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.)

Realizing vector meson dominance with transverse charge densities

The transverse charge density in a fast-moving nucleon is represented as a dispersion integral of the imaginary part of the Dirac form factor in the timelike region (spectral function). At a given transverse distance b the integration effectively extends over energies in a range sqrt{t} ~< 1/b, with exponential suppression of larger values. The transverse charge density at peripheral distances thus acts as a low-pass filter for the spectral function and allows one to select energy regions dominated by specific t-channel states, corresponding to definite exchange mechanisms in the spacelike form factor. We show that distances b ~ 0.5 - 1.5 fm in the isovector density are maximally sensitive to the rho meson region, with only a ~10% contribution from higher-mass states. Soft-pion exchange governed by chiral dynamics becomes relevant only at larger distances. In the isoscalar density higher-mass states beyond the omega are comparatively more important. The dispersion approach suggests that the positive transverse charge density in the neutron at b ~ 1 fm, found previously in a Fourier analysis of spacelike form factor data, could serve as a sensitive test of the the isoscalar strength in the ~1 GeV mass region. In terms of partonic structure, the transverse densities in the vector meson region b ~ 1 fm support an approximate mean-field picture of the motion of valence quarks in the nucleon.Comment: 14 pages, 12 figure

Development Of Third Harmonic Generation As A Short Pulse Probe Of Shock Heated Material

We are studying high-pressure laser produced shock waves in silicon (100). To examine the material dynamics, we are performing pump-probe style experiments utilizing 600 ps and 40 fs laser pulses from a Ti:sapphire laser. Two-dimensional interferometry reveals information about the shock breakout, while third harmonic light generated at the rear surface is used to infer the crystalline state of the material as a function of time. Sustained third harmonic generation (THG) during a similar to 100 kbar shock breakout indicate that the rear surface remains crystalline for at least 3 ns. However, a decrease in THG during a similar to 300 kbar shock breakout suggests a different behavior, which could include a change in crystalline structure.Mechanical Engineerin

A note on compactly generated co-t-structures

The idea of a co-t-structure is almost "dual" to that of a t-structure, but with some important differences. This note establishes co-t-structure analogues of Beligiannis and Reiten's corresponding results on compactly generated t-structures.Comment: 10 pages; details added to proofs, small correction in the main resul

Dynamic autonomous intelligent control of an asteroid lander

One of the future flagship missions of the European Space Agency (ESA) is the asteroid sample return mission Marco-Polo. Although there have been a number of past missions to asteroids, a sample has never been successfully returned. The return of asteroid regolith to the Earth's surface introduces new technical challenges. This paper develops attitude control algorithms for the descent phase onto an asteroid in micro-gravity conditions and draws a comparison between the algorithms considered. Two studies are also performed regarding the Fault Detection Isolation and Recovery (FDIR) of the control laws considered. The potential of using Direct Adaptive Control (DAC) as a controller for the surface sampling process is also investigated. Use of a DAC controller incorporates increased levels of robustness by allowing realtime variation of control gains. This leads to better response to uncertainties encountered during missions