research papers
Converting threespace matrices to equivalent sixspace matrices for Delone scalars in S^{6}
^{a}Ronin Institute, 9515 NE 137th Street, Kirkland, WA 980341820, USA, ^{b}Ronin Institute, c/o NSLSII, Brookhaven National Laboratory, Upton, NY 11973, USA, and ^{c}Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
^{*}Correspondence email: lawrence.andrews@ronininstitute.org
The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as threebythree matrices that transform threespace lattice vectors. Using those threebythree matrices when working in the sixdimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the threespace representation, apply the threebythree matrices and then backtransform to the sixspace representation, but it is much simpler to have the equivalent sixbysix matrices and apply them directly. The general form of the transformation from the threespace matrix to the corresponding matrix operating on Selling scalars (expressed in space S^{6}) is derived, and the particular S^{6}matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)
Keywords: Delaunay; Delone; centering transformations; centered lattices; reduced cells; lattice centering; Niggli; Selling; matrix transformations.
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 threespace lattice vectors (Burzlaff & Zimmermann, 1985; Burzlaff et al., 1992). However, for both the major cell reductions [Niggli (1928) and Delone (1933)], it is convenient to work in a higherdimension space than E^{3}, as reported by Andrews & Bernstein (1988) for G^{6} reduction and Andrews et al. (2019) for S^{6}. Therefore, as we did for G^{6} (Andrews & Bernstein, 1988), we need to provide the mathematically equivalent sixbysix matrices for centering in S^{6}. This reduces the need to convert repeatedly from S^{6} into threespace vectors, transform the threespace vectors, and then transform back into S^{6}. We derive the general form and list the particular matrices for converting from the 24 canonical Delone types to centered lattices in S^{6}.
2. Background and notation
2.1. The space
Andrews et al. (2019) introduced the space S^{6} as an alternative representation of crystallographic lattices. The space is defined in terms of the `Selling scalars' used in Selling reduction (Selling, 1874) and by Delone (1933) for the classification of lattices. A point s in S^{6} is defined by
where d = −a − b − c.
2.2. The space
A crystallographic 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 threedimensional space of real numbers E^{3} (also written as R^{3}) with each cell expressed as a 3 × 3 matrix of real numbers, i.e. as an element of E^{3} × E^{3} (see Burzlaff et al., 1992). An issue with this approach is that we should get the same crystallographic after any proper rotation of E^{3}, 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 c ∈ E^{3} × E^{3} as equivalent to rc, for all r ∈ SO(3), and work in the space of (E^{3} × E^{3})/SO(3). We call this space of equivalence classes E^{3×3}.
is commonly represented as three cell edge lengths and three angles, [Because the matrices that multiply cells represented in E^{3×3} are indistinguishable from ordinary 3 × 3 matrices, we will designate them as M_{E3}, with the understanding that they may be applied in either space E^{3} or space E^{3×3.}
The convention in E^{3} 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 righthanded setting. The convention in S^{6} 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 M_{YX} mapping
and a not necessarily invertible mapping
and a transformation of X into X
We compose the mappings and the transformation to define a new transformation U of Y into Y
If Y is a finitedimensional linear 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 E^{3×3} and Y is in S^{6}, we can map E^{3×3} to S^{6},
Because M_{XY} is invariant under rotation, there are infinitely many choices for the inverse. We can choose, for example,
where
which is applicable for a reasonable set of valid S^{6} 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 nonzero component, letting X be in E^{3×3}, Y be in S^{6} and M_{XY} be E^{3}toS^{6} (see Section 3.1).
3. Converting an matrix to an matrix
If we represent the cell as an S^{6} vector (Andrews et al., 2019), we can define an operator E^{3}toS^{6} where E^{3}toS^{6}(a, b, c) = [b · c, a · c, a · b, a · d, b · d, c · d], where d = −a − b − c.
We form a matrix operating in E^{3×3}, M_{E3} = [[m_{1,1}, m_{1,2}, m_{1,3}], [m_{2,1}, m_{2,2}, m_{2,3}], [m_{3,1}, m_{3,2}, m_{3,3}]]. We need to compute a 6 × 6 matrix, M_{S6}, to operate on S^{6} vectors.
