr &\quad\quad-m_{1,2}m_{3,2}]. &(12)}]

4. Conversion of reduced primitive lattices to centered lattices

Tables 2[link] and 3[link] list the matrices (Burzlaff & Zimmermann, 1985[Burzlaff, H. & Zimmermann, H. (1985). Z. Kristallogr. 170, 247-262.]) for converting from primitive to standard centered lattices, computed from the above derivations. The designations of the 24 Delone types are slight modifications of the symbols of Delone (1933[Delone, B. N. (1933). Z. Krystallogr. 84, 109-149.]) to more modern forms. His cubic lattices are changed from `K' to `C', tetragonal from `Q' to `T' and triclinic from `T' to `A'. For each lattice, the E3×3 matrix is listed, followed on the next line by the corresponding S6 matrix.

Table 2
The first eight of the transformation matrices for each of the 24 Delone types

The E3×3 and S6 matrices are both listed in each case. The remaining 16 cases are in Table 3[link].

Type Lattice ME3 and MS6
C1 cI [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
    [[[1,0,0,\overline{1},0,0]], [[0,1,0,0,\overline{1},0]], [[0,0,1,0,0,\overline{1}]], [0, 0, 0, 0, 2, 2], [0, 0, 0, 2, 0, 2], [0, 0, 0, 2, 2, 0]]
C3 cF [[1, 1, 0], [[\overline{1},1,0]], [1, 1, 2]]
    [[[1,\overline{1},0,1,\overline{1},0]], [[1,1,0,\overline{1},\overline{1},0]], [[\overline{1},1,0,1,\overline{1},0]], [[1,\overline{1},0,1,3,0]], [[1,1,4,\overline{1},3,0]], [[\overline{1},1,0,1,3,4]]]
C5 cP Identity
R1 hR [[[1,\overline{1},0]], [[0,1,\overline{1}]], [1, 1, 1]]
    [[[0,0,0,0,\overline{1},1]], [[0,0,0,\overline{1},1,0]], [[2,\overline{1},2,0,1,0]], [[\overline{1},2,2,2,\overline{1},0]], [[2,2,\overline{1},0,1,0]], [0, 0, 0, 2, 1, 0]]
R3 hR [[1, 0, 0], [0, 0, 1], [1, 3, 2]]
    [[[1,\overline{1},0,0,0,\overline{2}]], [[0,1,2,\overline{1},0,0]], [0, 1, 0, 0, 0, 0], [[0,\overline{1},\overline{1},2,0,0]], [0, 1, 0, 0, 0, 3], [0, 1, 2, 2, 9, 6]]
T1 tI [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
    [[[1,0,0,\overline{1},0,0]], [[0,1,0,0,\overline{1},0]], [[0,0,1,0,0,\overline{1}]], [0, 0, 0, 0, 2, 2], [0, 0, 0, 2, 0, 2], [0, 0, 0, 2, 2, 0]]
T2 tI [[1, 0, 0], [0, 1, 0], [1, 1, 2]]
    [[[1,0,0,0,\overline{1},0]], [[0,1,0,\overline{1},0,0]], [0, 0, 1, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 2, 2, 4]]
T5 tP Identity

Table 3
The second 16 of the transformation matrices for each of the 24 Delone types

The E3×3 and S6 matrices are both listed in each case. The first eight cases are in Table 2[link].

Type Lattice ME3 and MS6
O1A oF [[1, 1, 0], [[\overline{1},1,0]], [1, 1, 2]]
    [[[1,\overline{1},0,1,\overline{1},0]], [[1,1,0,\overline{1},\overline{1},0]], [[\overline{1},1,0,1,\overline{1},0]], [[1,\overline{1},0,1,3,0]], [[1,1,4,\overline{1},3,0]], [[\overline{1},1,0,1,3,4]]]
O1B oI [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
    [[[1,0,0,\overline{1},0,0]], [[0,1,0,0,\overline{1},0]], [[0,0,1,0,0,\overline{1}]], [0, 0, 0, 0, 2, 2], [0, 0, 0, 2, 0, 2], [0, 0, 0, 2, 2, 0]]
O2 oI [[1, 0, 0], [0, 1, 0], [1, 1, 2]]
    [[[1,0,0,0,\overline{1},0]], [[0,1,0,\overline{1},0,0]], [0, 0, 1, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 2, 2, 4]]
O3 oI [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
    [[[1,0,0,\overline{1},0,0]], [[0,1,0,0,\overline{1},0]], [[0,0,1,0,0,\overline{1}]], [0, 0, 0, 0, 2, 2], [0, 0, 0, 2, 0, 2], [0, 0, 0, 2, 2, 0]]
O4 oS [[[1,\overline{1},0]], [1, 1, 0], [0, 0, 1]]
    [[1, 1, 0, 0, 0, 0], [[\overline{1},1,0,0,0,0]], [[1,\overline{1},0,\overline{1},1,0]], [1, 1, 4, 2, 0, 0], [[\overline{1},1,0,2,0,0]], [[1,\overline{1},0,0,0,1]]]
O5 oP Identity
M1A mS [[[\overline{1},\overline{1},\overline{1}]], [[1,\overline{1},0]], [0, 0, 1]]
    [[[\overline{1},1,0,0,0,0]], [0, 0, 0, 0, 0, 1], [[0,0,0,1,\overline{1},0]], [0, 0, 0, 0, 2, 0], [2, 0, 4, 0, 2, 0], [2, 0, 0, 0, 0, 0]]
M1B mS [[0, 1, 1], [1, 1, 0], [[\overline{1},0,\overline{1}]]]
    [[[\overline{1},0,0,1,0,0]], [[0,0,\overline{1},0,0,1]], [[0,1,0,0,\overline{1},0]], [0, 0, 2, 0, 2, 0], [2, 0, 0, 0, 2, 0], [2, 0, 2, 0, 0, 0]]
M2A mS [[[\overline{1},\overline{1},\overline{2}]], [0, 1, 0], [1, 0, 0]]
    [[0, 0, 1, 0, 0, 0], [[0,\overline{1},0,1,0,0]], [[\overline{1},0,0,0,1,0]], [2, 2, 0, 0, 0, 4], [2, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0]]
M2B mS [[0, 1, 1], [1, 1, 0], [[\overline{1},0,\overline{1}]]]
    [[[\overline{1},0,0,1,0,0]], [[0,0,\overline{1},0,0,1]], [[0,1,0,0,\overline{1},0]], [0, 0, 2, 0, 2, 0], [2, 0, 0, 0, 2, 0], [2, 0, 2, 0, 0, 0]]
M3 mS [[[\overline{1},\overline{1},\overline{2}]], [0, 1, 0], [1, 0, 0]]
    [[0, 0, 1, 0, 0, 0], [[0,\overline{1},0,1,0,0]], [[\overline{1},0,0,0,1,0]], [2, 2, 0, 0, 0, 4], [2, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0]]
M4 mP Identity
A1 aP Identity
A2 aP Identity
A3 aP Identity
H4 hP Identity

