Converting three-space matrices to equivalent six-space matrices for Delone scalars in S 6

• Published in: Acta crystallographica. Section A, Foundations and advances Vol. 76; no. Pt 1; pp. 79 - 83
• Main Authors: Andrews, Lawrence C, Bernstein, Herbert J, Sauter, Nicholas K
• Format: Journal Article
• Language: English
• Published: United States International Union of Crystallography (IUCr) 01.01.2020
• Subjects: centered lattices; centering transformations; Delaunay; Delone; lattice centering; matrix transformations; Niggli; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; reduced cells; Selling; Delone; centered lattices; matrix transformations; Selling; lattice centering; centering transformations; Delaunay; Niggli; reduced cells
• ISSN: 2053-2733; 2053-2733
• DOI: 10.1107/S2053273319014542

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 S ) is derived, and the particular S matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.).