This paper studies the symmetry group of two special types of carbon nanotori. The construction is motivated by a group-theoretical result.

The advantages of convergent-beam electron diffraction for symmetry determination at the scale of a few nm are well known. In practice, the approach is often limited due to the restriction on the angular range of the electron beam imposed by the small Bragg angle for high-energy electron diffraction, i.e. a large convergence angle of the incident beam results in overlapping information in the diffraction pattern. Techniques have been generally available since the 1980s which overcome this restriction for individual diffracted beams, by making a compromise between illuminated area and beam convergence. Here a simple technique is described which overcomes all of these problems using computer control, giving electron diffraction data over a large angular range for many diffracted beams from the volume given by a focused electron beam (typically a few nm or less). The increase in the amount of information significantly improves the ease of interpretation and widens the applicability of the technique, particularly for thin materials or those with larger lattice parameters.

A previous paper detailed a novel algorithm, ɛ-machine spectral reconstruction theory (ɛMSR), that infers pattern and disorder in planar-faulted, close-packed structures directly from X-ray diffraction patterns [Varn et al. (2013). Acta Cryst. A69, 197–206]. Here ɛMSR is applied to simulated diffraction patterns from four close-packed crystals. It is found that, for stacking structures with a memory length of three or less, ɛMSR reproduces the statistics of the stacking structure; the result being in the form of a directed graph called an ɛ-machine. For stacking structures with a memory length larger than three, ɛMSR returns a model that captures many important features of the original stacking structure. These include multiple stacking faults and multiple crystal structures. Further, it is found that ɛMSR is able to discover stacking structure in even highly disordered crystals. In order to address issues concerning the long-range order observed in many classes of layered materials, several length parameters are defined, calculable from the ɛ-machine, and their relevance is discussed.

It is shown that the Euclidean group of translations, when treated as a Lie group, generates translations not only in Euclidean space but on any space, curved or not. Translations are then not necessarily vectors (straight lines); they can be any curve compatible with the parameterization of the considered space. In particular, attention is drawn to the fact that one and only one finite and free module of the Lie algebra of the group of translations can generate both modulated and non-modulated lattices, the modulated character being given only by the parameterization of the space in which the lattice is generated. Moreover, it is shown that the diffraction pattern of a structure is directly linked to the action of that free and finite module. In the Fourier transform of a whole structure, the Fourier transform of the electron density of one unit cell (i.e. the structure factor) appears concretely, whether the structure is modulated or not. Thus, there exists a neat separation: the geometrical aspect on the one hand and the action of the group on the other, without requiring additional dimensions.

δ Recycling is a simple procedure for directly extracting phase information from Patterson-type functions [Rius (2012). Acta Cryst. A68, 399–400]. This new phasing method has a clear theoretical basis and was developed with ideal single-crystal X-ray diffraction data. On the other hand, introduction of the automated diffraction tomography (ADT) technique has represented a significant advance in electron diffraction data collection [Kolb et al. (2007). Ultramicroscopy, 107, 507–513]. When combined with precession electron diffraction, it delivers quasi-kinematical intensity data even for complex inorganic compounds, so that single-crystal diffraction data of nanometric volumes are now available for structure determination by direct methods. To check the tolerance of δ recycling to missing data-collection corrections and to deviations from kinematical behaviour of ADT intensities, δ recycling has been applied to differently shaped nanocrystals of various inorganic materials. The results confirm that it can phase ADT data very efficiently. In some cases even more complete structure models than those derived from conventional direct methods and least-squares refinement have been found. During this study it has been demonstrated that the Wilson-plot scaling procedure is largely insensitive to sample thickness variations and missing absorption corrections affecting electron ADT intensities.

A new study of the σ_A parameter has been undertaken to understand its behaviour when the diffraction amplitude distributions are far from the standard Wilson distributions. The study has led to the formulation of a new statistical interpretation of σ_A, expressed in terms of a correlation factor. The new formulas allow a more accurate use of σ_A in electron-density modification procedures.

