The Geometry of Niggli Reduction III: SAUC -- Search of Alternate Unit Cells

A crystallographic cell is a representation of a lattice, but each lattice can be represented just as well by any of an infinite number of such unit cells. Searching for matches to an experimentally determined crystallographic unit cell in a large collection of previously determined unit cells is a useful verification step in synchrotron data collection and can be a screen for "similar" structures, but it is more useful to search for a match to the lattice represented by the experimentally determined cell. For identification of substances with small cells, a unit cell match may be sufficient for unique identification. Due to experimental error and multiple choices of cells and differing choices of lattice centering representing the same lattice, simple searches based on raw cell edges and angles can miss similarities among lattices. A database of lattices using the G6 representation of the Niggli-reduced cell as the search key provides a more robust and complete search. Searching is implemented by finding the distance from the probe cell to related cells using a topological embedding of the Niggli reduction in G6, so that all cells representing similar lattices will be found. Comparison of results with those from older cell-based search algorithms suggests significant value in the new approach.