Date Published: May 01, 2019
Publisher: International Union of Crystallography
Author(s): Lawrence C. Andrews, Herbert J. Bernstein, Nicholas K. Sauter.
Algorithms for defining the difference between two lattices are described. They are based on the work of Selling and Delone (Delaunay).
Andrews et al. (2019 ▸) discuss the simplification resulting from using Selling reduction as opposed to using Niggli reduction. Here we continue that discussion with information on the space of unit cells and the subspace of reduced cells as the six-dimensional space of Selling inner products.
For a Bravais tetrahedron (Bravais, 1850 ▸) with defining vectors , , , (the edge vectors of the unit cell plus the negative sum of them), a point in isA simple example is the orthorhombic unit cell (10, 12, 20, 90, 90, 90) (a, b, c, α, β, γ). The corresponding vector is The scalars in are of a single type, unlike cell parameters (lengths and angles) and unlike (squared lengths and dot products). Delone et al. (1975 ▸) state ‘The Selling parameters are geometrically fully homogeneous’.
Alternatively, the space can be as represented as , a space of three complex axes. has advantages for understanding some of the properties of the space. When we compose of the scalars , the components of are the pairs of ‘opposite’ (Delone et al., 1975 ▸) scalars. In terms of the elements of , a unit cell in is . The presentation of the vector (1) from Section 2 is .
We require a distance metric that defines the shortest path among all the representations of two points (lattices). Common uses of a metric for lattices are searching in databases of unit-cell parameters, finding possible Bravais lattice types, locating possible suitable molecular replacement candidates and, recently, clustering of the images from serial crystallography.
This is a time of disruptive change in the image-clustering methods used in structural biology to understand polymorphs and dynamics at X-ray free-electron lasers and at synchrotrons. Serial crystallography is an essential technique at X-ray free-electron laser (XFEL) light sources and has become an important technique at synchrotrons as well (Rossmann, 2014 ▸), especially at newer high-brilliance beamlines. Methods that distribute the many diffraction images into clusters that likely represent crystals composed of proteins in similar states allow one to separate polymorphs and to categorize their dynamics. The inexorable increases in brilliance of these sources drives us to seek continual improvement in our algorithms and pipelines.
We have presented representations of a space (parameterized as and ) based on the Selling parameters and using the Selling reduction. Geometrically, this represents a significant simplification compared with the complex, non-convex asymmetric unit of Niggli reduction and .
The C++ code for distance calculations in is available using https://github.com/; for CS6Dist.h, use https://github.com/yayahjb/ncdist; for PointDistanceFollower (Follower implementation), S6Dist.h and .cpp, use https://github.com/duck10/LatticeRepLib.