The obvious basis vectors for S^{6}, [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 E^{3}, since a dot product of 1 for real nonzero unit basis vectors would imply an angle between them of zero, i.e. that they are identical, but if two unit basis E^{3} vectors, say a and b, are identical, and one, say c, is perpendicular to both a and b, then the d = −a − b − c vector cannot be perpendicular to a or b, because a · d = b · d = −a · a − a · b − a · c = −2a · a, which cannot be zero. Therefore we use the negatives of those S^{6} basis vectors.
The E^{3×3} basis vectors we choose are shown in Table 1 with the corresponding S^{6} vectors.

3.1. Relationship to
We use the operator E^{3}toS^{6} that converts a vector in E^{3×3} to one in S^{6} (see above). In our case, we are starting from reduced unit cells, which means that in S^{6} all six scalars are zero or negative. We choose the S^{6} basis vectors to have zero scalars except for a single −1 in each. In E^{3×3} we choose an orthogonal set (see above), where for each E^{3×3} vector E^{3}toS^{6} produces only the corresponding S^{6} basis vector.
As an illustrative example, we choose the first basis vector [above, and matrix E in step (b) in Fig. 1],
We apply M_{E3} to that vector [step (a) in Fig. 1] and the corresponding M_{S6} to the corresponding S^{6} vector. Because only one element of the S^{6} basis vector is nonzero, the result of multiplying by M_{S6} produces only the elements of the corresponding column of M_{S6}, with the other elements being zero [step (c) in Fig. 1]. When we multiply that E^{3×3} [step (b) in Fig. 1] basis vector by M_{E3} and then convert to S^{6} using E^{3}toS^{6} [step (d) in Fig. 1], the resulting elements of S^{6} are the same S^{6} column elements expressed in terms of the elements of M_{E3}.
In each case, only one of the Selling scalars will be −1 and the others will be 0 [step (c) in Fig. 1]. Because S^{6} is invariant under rotations of E^{3}, we could have used any unit vector on E^{3} in place of [1, 0, 0], and we would have obtained the same set of S^{6} basis vectors.
The computer algebra system Maxima (Version 5.36.1; Chou & Schelter, 1986; https://maxima.sourceforge.net) was used to generate the following equations.
For simplicity, we show the definition of the first row of the S^{6} matrix (the complete matrix is listed in Section B of the supporting information). For any threebythree matrix, M_{E3}, the equation below computes the first column of the negative of the complete matrix (the supporting information has all the columns):
4. Conversion of reduced primitive lattices to centered lattices
Tables 2 and 3 list the matrices (Burzlaff & Zimmermann, 1985) 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) 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 E^{3×3} matrix is listed, followed on the next line by the corresponding S^{6} matrix.


Burzlaff & Zimmermann (1985) renumbered the lattice types of Delone (1933). For example, the cubic lattices in Delone (1933) are K1, K3 and K5. In the reports by Burzlaff & Zimmermann (1985) and Burzlaff et al. (1992), 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. It is important to note that Burzlaff et al. (1992) showed the matrices as the transposes of the corresponding matrices of Burzlaff & Zimmermann (1985). We have chosen to start from the earlier paper. The M_{S6} matrices produced are then applied to the left of the S^{6} vectors. The International Tables for Crystallography (Burzlaff et al., 2016) use the same convention as Burzlaff & Zimmermann (1985).
5. Summary
This paper is a reference for researchers who need a method that applies to the space S^{6}, but for which only the matrices applicable to the edge vectors of the are available. In addition, we have provided a list of the matrices required for conversion of primitive cells in S^{6} to the more standard centered presentations.
6. Availability of code
The C++ code for distance calculations in S^{6} is available at github.com, both at https://github.com/duck10/LatticeRepLib.git and https://github.com/yayahjb/ncdist.git For E^{3}toS^{6} see LatticeRepLib/MatS6.cpp.
Supporting information
Discussion of linearity and complete definition of the transformation matrix. DOI: https://doi.org/10.1107/S2053273319014542/ae5074sup1.pdf
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. DEAC0298CH10886; grant No. ESC0012704); 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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