The new computer program SHELXT employs a novel dual‐space algorithm to solve the phase problem for single‐crystal reflection data expanded to the space group P1. Missing data are taken into account and the resolution extended if necessary. All space groups in the specified Laue group are tested to find which are consistent with the P1 phases. After applying the resulting origin shifts and space‐group symmetry, the solutions are subject to further dual‐space recycling followed by a peak search and summation of the electron density around each peak. Elements are assigned to give the best fit to the integrated peak densities and if necessary additional elements are considered. An isotropic refinement is followed for non‐centrosymmetric space groups by the calculation of a Flack parameter and, if appropriate, inversion of the structure. The structure is assembled to maximize its connectivity and centred optimally in the unit cell. SHELXT has already solved many thousand structures with a high success rate, and is optimized for multiprocessor computers. It is, however, unsuitable for severely disordered and twinned structures because it is based on the assumption that the structure consists of atoms.
This paper describes the mathematical basis for olex2.refine, the new refinement engine which is integrated within the Olex2 program. Precise and clear equations are provided for every computation performed by this engine, including structure factors and their derivatives, constraints, restraints and twinning; a general overview is also given of the different components of the engine and their relation to each other. A framework for adding multiple general constraints with dependencies on common physical parameters is described. Several new restraints on atomic displacement parameters are also presented.
An account is given of the development of the SHELX system of computer programs from SHELX‐76 to the present day. In addition to identifying useful innovations that have come into general use through their implementation in SHELX, a critical analysis is presented of the less‐successful features, missed opportunities and desirable improvements for future releases of the software. An attempt is made to understand how a program originally designed for photographic intensity data, punched cards and computers over 10000 times slower than an average modern personal computer has managed to survive for so long. SHELXL is the most widely used program for small‐molecule refinement and SHELXS and SHELXD are often employed for structure solution despite the availability of objectively superior programs. SHELXL also finds a niche for the refinement of macromolecules against high‐resolution or twinned data; SHELXPRO acts as an interface for macromolecular applications. SHELXC, SHELXD and SHELXE are proving useful for the experimental phasing of macromolecules, especially because they are fast and robust and so are often employed in pipelines for high‐throughput phasing. This paper could serve as a general literature citation when one or more of the open‐source SHELX programs (and the Bruker AXS version SHELXTL) are employed in the course of a crystal‐structure determination.
A strategy is described for regularizing ill posed structure and nanostructure scattering inverse problems (i.e. structure solution) from complex material structures. This paper describes both the philosophy and strategy of the approach, and a software implementation, DiffPy Complex Modeling Infrastructure (DiffPy‐CMI).
The Scherrer equation is a widely used tool to determine the crystallite size of polycrystalline samples. However, it is not clear if one can apply it to large crystallite sizes because its derivation is based on the kinematical theory of X‐ray diffraction. For large and perfect crystals, it is more appropriate to use the dynamical theory of X‐ray diffraction. Because of the appearance of polycrystalline materials with a high degree of crystalline perfection and large sizes, it is the authors' belief that it is important to establish the crystallite size limit for which the Scherrer equation can be applied. In this work, the diffraction peak profiles are calculated using the dynamical theory of X‐ray diffraction for several Bragg reflections and crystallite sizes for Si, LaB6 and CeO2. The full width at half‐maximum is then extracted and the crystallite size is computed using the Scherrer equation. It is shown that for crystals with linear absorption coefficients below 2117.3 cm−1 the Scherrer equation is valid for crystallites with sizes up to 600 nm. It is also shown that as the size increases only the peaks at higher 2θ angles give good results, and if one uses peaks with 2θ > 60° the limit for use of the Scherrer equation would go up to 1 µm. The maximum crystal size is determined for application of the Scherrer equation.
