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

Models for the Magnitude-Distribution of Brightest Cluster Galaxies

The brightest, or first-ranked, galaxies (BCGs) in rich clusters show a very small dispersion in luminosity, making them excellent standard candles. This small dispersion has raised questions about the nature of BCGs. Are they simply the extremes of normal galaxies formed via a stochastic process, or do they belong to a special class of atypical objects? Arguments have been proposed on both sides of the debate. Bhavsar (1989) suggested that the distribution in magnitudes can only be explained by a two-population model. Thus, a new controversy has arisen. Do first-ranked galaxies consist of one or two populations of objects? We examine an older and newer data set and present our results. Two-population models do better than do one-population models. A simple model where a random boost in the magnitude of a fraction of bright normal galaxies forms a class of atypical galaxies best describes the observed distribution of BCG magnitudes. Moreover, the parameters that describe the model and the parameters of the boost have a strong physical basis.Comment: Abstract submitted to AAS. Paper (6 pages, 4 figs.) to be published in the MNRAS; uses mn.st

The Detailed Chemical Abundance Patterns of M31 Globular Clusters

We present detailed chemical abundances for $>$20 elements in $\sim$30 globular clusters in M31. These results have been obtained using high resolution ($\lambda/\Delta\lambda\sim$24,000) spectra of their integrated light and analyzed using our original method. The globular clusters have galactocentric radii between 2.5 kpc and 117 kpc, and therefore provide abundance patterns for different phases of galaxy formation recorded in the inner and outer halo of M31. We find that the clusters in our survey have a range in metallicity of $-2.2<$[Fe/H]$<-0.11$. The inner halo clusters cover this full range, while the outer halo globular clusters at R$>$20 kpc have a small range in abundance of [Fe/H]$=-1.6 \pm 0.10$. We also measure abundances of alpha, r- and s-process elements. These results constitute the first abundance pattern constraints for old populations in M31 that are comparable to those known for the Milky Way halo.Comment: XII International Symposium on Nuclei in the Cosmos August 5-12, 2012 Cairns, Australia. To appear in Proceedings of Scienc

Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields

For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where $C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$) and the fundamental solution can be written down. Likewise, if all the $X_{i}$'s and $t$ are non-negative then the continued fraction expansion of $\sqrt{F_{i}}$ can be written down. Furthermore, the congruence class modulo 4 of $F_{i}$ depends in a simple way on the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$ so that the fundamental unit can be written down for a large class of real quadratic fields. Along the way a complete solution is given to the problem of determining for which symmetric strings of positive integers $a_{1},..., a_{n}$ do there exist positive integers $D$ and $a_{0}$ such that $\sqrt{D} = [ a_{0};\bar{a_{1}, >..., a_{n},2a_{0}}]$.Comment: 13 page