Burzlaff & Zimmermann (1985[Burzlaff, H. & Zimmermann, H. (1985). Z. Kristallogr. 170, 247-262.]) renumbered the lattice types of Delone (1933[Delone, B. N. (1933). Z. Krystallogr. 84, 109-149.]). For example, the cubic lattices in Delone (1933[Delone, B. N. (1933). Z. Krystallogr. 84, 109-149.]) are K1, K3 and K5. In the reports by Burzlaff & Zimmermann (1985[Burzlaff, H. & Zimmermann, H. (1985). Z. Kristallogr. 170, 247-262.]) and Burzlaff et al. (1992[Burzlaff, H., Zimmermann, H. & de Wolff, P. M. (1992). Crystal Lattices. International Tables for Crystallography, Vol. A, ch. 9, pp. 734-744. Dordrecht: Kluwer Academic Publishers.]), they are listed as K1, K2 and K3. Here they are listed as C1, C3 and C5. The full enumeration of the types is shown in Fig. 2[link]. It is important to note that Burzlaff et al. (1992[Burzlaff, H., Zimmermann, H. & de Wolff, P. M. (1992). Crystal Lattices. International Tables for Crystallography, Vol. A, ch. 9, pp. 734-744. Dordrecht: Kluwer Academic Publishers.]) showed the matrices as the transposes of the corresponding matrices of Burzlaff & Zimmermann (1985[Burzlaff, H. & Zimmermann, H. (1985). Z. Kristallogr. 170, 247-262.]). We have chosen to start from the earlier paper. The MS6 matrices produced are then applied to the left of the S6 vectors. The International Tables for Crystallography (Burzlaff et al., 2016[Burzlaff, H., Grimmer, H., Gruber, B., deWolff, P. & Zimmermann, H. (2016). Crystal Lattices. International Tables for Crystallography, Vol. A, ch. 3.1, pp. 698-718. Chester: International Union of Crystallography.]) use the same convention as Burzlaff & Zimmermann (1985[Burzlaff, H. & Zimmermann, H. (1985). Z. Kristallogr. 170, 247-262.]).

[Figure 2]
Figure 2
Delone's table of the 24 canonical types (modified).

5. Summary

This paper is a reference for researchers who need a method that applies to the space S6, but for which only the matrices applicable to the edge vectors of the unit cell are available. In addition, we have provided a list of the matrices required for conversion of primitive cells in S6 to the more standard centered presentations.

6. Availability of code

The C++ code for distance calculations in S6 is available at github.com, both at https://github.com/duck10/LatticeRepLib.git and https://github.com/yayahjb/ncdist.git For E3toS6 see LatticeRepLib/MatS6.cpp.

Supporting information


Acknowledgements

Careful copy editing and corrections by Frances C. Bernstein are gratefully acknowledged. Our thanks to Jean Jakoncic and Alexei Soares for helpful conversations and access to data and facilities at Brookhaven National Laboratory.

Funding information

Funding for this research was provided in part by: Dectris Ltd, US Department of Energy Offices of Biological and Environmental Research and of Basic Energy Sciences (grant No. DE-AC02-98CH10886; grant No. E-SC0012704); US National Institutes of Health (grant No. P41RR012408; grant No. P41GM103473; grant No. P41GM111244; grant No. R01GM117126; grant No. P30GM133893; grant No. R21GM129570).

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