Until recently, structure determination by transmission electron microscopy of beam‐sensitive three‐dimensional nanocrystals required electron diffraction tomography data collection at liquid‐nitrogen temperature, in order to reduce radiation damage. Here it is shown that the novel Timepix detector combines a high dynamic range with a very high signal‐to‐noise ratio and single‐electron sensitivity, enabling ab initio phasing of beam‐sensitive organic compounds. Low‐dose electron diffraction data (∼0.013 e− Å−2 s−1) were collected at room temperature with the rotation method. It was ascertained that the data were of sufficient quality for structure solution using direct methods using software developed for X‐ray crystallography (XDS, SHELX) and for electron crystallography (ADT3D/PETS, SIR2014). A specialized quantum area detector for electron diffraction studies makes it possible to solve the structure of small organic compound nanocrystals in non‐cryo conditions by direct methods.
Accurate structure refinement from electron‐diffraction data is not possible without taking the dynamical‐diffraction effects into account. A complete three‐dimensional model of the structure can be obtained only from a sufficiently complete three‐dimensional data set. In this work a method is presented for crystal structure refinement from the data obtained by electron diffraction tomography, possibly combined with precession electron diffraction. The principle of the method is identical to that used in X‐ray crystallography: data are collected in a series of small tilt steps around a rotation axis, then intensities are integrated and the structure is optimized by least‐squares refinement against the integrated intensities. In the dynamical theory of diffraction, the reflection intensities exhibit a complicated relationship to the orientation and thickness of the crystal as well as to structure factors of other reflections. This complication requires the introduction of several special parameters in the procedure. The method was implemented in the freely available crystallographic computing system Jana2006.
The Bilbao Crystallographic Server is a web site with crystallographic programs and databases freely available on‐line (http://www.cryst.ehu.es). The server gives access to general information related to crystallographic symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones etc.). Apart from the simple tools for retrieving the stored data, there are programs for the analysis of group–subgroup relations between space groups (subgroups and supergroups, Wyckoff‐position splitting schemes etc.). There are also software packages studying specific problems of solid‐state physics, structural chemistry and crystallography. This article reports on the programs treating representations of point and space groups. There are tools for the construction of irreducible representations, for the study of the correlations between representations of group–subgroup pairs of space groups and for the decompositions of Kronecker products of representations.
MicroED, a method at the intersection of X‐ray crystallography and electron cryo‐microscopy, has rapidly progressed by exploiting advances in both fields and has already been successfully employed to determine the atomic structures of several proteins from sub‐micron‐sized, three‐dimensional crystals. A major limiting factor in X‐ray crystallography is the requirement for large and well ordered crystals. By permitting electron diffraction patterns to be collected from much smaller crystals, or even single well ordered domains of large crystals composed of several small mosaic blocks, MicroED has the potential to overcome the limiting size requirement and enable structural studies on difficult‐to‐crystallize samples. This communication details the steps for sample preparation, data collection and reduction necessary to obtain refined, high‐resolution, three‐dimensional models by MicroED, and presents some of its unique challenges.
The Brillouin‐zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) offers k‐vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so‐called reciprocal‐space groups and this classification scheme is compared with the classification of Cracknell et al. [Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space‐group representations by specifying the independent parameter ranges of general and special k vectors. Guides to the k‐vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin‐zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.
The TOPOS program package was used to generate all subnets of 3‐ to 12‐coordinated binodal nets taken from the Reticular Chemistry Structure Resource database. 38 304 binodal nets with novel topologies were revealed and stored in the TTD collection. A new invariant, the adjacency matrix of the shell graph of a node, is proposed to distinguish the node local topology. With this invariant, the first six examples of binodal‐quasi‐uninodal nets were discovered. 4604 organic and metal‐organic frameworks were analyzed to find examples of the topologies generated. It was shown that many edge‐transitive nets as well as unknown topologies occur in crystal structures.
