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The main goal of the paper is to contribute to the agenda of developing an algorithmic model for crystallization and measuring the complexity of crystals by constructing embeddings of 3D parallelohedra into a primitive cubic network (pcu net). It is proved that any parallelohedron P as well as tiling by P, except the rhombic dodecahedron, can be embedded into the 3D pcu net. It is proved that for the rhombic dodecahedron embedding into the 3D pcu net does not exist; however, embedding into the 4D pcu net exists. The question of how many ways the embedding of a parallelohedron can be constructed is answered. For each parallelohedron, the deterministic finite automaton is developed which models the growth of the crystalline structure with the same combinatorial type as the given parallelohedron.Direct electron detection provides high detective quantum efficiency, significantly improved point spread function and fast read-out which have revolutionized the field of cryogenic electron microscopy. However, these benefits for high-resolution electron microscopy (HREM) are much less exploited, especially for in situ study where major impacts on crystallographic structural studies could be made. By using direct detection in electron counting mode, rutile nanocrystals have been imaged at high temperature inside an environmental transmission electron microscope. The improvements in image contrast are quantified by comparison with a charge-coupled device (CCD) camera and by image matching with simulations using an automated approach based on template matching. Together, these approaches enable a direct measurement of 3D shape and mosaicity (∼1°) of a vacuum-reduced TiO2 nanocrystal about 50 nm in size. Thus, this work demonstrates the possibility of quantitative HREM image analysis based on direct electron detection.A linear isometry R of \bb R^d is called a similarity isometry of a lattice \Gamma\subseteq\bb R^d if there exists a positive real number β such that βRΓ is a sublattice of (finite index in) Γ. The set βRΓ is referred to as a similar sublattice of Γ. A (crystallographic) point packing generated by a lattice Γ is a union of Γ with finitely many shifted copies of Γ. In this study, the notion of similarity isometries is extended to point packings. check details A characterization for the similarity isometries of point packings is provided and the corresponding similar subpackings are identified. Planar examples are discussed, namely the 1 × 2 rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, similarity isometries of point packings about points different from the origin are considered by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing are computed and compared with those of the hexagonal packing.The deterioration of both the signal-to-noise ratio and the spatial resolution in the electron-density distribution reconstructed from diffraction intensities collected at different orientations of a sample is analysed theoretically with respect to the radiation damage to the sample and the variations in the X-ray intensities illuminating different copies of the sample. The simple analytical expressions and numerical estimates obtained for models of radiation damage and incident X-ray pulses may be helpful in planning X-ray free-electron laser (XFEL) imaging experiments and in analysis of experimental data. This approach to the analysis of partially coherent X-ray imaging configurations can potentially be used for analysis of other forms of imaging where the temporal behaviour of the sample and the incident intensity during exposure may affect the inverse problem of sample reconstruction.Laboratory X-ray diffraction contrast tomography (LabDCT) has recently been developed as a powerful technique for non-destructive mapping of grain microstructures in bulk materials. As the grain reconstruction relies on segmentation of diffraction spots, it is essential to understand the physics of the diffraction process and resolve all the spot features in detail. To this aim, a flexible and standalone forward simulation model has been developed to compute the diffraction projections from polycrystalline samples with any crystal structure. The accuracy of the forward simulation model is demonstrated by good agreements in grain orientations, boundary positions and shapes between a virtual input structure and that reconstructed based on the forward simulated diffraction projections of the input structure. Further experimental verification is made by comparisons of diffraction spots between simulations and experiments for a partially recrystallized Al sample, where a satisfactory agreement is found for the spot positions, sizes and intensities. Finally, applications of this model to analyze specific spot features are presented.The previously reported exact potential and multipole moment (EP/MM) method for fast and accurate evaluation of the intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density [Volkov, Koritsanszky & Coppens (2004). Chem. Phys. Lett. 391, 170-175; Nguyen, Kisiel & Volkov (2018). Acta Cryst. A74, 524-536; Nguyen & Volkov (2019). Acta Cryst. A75, 448-464] is extended to the calculation of electrostatic interaction energies in molecular crystals using two newly developed implementations (i) the Ewald summation (ES), which includes interactions up to the hexadecapolar level and the EP correction to account for short-range electron-density penetration effects, and (ii) the enhanced EP/MM-based direct summation (DS), which at sufficiently large intermolecular separations replaces the atomic multipole moment approximation to the electrostatic energy with that based on the molecular multipole moments. As in the previous study [Nguyen, Kisiel & Volkov (2018). Acta Cryssion and speed with the ES technique only for crystal structures of small molecules that do not carry a large molecular dipole moment. The electron-density penetration effects, correctly accounted for by the two described methods, contribute 28-64% to the total electrostatic interaction energy in the examined systems, and thus cannot be neglected.