In situ synchrotron small-angle X-ray scattering (SAXS) is a powerful tool for studying dynamic processes during material preparation and application. The processing and analysis of large data sets generated from in situ X-ray scattering experiments are often tedious and time consuming. However, data processing software for in situ experiments is relatively rare, especially for grazing-incidence small-angle X-ray scattering (GISAXS). This article presents an open-source software suite (SGTools) to perform data processing and analysis for SAXS and GISAXS experiments. The processing modules in this software include (i) raw data calibration and background correction; (ii) data reduction by multiple methods; (iii) animation generation and intensity mapping for in situ X-ray scattering experiments; and (iv) further data analysis for the sample with an order degree and interface correlation. This article provides the main features and framework of SGTools. The workflow of the software is also elucidated to allow users to develop new features. Three examples are demonstrated to illustrate the use of SGTools for dealing with SAXS and GISAXS data. Finally, the limitations and future features of the software are also discussed.
The ability to imagine symmetry and the spatial arrangement of atoms and molecules is crucial in chemistry in general. Teaching and understanding crystallography and the composition of the solid state therefore require understanding of symmetry elements and their relationships. To foster the student's spatial imagination, models representing a range of concepts from individual rotation axes to complete space groups have been designed and built. These models are robust and large enough to be presented and operated in a lecture hall, and they enable students to translate conventional 2D notations into 3D objects and vice versa. Tackling them hands-on means understanding them.
Laboratory diffraction contrast tomography (LabDCT) is a recently developed technique to map crystallographic orientations of polycrystalline samples in three dimensions non-destructively using a laboratory X-ray source. In this work, a new theoretical procedure, named LabXRS, expanding LabDCT to include mapping of the deviatoric strain tensors on the grain scale, is proposed and validated using simulated data. For the validation, the geometries investigated include a typical near-field LabDCT setup utilizing Laue focusing with equal source-to-sample and sample-to-detector distances of 14 mm, a magnified setup where the sample-to-detector distance is increased to 200 mm, a far-field Laue focusing setup where the source-to-sample distance is also increased to 200 mm, and a near-field setup with a source-to-sample distance of 200 mm. The strain resolution is found to be in the range of 1–5 × 10−4, depending on the geometry of the experiment. The effects of other experimental parameters, including pixel binning, number of projections and imaging noise, as well as microstructural parameters, including grain position, grain size and grain orientation, on the strain resolution are examined. The dependencies of these parameters, as well as the implications for practical experiments, are discussed.
This article presents the design and manufacture of an automated scale model of a four-circle single-crystal X-ray diffractometer that can be used for scientific dissemination. The purpose of this device is to reach out to the wider public and students to introduce them in an entertaining way to one of the laboratory apparatuses to which they do not usually have access, to talk to them about crystallography in the broadest sense, to develop concepts in various fields of science and technology, and to initiate interest and discussions. The main technical aspects of the project are described, with the expectation that such an approach could be useful to anyone involved in scientific dissemination and could be developed for other laboratory equipment and other disciplines. This kind of device can also be the subject of scientific and technological projects in close collaboration with educational institutions.
In situ neutron diffraction is an important characterization technique for the investigation of many functional materials, e.g. for hydrogen uptake and release in hydrogen storage materials. A new sapphire single-crystal gas-pressure cell for elastic neutron scattering has been developed and evaluated; it allows conditions of 298 K and 9.5 MPa hydrogen pressure and 1110 K at ambient pressure. The pressure vessel consists of a sapphire single-crystal tube of 35 mm radius and a sapphire single-crystal crucible as sample holder. Heating is realized by two 100 W diode lasers. It is optimized for the D20 diffractometer, ILL, Grenoble, France, and requires the use of a radial oscillating collimator. Its advantages over earlier sapphire single-crystal gas-pressure cells are higher maximum temperatures and lower background at low and high diffraction angles. The deuterium uptake in palladium was followed in situ for validation, proving the potential of the type-III gas-pressure cell for in situ neutron diffraction on solid–gas reactions.
The stable ϕ phase that forms below ∼923 K around the Al69.2Cu20.0Cr10.8 composition was found to be hexagonal [P63, a = 11.045 (2), c = 12.688 (2) Å] and isostructural to the earlier reported Al6.2Cu2Re X phase [Samuha, Grushko & Meshi (2016). J. Alloys Compd.
670, 18–24]. Using the structural model of the latter, a successful Rietveld refinement of the XRD data for Al69.5Cu20.0Cr10.5 was performed. Both ϕ and X were found to be structurally related to the Al72.6Cu11.0Cr16.4 ζ phase [P63/m, a = 17.714, c = 12.591 Å; Sugiyama, Saito & Hiraga (2002). J. Alloys Compd.
342, 148–152], with a close lattice parameter c and a τ-times-larger lattice parameter a (τ is the golden mean). The structural relationship between ζ and ϕ was established on the basis of the similarity of their layered structures and common features. Additionally, the strong-reflections approach was successfully applied for the modeling of the ϕ phase based on the structural model of the ζ phase. The latter and the experimental structural model (retrieved following Rietveld refinement) were found to be essentially identical.
A versatile generic framework for parent grain reconstruction from fully or partially transformed child microstructures has been integrated into the open-source crystallographic toolbox MTEX. The framework extends traditional parent grain reconstruction, phase transformation and variant analysis to all parent–child crystal symmetry combinations. The inherent versatility of the universally applicable parent grain reconstruction methods and the ability to conduct in-depth variant analysis are showcased via example workflows that can be programmatically modified by users to suit their specific applications. This is highlighted by three applications, namely α′-to-γ reconstruction in a lath martensitic steel, α-to-β reconstruction in a Ti alloy, and a two-step reconstruction from α′ to ɛ to γ in a twinning and transformation-induced plasticity steel. Advanced orientation relationship discovery and analysis options, including variant analysis, are demonstrated via the add-on function library ORTools.
The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V − E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χm) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χm has an elegant topological interpretation through the concept of orbifolds. Alternatively, χm can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.
A method is reported to determine the phase and amplitude of sinusoidally modulated event rates, binned into four bins per oscillation, based on data generated at the resonant neutron spin-echo spectrometer RESEDA at FRM-II. The presented algorithm relies on a reconstruction of the unknown parameters. It omits a calculation-intensive fitting procedure and avoids contrast reduction due to averaging effects. It allows the current data acquisition bottleneck at RESEDA to be relaxed by a factor of four and thus increases the potential time resolution of the detector by the same factor. The approach is explained in detail and compared with the established fitting procedures of time series having four and 16 time bins per oscillation. In addition the empirical estimates of the errors of the three methods are presented and compared with each other. The reconstruction is shown to be unbiased, asymptotic and efficient for estimating the phase. Reconstructing the contrast increases the error bars by roughly 10% as compared with fitting 16 time-binned oscillations. Finally, the paper gives heuristic, analytical equations to estimate the error for phase and contrast as a function of their initial values and counting statistics.