326 research outputs found
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
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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
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