Early neutron diffraction experiments performed in 1944 using the first nuclear reactors are described.

Although neutron diffraction was first observed using radioactive decay sources shortly after the discovery of the neutron, it was only with the availability of higher intensity neutron beams from the first nuclear reactors, constructed as part of the Manhattan Project, that systematic investigation of Bragg scattering became possible. Remarkably, at a time when the war effort was singularly focused on the development of the atomic bomb, groups working at Oak Ridge and Chicago carried out key measurements and recognized the future utility of neutron diffraction quite independent of its contributions to the measurement of nuclear cross sections. Ernest O. Wollan, Lyle B. Borst and Walter H. Zinn were all able to observe neutron diffraction in 1944 using the X-10 graphite reactor and the CP-3 heavy water reactor. Subsequent work by Wollan and Clifford G. Shull, who joined Wollan’s group at Oak Ridge in 1946, laid the foundations for widespread application of neutron diffraction as an important research tool.

The standard settings of (3 + d)-dimensional superspace groups are determined for a series of modulated compounds, especially concentrating on d = 2 and 3. The coordinate transformation in superspace is discussed in view of its implications in physical space.

An algorithm is presented which determines the equivalence of two settings of a (3 + d)-dimensional superspace group (d = 1, 2, 3). The algorithm has been implemented as a web tool on , providing the transformation of any user-given superspace group to the standard setting of this superspace group in . It is shown how the standard setting of a superspace group can be directly obtained by an appropriate transformation of the external-space lattice vectors (the basic structure unit cell) and a transformation of the internal-space lattice vectors (new modulation wavevectors are linear combinations of old modulation wavevectors plus a three-dimensional reciprocal-lattice vector). The need for non-standard settings in some cases and the desirability of employing standard settings of superspace groups in other cases are illustrated by an analysis of the symmetries of a series of compounds, comparing published and standard settings and the transformations between them. A compilation is provided of standard settings of compounds with two- and three-dimensional modulations. The problem of settings of superspace groups is discussed for incommensurate composite crystals and for chiral superspace groups.

B eq is a well defined entity that represents one property, mean-square displacement, of the anisotropic atomic displacement parameter tensor from which it is derived. B eq is not, however, the best estimate of the B factor that would result from isotropic model refinement. A new entity B est is proposed to serve this purpose.

Crystallographic structural models for macromolecules have typically included an isotropic displacement parameter B iso for each atom. In cases where the structural model instead includes anisotropic displacement parameters U ij, the derived quantity B eq can be substituted for B iso for many purposes. B eq is not, however, the best predictor of the value B iso that would hypothetically have been obtained by direct refinement of an isotropic model. A new entity B est is proposed that represents an estimate for B iso that minimizes the Kullback–Leibler divergence from a paired anisotropic model. In general B est/B eq < 1, with the difference between the two values becoming larger for atoms that are more anisotropic. Although this difference does not affect direct refinement of either isotropic or anisotropic models, it is relevant to any analysis that compares isotropic and anisotropic models of the same underlying structure. In particular, it may lead to improved selection of multi-group TLS models based on analysis of an initial isotropic refinement.

Search spaces in the method of molecular replacement are shown to be coset spaces of the Lie group of rigid-body motions by the chiral space group of a crystal. The resulting ‘motion space’ can be endowed with a quasigroup operation that has interesting properties which are explored here.

Molecular replacement (MR) is a well established method for phasing of X-ray diffraction patterns for crystals composed of biological macromolecules of known chemical structure but unknown conformation. In MR, the starting point is known structural domains that are presumed to be similar in shape to those in the macromolecular structure which is to be determined. A search is then performed over positions and orientations of the known domains within a model of the crystallographic asymmetric unit so as to best match a computed diffraction pattern with experimental data. Unlike continuous rigid-body motions in Euclidean space and the discrete crystallographic space groups, the set of motions over which molecular replacement searches are performed does not form a group under the operation of composition, which is shown here to lack the associative property. However, the set of rigid-body motions in the asymmetric unit forms another mathematical structure called a quasigroup, which can be identified with right-coset spaces of the full group of rigid-body motions with respect to the chiral space group of the macromolecular crystal. The algebraic properties of this space of motions are articulated here.

A modified Laue technique suitable for time-resolved diffraction is described in which profile-independent integration is used, the RATIO method is applied and multi-crystal data are normalized to a common scale. The method is applied in single-pulse pump–probe studies of a binuclear Rh complex, showing Rh—Rh bond shortening of 0.136 (8) Å on excitation.

A modified Laue method is shown to produce excited-state structures at atomic resolution of a quality competitive with those from monochromatic experiments. The much faster data collection allows the use of only one or a few X-ray pulses per data frame, which minimizes crystal damage caused by laser exposure of the samples and optimizes the attainable time resolution. The method has been applied to crystals of the α-modification of Rh2(μ-PNP)2(PNP)2 (BPh4)2 [PNP = CH3N(P(OCH3)2)2, Ph = phenyl]. The experimental results show a shortening of the Rh—Rh distance in the organometallic complex of 0.136 (8) Å on excitation and are quantitatively supported by quantum-mechanical (QM)/molecular-mechanics (MM) theoretical calculations which take into account the confining effect of the crystal environment, but not by theoretical results on the isolated complex, demonstrating the defining effect of the crystal matrix.

