Comparative study of X-ray charge-density data on CoSb3

CoSb3is an example of a highly challenging case for experimental charge-density analysis due to the heavy elements (suitability factor of ∼0.01), the perfect crystallinity and the high symmetry of the compound. It is part of a family of host–guest structures that are potential candidates for use as high-performance thermoelectric materials. Obtaining and analysing accurate charge densities of the undoped host structure potentially can improve the understanding of the thermoelectric properties of this family of materials. In a previous study, analysis of the electron density gave a picture of covalent Co–Sb and Sb–Sb interactions together with relatively low atomic charges based on state-of-the-art experimental and theoretical data. In the current study, several experimental X-ray diffraction data sets collected on the empty CoSb3framework are compared in order to probe the experimental requirements for obtaining data of high enough quality for charge-density analysis even in the case of very unsuitable crystals. Furthermore, the quality of the experimental structure factors is tested by comparison with theoretical structure factors obtained from periodic DFT calculations. The results clearly show that, in the current study, the data collected on high-intensity, high-energy synchrotron sources and very small crystals are superior to data collected at conventional sources, and in fact necessary for a meaningful charge-density study, primarily due to greatly diminished effects of extinction and absorption which are difficult to correct for with sufficient accuracy.

Spin line groups

Spin line groups describe the symmetries of spin arrangements in quasi-one-dimensional systems. These groups are derived for the first family of line groups. Among them, magnetic groups are singled out as a special case. Spin arrangements generated by the derived groups are first discussed for single-orbit systems and then the conclusions are extended to multi-orbit cases. The results are illustrated by the examples of a CuO2zigzag chain, a13C nanotube and the hexaferrite Ba2Mg2Fe12O22. Applications to neutron diffraction and classical ground-state determination are indicated.

Direct phasing of nanocrystal diffraction

Recent experiments at free-electron laser X-ray sources have been able to resolve the intensity distributions about Bragg peaks in nanocrystals of large biomolecules. Information derived from small shifts in the peak positions augment the Bragg samples of the particle intensity with samples of its gradients. Working on the assumption that the nanocrystal is entirely generated by lattice translations of a particle, an algorithm is developed that reconstructs the particle from intensities and intensity gradients. Unlike traditional direct phasing methods that require very high resolution data in order to exploit sparsity of the electron density, this method imposes no constraints on the contrast other than positivity and works well at low resolution. Successful reconstructions are demonstrated with simulatedP1 lysozyme nanocrystal data down to a signal-to-noise ratio of 2 in the intensity gradients.

More about residual values

The usual residual values are complemented by expectation values based solely on the experimental data and the number of model parameters. These theoreticalRvalues serve as benchmark values when all of the basic assumptions for a least-squares refinement,i.e.no systematic errors and a fully adequate model capable of describing the data, are fulfilled. The prediction ofRvalues as presented here is applicable to any field where model parameters are fitted to data with known precision. For crystallographic applications,F2-based residual benchmark values are given. They depend on the first and second moments of variance, intensity and significance distributions, 〈σ2〉, 〈Io2〉, 〈Io2/σ2〉. Possible applications of the theoreticalRvalues are, for example, as a data-quality measure or the detection of systematic deviations between experimental data and model predicted data, although the theoreticalRvalues cannot identify the origin of these systematic deviations. The change inRvalues due to application of a weighting scheme is quantified with the theoreticalRvalues.

Distribution rules of systematic absences on the Conway topograph and their application to powder auto-indexing

This paper presents several general properties of systematic absences that are available before unit-cell parameters and the space group have been determined. The properties are given in the form of distribution rules of Miller indices corresponding to systematic absences on atopograph. A topograph is a graph whose edges are associated with a set of four lattice vectors satisfying Ito's equation 2(|l1*|2+ |l2*|2) = |l1*+l2*|2+ |l1*−l2*|2. It is possible to integrate global information about extinct reflections by using topographs. As an example of the application of these rules, a new powder auto-indexing algorithm is introduced, focusing on its theoretical aspects.

Homometry in the light of coherent beams

Two systems are homometric if they are indistinguishable by diffraction. A distinction is first made between Bragg and diffuse scattering homometry, and it is shown that in the last case coherent diffraction can allow the diffraction diagrams to be differentiated. The study of the Rudin–Shapiro sequence, homometric to random sequences, allows one to manipulate independently two-point and four-point correlation functions, and to show their effect on the statistics of speckle patterns. This study provides evidence that long-range order in high-order correlation functions has a measurable effect on the speckle statistics.

Nets with collisions (unstable nets) and crystal chemistry

Nets in which different vertices have identical barycentric coordinates (i.e.have collisions) are called unstable. Some such nets have automorphisms that do not correspond to crystallographic symmetries and are called non-crystallographic. Examples are given of nets taken from real crystal structures which have embeddings with crystallographic symmetry in which colliding nodes either are, or are not, topological neighbors (linked) and in which some links coincide. An example is also given of a crystallographic net of exceptional girth (16), which has collisions in barycentric coordinates but which also has embeddings without collisions with the same symmetry. In this last case the collisions are termedunforced.