A detailed set of synthetic and crystallographic guidelines for the crystalline sponge method based upon the analysis of expediently synthesized crystal sponges using third‐generation synchrotron radiation are reported. The procedure for the synthesis of the zinc‐based metal–organic framework used in initial crystal sponge reports has been modified to yield competent crystals in 3 days instead of 2 weeks. These crystal sponges were tested on some small molecules, with two being unexpectedly difficult cases for analysis with in‐house diffractometers in regard to data quality and proper space‐group determination. These issues were easily resolved by the use of synchrotron radiation using data‐collection times of less than an hour. One of these guests induced a single‐crystal‐to‐single‐crystal transformation to create a larger unit cell with over 500 non‐H atoms in the asymmetric unit. This led to a non‐trivial refinement scenario that afforded the best Flack x absolute stereochemical determination parameter to date for these systems. The structures did not require the use of PLATON/SQUEEZE or other solvent‐masking programs, and are the highest‐quality crystalline sponge systems reported to date where the results are strongly supported by the data. A set of guidelines for the entire crystallographic process were developed through these studies. In particular, the refinement guidelines include strategies to refine the host framework, locate guests and determine occupancies, discussion of the proper use of geometric and anisotropic displacement parameter restraints and constraints, and whether to perform solvent squeezing/masking. The single‐crystal‐to‐single‐crystal transformation process for the crystal sponges is also discussed. The presented general guidelines will be invaluable for researchers interested in using the crystalline sponge method at in‐house diffraction or synchrotron facilities, will facilitate the collection and analysis of reliable high‐quality data, and will allow construction of chemically and physically sensible models for guest structural determination.
The topological complexity of a crystal structure can be quantitatively evaluated using complexity measures of its quotient graph, which is defined as a projection of a periodic network of atoms and bonds onto a finite graph. The Shannon information‐based measures of complexity such as topological information content, IG, and information content of the vertex‐degree distribution of a quotient graph, Ivd, are shown to be efficient for comparison of the topological complexity of polymorphs and chemically related structures. The IG measure is sensitive to the symmetry of the structure, whereas the Ivd measure better describes the complexity of the bonding network.
Data collected during dynamic structure pump–probe crystallography experiments require appropriate indicators of agreement and tools to visualize the electron‐density distribution changes. Agreement factors based on the ratio of intensities R with and without the external perturbation are shown to be analogous to the 1 and w2 factors widely used in standard crystallographic refinements. The η‐based factors, normalized by the average relative intensity change, are significantly larger than R‐based values. It is shown that the relative intensity change η‐based factors are not suitable for comparing different data sets. Fourier photodifference maps allow the visualization of the externally induced structural changes in the crystal, but also can be used during refinement to observe residual peaks not yet accounted for by the model and thus monitor the progress of the refinement. The photodeformation maps are a complementary tool to confirm the validity of the final model. Photodeformation maps with equalized laser‐on and laser‐off thermal motion are used to highlight the structural changes.
The Scherrer equation is a widely used tool to obtain crystallite size from polycrystalline samples. Its limit of applicability has been determined recently, using computer simulations, for a few structures and it was proposed that it is directly dependent on the linear absorption coefficient (μ0) and Bragg angle (gθB). In this work, a systematic study of the Scherrer limit is presented, where it is shown that it is equal to approximately 11.9% of the extinction length. It is also shown that absorption imposes a maximum value on it and that this maximum is directly proportional to sin gθB/μ0. Study of the limit of applicability of the Scherrer equation has found it is approximately 11.9% of the extinction length and has a maximum value because of absorption.