Good agreement between the high-resolution experimental results for (222, 113) three-beam diffraction in Ge and computer simulations based on the dynamical multiple diffraction theory are presented.

The results of high-resolution analysis of the (222, >113) three-beam diffraction in Ge are presented. For monochromatization and angular collimation of the incident synchrotron beam a multi-crystal arrangement in a dispersive setup in both vertical and horizontal planes was used in an attempt to experimentally approach plane-wave incident conditions. Using this setup, for various azimuthal angles the polar angular curves which are very close to theoretical computer simulations for the plane monochromatic wave were measured. The effect of the strong two-beam 222 diffraction was observed for the first time with the maximum reflectivity close to 60% even though the total reflection of the incident beam into a forbidden reflection was not achieved owing to absorption. The structure factor of the 222 reflection in Ge was experimentally determined.

An extension is proposed to the rigid-bond description of atomic thermal motion in crystals.

The rigid-bond model [Hirshfeld (1976 ▶). Acta Cryst. A32, 239–244] states that the mean-square displacements of two atoms are equal in the direction of the bond joining them. This criterion is widely used for verification (as intended by Hirshfeld) and also as a restraint in structure refinement as suggested by Rollett [Crystallographic Computing (1970 ▶), edited by F. R. Ahmed et al., pp. 167–181. Copenhagen: Munksgaard]. By reformulating this condition, so that the relative motion of the two atoms is required to be perpendicular to the bond, the number of restraints that can be applied per anisotropic atom is increased from about one to about three. Application of this condition to 1,3-distances in addition to the 1,2-distances means that on average just over six restraints can be applied to the six anisotropic displacement parameters of each atom. This concept is tested against very high resolution data of a small peptide and employed as a restraint for protein refinement at more modest resolution (e.g. 1.7 Å).

A new algorithm is developed for reconstructing the high-resolution three-dimensional diffraction intensity function of a globular biological macromolecule from many quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The structural resolution is expressed as a function of the incident X-ray intensity and quantities characterizing the target molecule.

A new two-step algorithm is developed for reconstructing the three-dimensional diffraction intensity of a globular biological macromolecule from many experimentally measured quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The first step is classification of the two-dimensional patterns into groups according to the similarity of direction of the incident X-rays with respect to the molecule and an averaging within each group to reduce the noise. The second step is detection of common intersecting circles between the signal-enhanced two-dimensional patterns to identify their mutual location in the three-dimensional wavenumber space. The newly developed algorithm enables one to detect a signal for classification in noisy experimental photon-count data with as low as ∼0.1 photons per effective pixel. The wavenumber of such a limiting pixel determines the attainable structural resolution. From this fact, the resolution limit due to the quantum noise attainable by this new method of analysis as well as two important experimental parameters, the number of two-dimensional patterns to be measured (the load for the detector) and the number of pairs of two-dimensional patterns to be analysed (the load for the computer), are derived as a function of the incident X-ray intensity and quantities characterizing the target molecule.

A complete set of structure factors has been extracted from hundreds of thousands of femtosecond X-ray diffraction patterns from randomly oriented Photosystem I membrane protein nanocrystals, using the Monte Carlo method of intensity integration. The data, collected at the Linac Coherent Light Source, are compared with conventional single-crystal data collected at a synchrotron source, and the quality of each data set was found to be similar.

A complete set of structure factors has been extracted from hundreds of thousands of femtosecond single-shot X-ray microdiffraction patterns taken from randomly oriented nanocrystals. The method of Monte Carlo integration over crystallite size and orientation was applied to experimental data from Photosystem I nanocrystals. This arrives at structure factors from many partial reflections without prior knowledge of the particle-size distribution. The data were collected at the Linac Coherent Light Source (the first hard-X-ray laser user facility), to which was fitted a hydrated protein nanocrystal injector jet, according to the method of serial crystallography. The data are single ‘still’ diffraction snapshots, each from a different nanocrystal with sizes ranging between 100 nm and 2 µm, so the angular width of Bragg peaks was dominated by crystal-size effects. These results were compared with single-crystal data recorded from large crystals of Photosystem I at the Advanced Light Source and the quality of the data was found to be similar. The implications for improving the efficiency of data collection by allowing the use of very small crystals, for radiation-damage reduction and for time-resolved diffraction studies at room temperature are discussed.

The modelling of grazing-incidence small-angle X-ray scattering (GISAXS) from three-dimensional quantum dot lattices is described.

The ordering of quantum dots in three-dimensional quantum dot lattices is investigated by grazing-incidence small-angle X-ray scattering (GISAXS). Theoretical models describing GISAXS intensity distributions for three general classes of lattices of quantum dots are proposed. The classes differ in the type of disorder of the positions of the quantum dots. The models enable full structure determination, including lattice type, lattice parameters, the type and degree of disorder in the quantum dot positions and the distributions of the quantum dot sizes. Applications of the developed models are demonstrated using experimentally measured data from several types of quantum dot lattices formed by a self-assembly process.

