The Geometry of Niggli Reduction I: The Boundary Polytopes of the Niggli Cone

Correct identification of the Bravais lattice of a crystal is an important step in structure solution. Niggli reduction is a commonly used technique. We investigate the boundary polytopes of the Niggli-reduced cone in the six-dimensional space G6 by algebraic analysis and organized random probing of regions near 1- through 8-fold boundary polytope intersections. We limit consideration of boundary polytopes to those avoiding the mathematically interesting but crystallographically impossible cases of 0 length cell edges. Combinations of boundary polytopes without a valid intersection in the closure of the Niggli cone or with an intersection that would force a cell edge to 0 or without neighboring probe points are eliminated. 216 boundary polytopes are found: 15 5-D boundary polytopes of the full G6 Niggli cone, 53 4-D boundary polytopes resulting from intersections of pairs of the 15 5-D boundary polytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-fold intersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopes resulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-D boundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higher intersections of the 15 5-D boundary polytopes. All primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes. All non-primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes and of the 7 special-position subspaces of the 5-D boundary polytopes. This study provides a new, simpler and arguably more intuitive basis set for the classification of lattice characters and helps to illuminate some of the complexities in Bravais lattice identification. The classification is intended to help in organizing database searches and in understanding which lattice symmetries are "close" to a given experimentally determined cell

The Geometry of Niggli Reduction II: BGAOL -- Embedding Niggli Reduction

Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program, BGAOL, for Bravais lattice determination. Results from BGAOL are compared to the results from other metric-based Bravais lattice determination algorithms

Converting three-space matrices to equivalent six-space matrices for Delone scalars in S6.

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 S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)

Use of entanglement in quantum optics

Several recent demonstrations of two-particle interferometry are reviewed and shown to be examples of either color entanglement or beam entanglement. A device, called a number filter, is described and shown to be of value in preparing beam entanglements. Finally, we note that all three concepts (color and beam entaglement, and number filtering) may be extended to three or more particles

An Invertible Seven-Dimensional Dirichlet Cell Characterization of Lattices

Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on the three shortest non-coplanar lattice edge vectors) or by Delaunay-reduced cells (based on four edge vectors summing to zero and all meeting at obtuse or right angles) are commonly used. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner-Seitz (or Dirichlet, or Voronoi) cell of the lattice, which consists of the points at least as close to a chosen lattice point than they are to any other lattice point. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by thirteen lattice half-edges: the midpoints of the three Niggli cell edges, the six Niggli cell face diagonals and the four body-diagonals, but seven of the edge lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths and the shortest body-diagonal length, from which the Niggli-reduced cell may be recovered.Comment: 29 pages, 2 figure