The description of displacive distorted structures in terms of symmetry‐adapted modes is reviewed. A specific parameterization of the symmetry‐mode decomposition of these pseudosymmetric structures defined on the setting of the experimental space group is proposed. This approach closely follows crystallographic conventions and permits a straightforward transformation between symmetry‐mode and conventional descriptions of the structures. Multiple examples are presented showing the insight provided by the symmetry‐mode approach. The methodology is shown at work, illustrating its various possibilities for improving the characterization of distorted structures, for example: detection of hidden structural correlations, identification of fundamental and marginal degrees of freedom, reduction of the effective number of atomic positional parameters, quantitative comparison of structures with the same or different space group, detection of false refinement minima, systematic characterization of thermal behavior, rationalization of phase diagrams and various symmetries in families of compounds etc. The close relation of the symmetry‐mode description with the superspace formalism applied to commensurate superstructures is also discussed. Finally, the application of this methodology in the field of ab initio or first‐principles calculations is outlined. At present, there are several freely available user‐friendly computer tools for performing automatic symmetry‐mode analyses. The use of these programs does not require a deep knowledge of group theory and can be applied either a posteriori to analyze a given distorted structure or a priori to parameterize the structure to be determined. It is hoped that this article will encourage the use of these tools. All the examples presented here have been worked out using the program AMPLIMODES [Orobengoa et al. (2009). J. Appl. Cryst.42, 820–833].
Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. This article presents a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point, factor and space groups, site symmetries and Wyckoff positions. The protocol determines a system‐specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self‐consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space‐group information provided for over 54 000 entries in the Inorganic Crystal Structure Database (ICSD). Subsequently, a complete symmetry analysis is applied to all 1.7+ million entries of the AFLOW data repository. The AFLOW‐SYM package has been implemented in, and made available for, public use through the automated ab initio framework AFLOW. AFLOW‐SYM is a comprehensive crystal‐symmetry analysis suite catering to automated computational workflows and presenting a wealth of symmetry descriptions. This platform employs robust mapping techniques, validates calculated symmetry elements and resolves self‐consistent tolerances for each system to yield results more commensurate with experiments compared with other common symmetry packages.
More than 35 years and 11 000 publications after the discovery of quasicrystals by Dan Shechtman, quite a bit is known about their occurrence, formation, stability, structures and physical properties. It has also been discovered that quasiperiodic self‐assembly is not restricted to intermetallics, but can take place in systems on the meso‐ and macroscales. However, there are some blank areas, even in the centre of the big picture. For instance, it has still not been fully clarified whether quasicrystals are just entropy‐stabilized high‐temperature phases or whether they can be thermodynamically stable at 0 K as well. More studies are needed for developing a generally accepted model of quasicrystal growth. The state of the art of quasicrystal research is briefly reviewed and the main as‐yet unanswered questions are addressed, as well as the experimental limitations to finding answers to them. The focus of this discussion is on quasicrystal structure analysis as well as on quasicrystal stability and growth mechanisms. The state of the art of quasicrystal research is critically reviewed. Fundamental questions that are still unanswered are discussed and experimental limitations are considered.
This paper summarizes the current state of charge flipping, a recently developed algorithm of ab initio structure determination. Its operation is based on the perturbation of large plateaus of low electron density but not directly on atomicity. Such a working principle radically differs from that of classical direct methods and offers complementary applications. The list of successful structure‐solution cases includes periodic and aperiodic crystals using single‐crystal and powder diffraction data measured with X‐ray and neutron radiation. Apart from counting applications, the paper mainly deals with algorithmic issues: it describes and compares new variants of the iteration scheme, helps to identify and improve solutions, discusses the required data and the use of known information. Finally, it tries to foretell the future of such an alternative among well established direct methods.
Compared with X‐rays, electron diffraction faces a crucial challenge: dynamical electron scattering compromises structure solution and its effects can only be modelled in specific cases. Dynamical scattering can be reduced experimentally by decreasing crystal size but not without a penalty, as it also reduces the overall diffracted intensity. In this article it is shown that nanometre‐sized crystals from organic pharmaceuticals allow positional refinement of the hydrogen atoms, even whilst ignoring the effects of dynamical scattering during refinement. To boost the very weak diffraction data, a highly sensitive hybrid pixel detector was employed. A general likelihood‐based computational approach was also introduced for further reducing the adverse effects of dynamic scattering, which significantly improved model accuracy, even for protein crystal data at substantially lower resolution. Experimental and computational reduction of dynamical electron scattering allows for visualizing of individual hydrogen atoms.