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atom bumping  crystallographic  group  maxn  minn  models  new concept  space  structure  theory  unit cell  wavevector substar  wavevector 
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Preview: Acta Crystallographica Section A

Acta Crystallographica Section A

Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination

Published: 2017-09-01


Close-packed structure dynamics with finite-range interaction: computational mechanics with individual layer interaction


This is the second contribution in a series of papers dealing with dynamical models in equilibrium theories of polytypism. A Hamiltonian introduced by Ahmad & Khan [Phys. Status Solidi B (2000), 218, 425–430] avoids the unphysical assignment of interaction terms to fictitious entities given by spins in the Hägg coding of the stacking arrangement. In this paper an analysis of polytype generation and disorder in close-packed structures is made for such a Hamiltonian. Results are compared with a previous analysis using the Ising model. Computational mechanics is the framework under which the analysis is performed. The competing effects of disorder and structure, as given by entropy density and excess entropy, respectively, are discussed. It is argued that the Ahmad & Khan model is simpler and predicts a larger set of polytypes than previous treatments.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces


In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

The wavevector substar group in reciprocal space and its representation


A new concept of the wavevector substar group is established which, in the study of translational symmetry breaking of a crystal, only considers the particular arms of the wavevector star taking part in the phase transition; this is in contrast with the traditional Landau theory which considers all of the arms of the wavevector star. It is shown that this new concept can be used effectively to investigate the interesting physical properties of crystals associated with translational symmetry breaking. It is shown that studies on complicated phase transitions related to reducible representations, such as those in perovskite KMnF3 multiferroics and the high-temperature superconductor La2/3Mg1/2W1/2O3 (La4Mg3W3O18), are much simplified by the new concept. The theory of the wavevector substar group and its representation is a powerful mathematical tool for the study of various symmetry-breaking phenomena in solid-state crystals.

Real-time detection and resolution of atom bumping in crystallographic models


A basic principle in crystal structure determination is that there should be proper distances between adjacent atoms. Therefore, detection of atom bumping is of fundamental significance in structure determination, especially in the direct-space method where crystallographic models are just randomly generated. Presented in this article is an algorithm that detects atom bonding in a unit cell based on the sweep and prune algorithm of axis-aligned bounding boxes and running in the O(n log n) time bound, where n is the total number of atoms in the unit cell. This algorithm only needs the positions of individual atoms in the unit cell and does not require any prior knowledge of existing bonds, and is thus suitable for modelling of inorganic crystals where the bonding relations are often unknown a priori. With this algorithm, computation routines requiring bonding information, e.g. anti-bumping and computation of coordination numbers and valences, can be performed efficiently. As an example application, an evaluation function for atom bumping is proposed, which can be used for real-time elimination of crystallographic models with unreasonably short bonds during the procedure of global optimization in the direct-space method.

Accelerated scattering of convex polyhedra


The formulas for the minimum (minn) and maximum (maxn) names in the classes of convex n-acra (i.e. n-vertex polyhedra) are found for any n. The asymptotic behaviour (as n → ∞) for maxn+1/maxn, minn+1/minn, minn+1/maxn and maxn/minn is established. They characterize in detail the accelerated scattering of [minn, maxn] ranges on a real line.