Abstract
Algorithms for quantifying the differences between two lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial crystallography. It is particularly desirable for the differences between lattices to be computed as a perturbation-stable metric, i.e. as distances that satisfy the triangle inequality, so that standard tree-based nearest-neighbor algorithms can be used, and for which small changes in the lattices involved produce small changes in the distances computed. A perturbation-stable metric space related to the reduction algorithm of Selling and to the Bravais lattice determination methods of Delone is described. Two ways of representing the space, as six-dimensional real vectors or equivalently as three-dimensional complex vectors, are presented and applications of these metrics are discussed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.).
Highlights
Andrews et al (2019) discuss the simplification resulting from using Selling reduction as opposed to using Niggli reduction
Algorithms for quantifying the differences among lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial crystallography
Andrews et al (1980) discussed V7, a perturbation-stable space in which, using real- and reciprocalspace Niggli reduction, a lattice is represented by three cell edge lengths, three reciprocal cell edge lengths and the cell volume, which was proposed for cell database searches, but which has difficulties when used for lattice determination
Summary
Andrews et al (2019) discuss the simplification resulting from using Selling reduction as opposed to using Niggli reduction. Andrews et al (1980) discussed V7, a perturbation-stable space in which, using real- and reciprocalspace Niggli reduction, a lattice is represented by three cell edge lengths, three reciprocal cell edge lengths and the cell volume, which was proposed for cell database searches, but which has difficulties when used for lattice determination. Minimizing among distances computed from alternate paths between Selling-reduced cells with appropriate sewing at the six boundaries of the Selling-reduced fundamental region of S6 yields a computationally sound metric space within which to do lattice identification, cell database searching and serial crystallography clustering. The objective of this paper is to explain how to compute the distances between lattices using S6 and C3
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Acta crystallographica. Section A, Foundations and advances
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