Femtosecond X-ray pulses from X-ray free-electron laser sources make it feasible to conduct room-temperature solution scattering experiments far below molecular rotational diffusion timescales. Owing to the ultra-short duration of each snapshot in these fluctuation scattering experiments, the particles are effectively frozen in space during the X-ray exposure. In contrast to standard small-angle scattering experiments, the resulting scattering patterns are anisotropic. The intensity fluctuations observed in the diffraction images can be used to obtain structural information embedded in the average angular correlation of the Fourier transform of the scattering species, of which standard small-angle scattering data are a subset. The additional information contained in the data of these fluctuation scattering experiments can be used to determine the structure of macromolecules in solution without imposing symmetry or spatial restraints during model reconstruction, reducing ambiguities normally observed in solution scattering studies. In this communication, a method that utilizes fluctuation X-ray scattering data to determine low-resolution solution structures is presented. The method is validated with theoretical data calculated from several representative molecules and applied to the reconstruction of nanoparticles from experimental data collected at the Linac Coherent Light Source.

New tables of irreducible representations (IRs) are introduced for the 230 crystallographic space groups (SGs) in three-dimensional space, at both special and non-special k vectors, and for their extensions to (3 + d)-dimensional superspace (`superspace-extended SGs' or SSESGs). Neither a tabulation of SG IR matrices for non-special k vectors nor a tabulation of SSESG IR matrices for d > 1 have been previously published. These tabulations are made possible by a new form in which the IR matrices of SGs are separated as a product of a translation part T and a point-operation part P, and where the IR matrices of SSESGs are separated as a product of a phase-shift part Q and a point-operation part P_s. Both T and Q have a simple prescribed form that does not need to be tabulated. Also, the new IR matrices are in a convenient block form which allows one to see by inspection which parts of the matrices and the associated order parameters belong to which arm of the star of k. In addition to complex IR matrices, real physically irreducible representation (PIR) matrices are tabulated. The new IR and PIR tables are available on the ISO-IR website (http://stokes.byu.edu/iso/irtables.php) in both convenient human-readable and computer-readable forms.

High-symmetry free tilings of the two-dimensional hyperbolic plane () can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.

Ordered arrays of cylinders, known as rod packings, are now widely used in descriptions of crystalline structures. These are generalized to include crystallographic packed arrays of filaments with circular cross sections, including curvilinear cylinders whose central axes are generic helices. A suite of the simplest such general rod packings is constructed by projecting line patterns in the hyperbolic plane () onto cubic genus-3 triply periodic minimal surfaces in Euclidean space (): the primitive, diamond and gyroid surfaces. The simplest designs correspond to `classical' rod packings containing conventional cylindrical filaments. More complex packings contain three-dimensional arrays of mutually entangled filaments that can be infinitely extended or finite loops forming three-dimensional weavings. The concept of a canonical `ideal' embedding of these weavings is introduced, generalized from that of knot embeddings and found algorithmically by tightening the weaving to minimize the filament length to volume ratio. The tightening algorithm builds on the SONO algorithm for finding ideal conformations of knots. Three distinct classes of weavings are described.

Periodic nets are commonly used to represent the topology of crystal structures. Non-crystallographic (NC) nets are p-periodic nets whose automorphism groups are not isomorphic to any isometry group in the Euclidean space. This work deals with the special class of NC nets possessing non-trivial finite blocks of imprimitivity for bounded automorphisms. It is shown that periodic, barycentric representations of NC nets with this property display vertex collisions, every block being represented as a single point. As a consequence, the labelled quotient graph of these nets shows an equitable partition that also respects the voltages over the edges, introduced as an equivoltage partition. Possible motions within linked blocks of imprimitivity are characterized as correlation groups. Some non-trivial examples of NC nets that have bounded automorphism groups with and without fixed points are explored from the viewpoint of equivoltage partitions and correlation groups, and a general algorithm is proposed to this end. It is shown that the group of bounded automorphisms of these nets can be described using wreath products of finite permutation groups by translation groups.

Nonlocality in spherical-aberration-corrected high-angle annular dark-field (HAADF) scanning transmission electron microscope (STEM) images is theoretically and experimentally examined using the absorption potential describing thermal diffuse scattering (TDS). A detailed comparison between the simulated and the experimentally obtained high-quality HAADF STEM images of an Si(110) bulk structure and a PbTiO₃(100)/SrTiO₃(100) interfacial structure unambiguously demonstrates the need to use a nonlocal TDS absorption potential. The nonlocality in the TDS absorption potential cannot be ignored in a detailed analysis of spherical-aberration-corrected HAADF STEM images of materials consisting of several heavy elements, although it can be completely disregarded for those consisting of only light elements.

