Acta Crystallographica Section A Foundations and Advances

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ISSN / EISSN : 2053-2733 / 2053-2733
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Acta Crystallographica Section A Foundations and Advances, Volume 78; https://doi.org/10.1107/s2053273322007215

Abstract:
X-ray scattering and diffraction phenomena are widely used as analytical tools in nanoscience. Size discrepancies between the two phenomena are commonly observed in crystalline nanoparticle systems. The root of the problem is that each phenomenon is affected by size distribution differently, causing contrasting shifts between the two methods. Once understood, the previously discrepant results lead to a simple formula for obtaining the nanoparticle size distribution.
Acta Crystallographica Section A Foundations and Advances, Volume 78; https://doi.org/10.1107/s2053273322006143

Abstract:
The dynamical theory of diffraction is used to analyse irregular X-ray mirage interference fringes observed in Si220 X-ray reflection topography from a weakly bent, thin crystal due to gravity. The origin of the irregular fringes is attributed to the interference between mirage diffracted beams and a reflected beam from the back surface, which is a new type of interference fringe. The irregular fringes are reproduced by calculating the reflected intensities numerically. The effects of absorption and thermal vibration are quite important for the reproduction. The result shows that the interference fringes depend on the strain as well as the thickness of the crystal, which indicates that the fringes should be useful for analysing weak strain in a crystal as an application.
Acta Crystallographica Section A Foundations and Advances, Volume 78; https://doi.org/10.1107/s2053273322006970

Abstract:
An indicator-based crystallographic phase retrieval method has been developed for diffraction data of bicontinuous cubic phases of lyotropic liquid crystals. Such liquid crystals have large structural disorder; the number of independent Bragg reflections that can be observed is limited. This paper proposes two indicators to identify plausible combination(s) of crystallographic phases, i.e. electron-density distribution. The indicators are based on the characteristics of the liquid crystals: amphiphilic molecules diffuse mainly in the direction parallel to polar–nonpolar interfaces and the electron density in the direction parallel to the interfaces is averaged. One indicator is the difference between the maximum and minimum electron density, and the other is calculated from the Hessian matrix of the electron density. Using test data, the electron densities were calculated for all possible phase combinations and indicators were obtained. The results indicated that the electron densities with the minimum indicators were close to the true electron density. Therefore, this method is effective for phase retrieval. The accuracy of the phase retrieval decreased when the volume fraction of the region including the triply periodic minimal surface increased.
Acta Crystallographica Section A Foundations and Advances, Volume 78; https://doi.org/10.1107/s2053273322006441

Abstract:
A semi-analytic formula for the temperature-dependent static structure factor S(k) of polycrystalline and amorphous solids applicable to the entire wavenumber (k) range is derived. The formula describes thermal diffuse scattering due to multiphonon processes entirely by a single kernel function without resorting to the standard perturbation expansion. It is analytically proven that S(k → 0) is determined from the one-phonon term, whereas the asymptotic limit S(k → ∞) = 1 can be reproduced through a Gaussian integral of the multiphonon term. The formula also reveals that an enhancement of the one-phonon scattering intensity at each Bragg point is expressed as a logarithmic singularity. Numerical examples for a face-centred cubic polycrystal near the melting point are shown. The present formula is computationally more efficient than other theoretical models, requiring only a one-dimensional integration to obtain S(k) once the elastic part of the structure factor and the Debye–Waller factor are given.
Acta Crystallographica Section A Foundations and Advances, Volume 78; https://doi.org/10.1107/s2053273322006325

Abstract:
Five datasets of high-temperature X-ray diffraction performed upon the decagonal phase of AlCuRh are used to derive the temperature-related structural changes. Two sets of atomic structure refinements are conducted, with four and five atomic surfaces, respectively. The fifth atomic surface emerges as a consequence either of the transition to a tiling with different local isomorphism than the Penrose tiling or of the structure being phason disordered. The correlation of the occupancy of the fifth atomic surface with the thermal dependence of the lattice constants is 80%. In addition, the structural refinement with five atomic surfaces allowed a better agreement with the experimental data to be obtained, compared with the model with four atomic surfaces.
Acta Crystallographica Section A Foundations and Advances, Volume 78; https://doi.org/10.1107/s2053273322004934

Abstract:
Calculating dynamical diffraction patterns for X-ray diffraction imaging techniques requires numerical integration of the Takagi–Taupin equations. This is usually performed with a simple, second-order finite difference scheme on a sheared computational grid in which two of the axes are aligned with the wavevectors of the incident and scattered beams. This dictates, especially at low scattering angles, an oblique grid of uneven step sizes. Here a finite difference scheme is presented that carries out this integration in slab-shaped samples on an arbitrary orthogonal grid by implicitly utilizing Fourier interpolation. The scheme achieves the expected second-order convergence and a similar error to the traditional approach for similarly dense grids.
Acta Crystallographica Section A Foundations and Advances, Volume 78, pp 371-383; https://doi.org/10.1107/s2053273322005058

Abstract:
Coordination sequences (also called growth functions) appear in various areas of chemistry and crystallography, such as ice crystals and zeolites, and various areas of mathematics, such as lattice theory and geometric group theory. Cannon's method of cone types is modified for finding the coordination sequence of the Cayley graph of a group. This method is then applied to compute the growth functions and the growth series of the Cayley graphs of the wallpaper groups (the 2D crystallographic groups). The paper has several tables and colored figures summarizing and illustrating the results.
Acta Crystallographica Section A Foundations and Advances, Volume 78, pp 364-370; https://doi.org/10.1107/s2053273322003898

Abstract:
A magnetic superspace group (MSSG) simultaneously constrains both the magnetic and non-magnetic (e.g. displacive, occupational, rotation and strain) degrees of freedom of an incommensurately modulated magnetic crystal. We present the first enumeration and tabulation of all non-equivalent (3+d)-dimensional magnetic superspace groups for d = 1, 2 and 3 independent incommensurate modulations, along with a number, symbol and reference setting for each group. We explain the process for generating an exhaustive set of inequivalent magnetic superspace groups, describe several examples, and show how the tables can be accessed via the ISO(3+d)D interface within the ISOTROPY Software Suite. We recommend that published incommensurate magnetic structures indicate a magnetic superspace-group number and symbol from these tables, as well as the transformation matrix from the published group setting to the reference setting used in these tables.
Acta Crystallographica Section A Foundations and Advances, Volume 78, pp 294-301; https://doi.org/10.1107/s2053273322005071

Abstract:
A method is proposed to reconstruct the 3D molecular structure from micrographs collected at just one sample tilt angle in the random conical tilt scheme in cryo-electron microscopy. The method uses autocorrelation analysis on the micrographs to estimate features of the molecule which are invariant under certain nuisance parameters such as the positions of molecular projections in the micrographs. This enables the molecular structure to be reconstructed directly from micrographs, completely circumventing the need for particle picking. Reconstructions are demonstrated with simulated data and the effect of the missing-cone region is investigated. These results show promise to reduce the size limit for single-particle reconstruction in cryo-electron microscopy.
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