Acta Crystallographica Section A: Foundations and Advances最新文献

The theoretical investigation of double-slit X-ray dynamical diffraction in curved crystals shows that Young's interference fringes are formed. An expression for the period of the fringes has been established which is polarization sensitive. The position of the fringes in the cross section of the beam depends on the deviation from the Bragg exact orientation for a perfect crystal, on the curvature radius and on the thickness of the crystal. This type of diffraction can be used for determination of the curvature radius by measuring the shift of the fringes from the centre of the beam.

This paper presents the basic tools used to describe the global symmetry of so-called bilayer structures obtained when two differently oriented crystalline monoatomic layers of the same structure are superimposed and displaced with respect to each other. The 2D nature of the layers leads to the use of complex numbers that allows for simple explicit analytical expressions of the symmetry properties involved in standard bicrystallography [Gratias & Portier (1982). J. Phys. Colloq. 43, C6-15-C6-24; Pond & Vlachavas (1983). Proc. R. Soc. Lond. Ser. A, 386, 95-143]. The focus here is on the twist rotations such that the superimposition of the two layers generates a coincidence lattice. The set of such coincidence rotations plotted as a function of the lengths of their coincidence lattice unit-cell nodes exhibits remarkable arithmetic properties. The second part of the paper is devoted to determination of the space groups of the bilayers as a function of the rigid-body translation associated with the coincidence rotation. These general results are exemplified with a detailed study of graphene bilayers, showing that the possible symmetries of graphene bilayers with a coincidence lattice, whatever the rotation and the rigid-body translation, are distributed in only six distinct types of space groups. The appendix discusses some generalized cases of heterophase bilayers with coincidence lattices due to specific lattice constant ratios, and mechanical deformation by elongation and shear of a layer on top of an undeformed one.

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 vectors) or by Delaunay-reduced cells (based on four non-coplanar vectors summing to zero and all meeting at obtuse or right angles) is commonly performed. 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 as they are to any other lattice point. The three non-coplanar lattice vectors chosen are here called the Niggli-reduced cell edges. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by 13 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 lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths, and the shortest body-diagonal length. These seven are sufficient to recover the Niggli-reduced cell.

Obituary for André Authier.

Analytical representations of X-ray atomic form factor data have been determined. The original data, f₀(s;Z), are reproduced to a high degree of accuracy. The mean absolute errors calculated for all s = sin θ/λ and Z values in question are primarily determined by the precision of the published data. The inverse Mott-Bethe formula is the underlying basis with the electron scattering factor expressed by an expansion in Gaussian basis functions. The number of Gaussians depends upon the element and the data and is in the range 6-20. The refinement procedure, conducted to obtain the parameters of the models, is carried out for seven different form factor tables published in the span Cromer & Mann [(1968), Acta Cryst. A24, 321-324] to Olukayode et al. [(2023), Acta Cryst. A79, 59-79]. The s ranges are finite, the most common span being [0.0, 6.0] Å^-1. Only one function for each element is needed to model the full range. This presentation to a large extent makes use of a detailed graphical account of the results.

Diffraction intensities from a crystallographic experiment include contributions from the entire unit cell of the crystal: the macromolecule, the solvent around it and eventually other compounds. These contributions cannot typically be well described by an atomic model alone, i.e. using point scatterers. Indeed, entities such as disordered (bulk) solvent, semi-ordered solvent (e.g. lipid belts in membrane proteins, ligands, ion channels) and disordered polymer loops require other types of modeling than a collection of individual atoms. This results in the model structure factors containing multiple contributions. Most macromolecular applications assume two-component structure factors: one component arising from the atomic model and the second one describing the bulk solvent. A more accurate and detailed modeling of the disordered regions of the crystal will naturally require more than two components in the structure factors, which presents algorithmic and computational challenges. Here an efficient solution of this problem is proposed. All algorithms described in this work have been implemented in the computational crystallography toolbox (CCTBX) and are also available within Phenix software. These algorithms are rather general and do not use any assumptions about molecule type or size nor about those of its components.

As an alternative approach to X-ray crystallography and single-particle cryo-electron microscopy, single-molecule electron diffraction has a better signal-to-noise ratio and the potential to increase the resolution of protein models. This technology requires collection of numerous diffraction patterns, which can lead to congestion of data collection pipelines. However, only a minority of the diffraction data are useful for structure determination because the chances of hitting a protein of interest with a narrow electron beam may be small. This necessitates novel concepts for quick and accurate data selection. For this purpose, a set of machine learning algorithms for diffraction data classification has been implemented and tested. The proposed pre-processing and analysis workflow efficiently distinguished between amorphous ice and carbon support, providing proof of the principle of machine learning based identification of positions of interest. While limited in its current context, this approach exploits inherent characteristics of narrow electron beam diffraction patterns and can be extended for protein data classification and feature extraction.

The paper by Gopalan [(2020). Acta Cryst. A76, 318-327] presented an enumeration of the 41 physical quantity types in non-relativistic physics, in arbitrary dimensions, based on the formalism of Clifford algebra. Gopalan considered three antisymmetries: spatial inversion, 1, time reversal, 1', and wedge reversion, 1^†. A consideration of the set of all seven antisymmetries (1, 1', 1^†, 1'^†, 1^†, 1', 1'^†) leads to an extension of the results obtained by Gopalan. It is shown that there are 51 types of physical quantities with distinct symmetry properties in total.

Automatic crystal orientation determination and orientation mapping are important tools for research on polycrystalline materials. The most common methods of automatic orientation determination rely on detecting and indexing individual diffraction reflections, but these methods have not been used for orientation mapping of quasicrystalline materials. The paper describes the necessary changes to existing software designed for orientation determination of periodic crystals so that it can be applied to quasicrystals. The changes are implemented in one such program. The functioning of the modified program is illustrated by an example orientation map of an icosahedral polycrystal.