Comprehensive theoretical and experimental studies on the importance and application of anomalous scattering factors can be found in the literature. The aim of this study was to determine the role and impact of anomalous scattering factors on the Rayleigh scattering of photons, particularly within the regions around the elemental absorption edges, using Monte Carlo sampling techniques. In doing so, an improved version of the already established Monte Carlo techniques for Rayleigh scattering is proposed. The improved version is capable of using the available state-of-the-art anomalous scattering factors, and illustrates and highlights their role in calculating accurate coherent scattering amplitudes. A substantial increase in the forward scattering by the neutral atoms of germanium, caesium and lead, which is a maximum around the K edges due to the inclusion of anomalous scattering factors, was observed at all the energies that were examined. The results show that the angular distribution of coherent scattering of the photons depended upon the anomalous scattering factors. Serious errors could be produced when measuring the exact scattering amplitudes, particularly within the regions around the elemental absorption edges, by ignoring the effects of coherently scattered photons in the Monte Carlo sampling. Furthermore, the improved model provides some extra information on elemental K-edge energies by producing dips in the plots of the calculated normalized cumulative probability distribution function against the energy of the incident photons for all three elements. In conclusion, the use of complex atomic form factors has produced an improved and fairly good approximation which is in very good agreement with the corresponding experimental and scattering-matrix results.

Fully periodic Hartree–Fock and density functional theory calculations have been used to compute the anisotropic displacement parameters (ADPs) of molecular crystals at different temperatures by using the CRYSTAL code. Crystalline urea was adopted as a benchmark system to investigate the dependence on basis set and Hamiltonian. The results were compared with ADPs derived from neutron diffraction experiments. The approach can estimate the internal ADPs, corresponding to the contributions of high-frequency intramolecular vibrations, and for these internal contributions the results are almost independent of the basis set and Hamiltonian. Much larger variations and discrepancies from neutron diffraction experiments are seen for the external, low-frequency modes, which become dominant at higher temperatures. The approach was then tested on benzene and urotropine. Finally, ADPs of L-alanine were predicted at the B3LYP/6-31G(d,p) level of theory. The total ADPs, including low-frequency external modes, are underestimated, but can be brought into good agreement with the experimental ADPs by introducing a Grüneisen parameter, which partly accounts for anharmonicity of the potential energy surface, but likely also contains contributions from other deficiencies of the calculations.

This paper presents a new, highly stable, periodic approximant to the Al-based F-type icosahedral quasicrystals, i-Al–Pd–TM (TM = transition metals). The structure of this intermetallic Al–Pd–Cr–Fe compound is determined ab initio using single-crystal X-ray diffraction, where the space group is identified to be and the lattice constant 40.5 Å. The structure is well described as a dense packing of clusters of two kinds, which are called the pseudo-Mackay-type and the mini-Bergman-type clusters. Adjacent clusters can be markedly interpenetrated, while the structure requires no glue atoms to fill in the gaps between the clusters. It is shown that the clusters are centred at the vertices of a canonical cell tiling, which corresponds to a 2 × 2 × 2 superstructure of Henley's cubic 3/2 packing, and that the parity of each vertex determines the kind of associated cluster. The proper quasi-lattice constant for describing the cluster packing is 1/τ (τ = golden mean) times the conventional one used to describe Al-based P-type icosahedral alloys. The superstructure ordering of the present approximant turns out to be of a different kind from the P-type superstructure ordering previously reported in i-Al–Pd–Mn. The present results will greatly improve the understanding of atomic structures of F-type icosahedral quasicrystals and their approximants.

The paper is devoted to the elaboration of a mathematical apparatus for studying second-order phase transitions, both commensurate and incommensurate, and the properties of emerging phases on the basis of the approach in equilibrium statistical mechanics proposed earlier by the author. It is shown that the preliminary symmetry analysis for a concrete crystal can be performed analogously with the one in the Landau phenomenological theory of phase transitions. The analysis enables one to deduce a set of transcendental equations that describe the emerging phases and corresponding phase transitions. The treatment of an incommensurate phase is substantially complicated because the symmetry of the phase cannot be described in terms of customary space groups. For this reason, a strategy of representing the incommensurate phase as the limit of a sequence of long-period commensurate phases whose period tends to infinity is worked out.

Two-dimensional point sets derived from pairs of quasirandom numbers generated by the bit-reversal method introduced by van der Corput exhibit features well known from the quasiperiodic binary substitution tilings derived from the rhombic tilings of Penrose and Ammann–Beenker. The concept of geometric discrepancy, a measure describing the uniformity of distribution of quasirandom sequences or point sets, is discussed from the perspective of structural chemistry.