Theoretical analysis and experimental validation prove that a multi-dataset data-collection strategy produces better diffraction data. The readiness test is a simple and sensitive method for X-ray data-collection system evaluation and a benchmark.

A multi-dataset (MDS) data-collection strategy is proposed and analyzed for macromolecular crystal diffraction data acquisition. The theoretical analysis indicated that the MDS strategy can reduce the standard deviation (background noise) of diffraction data compared with the commonly used single-dataset strategy for a fixed X-ray dose. In order to validate the hypothesis experimentally, a data-quality evaluation process, termed a readiness test of the X-ray data-collection system, was developed. The anomalous signals of sulfur atoms in zinc-free insulin crystals were used as the probe to differentiate the quality of data collected using different data-collection strategies. The data-collection results using home-laboratory-based rotating-anode X-ray and synchrotron X-ray systems indicate that the diffraction data collected with the MDS strategy contain more accurate anomalous signals from sulfur atoms than the data collected with a regular data-collection strategy. In addition, the MDS strategy offered more advantages with respect to radiation-damage-sensitive crystals and better usage of rotating-anode as well as synchrotron X-rays.

X-ray free-electron lasers are being used to determine the three-dimensional structure of objects from random snapshots. The two apparently very different Bayesian algorithms capable of performing this at ultra-low signal are fundamentally the same.

The advent of X-ray free-electron lasers promises the possibility to determine the structure of individual particles such as microcrystallites, viruses and biomolecules from single-shot diffraction snapshots obtained before the particle is destroyed by the intense femtosecond pulse. This program requires the ability to determine the orientation of the particle giving rise to each snapshot at signal levels as low as ~10−2 photons per pixel. Two apparently different approaches have recently demonstrated this capability. Here we show they represent different implementations of the same fundamental approach, and identify the primary factors limiting their performance.

The effect of the total reflection of the incident beam into the 222 reflected beam in the Renninger (222/113) case in Si was experimentally observed by using a highly monochromatic beam with high angular collimation in both the vertical and horizontal planes.

X-ray optical schemes capable of producing a highly monochromatic beam with high angular collimation in both the vertical and horizontal planes have been evaluated and utilized to study high-resolution diffraction phenomena in the Renninger (222/113) case of three-beam diffraction in silicon. The effect of the total reflection of the incident beam into the nearly forbidden reflected beam was observed for the first time with the maximum 222 reflectivity at the 70% level. We have demonstrated that the width of the 222 reflection can be varied many times by tuning the azimuthal angle by only a few µrad in the vicinity of the three-beam diffraction region. This effect, predicted theoretically more than 20 years ago, is explained by the enhancement of the 222 scattering amplitude due to the virtual two-stage 000 113 222 process which depends on the azimuthal angle.

A reference table of exact direct-space asymmetric units for the 230 crystallographic space groups is presented, based on a new geometric notation for asymmetric unit conditions.

It is well known that the direct-space asymmetric unit definitions found in the International Tables for Crystallography, Volume A, are inexact at the borders. Face- and edge-specific sub-conditions have to be added to remove parts redundant under symmetry. This paper introduces a concise geometric notation for asymmetric unit conditions. The notation is the foundation for a reference table of exact direct-space asymmetric unit definitions for the 230 crystallographic space-group types. The change-of-basis transformation law for the conditions is derived, which allows the information from the reference table to be used for any space-group setting. We also show how the vertices of an asymmetric unit can easily be computed from the information in the reference table.

A rapid, easy-to-perform translation calibration procedure has been developed for use with the EMBL/ESRF mini-κ goniometer head and for other inverse-kappa goniometers designed for macromolecular crystallography. Regular calibration ensures the precision of experiments that rely on many degrees of freedom in crystal reorientation.

Precise and convenient crystal reorientation is of experimental importance in macromolecular crystallography (MX). The development of multi-axis goniometers, such as the ESRF/EMBL mini-κ, necessitates the corresponding development of calibration procedures that can be used for the setup, maintenance and troubleshooting of such devices. While traditional multi-axis goniometers require all rotation axes to intersect the unique point of the sample position, recently developed miniaturized instruments for sample reorientation in MX are not as restricted. However, the samples must always be re-centred following a change in orientation. To overcome this inconvenience and allow the use of multi-axis goniometers without the fundamental restriction of having all axes intersecting in the same point, an automatic translation correction protocol has been developed for such instruments. It requires precise information about the direction and location of the rotation axes. To measure and supply this information, a general, easy-to-perform translation calibration (TC) procedure has also been developed. The TC procedure is routinely performed on most MX beamlines at the ESRF and some results are presented for reference.

Excited-state geometries determined by time-resolved synchrotron diffraction are summarized with emphasis on their comparison with a series of theoretical results. The relative merits of monochromatic and polychromatic (Laue) techniques are discussed.

Definitive experimental results on the geometry of fleeting species are at the time of writing still limited to monochromatic data collection, but methods for modifications of the polychromatic Laue data to increase their accuracy and their suitability for pump–probe experiments have been implemented and are reviewed. In the monochromatic experiments summarized, excited-state conversion percentages are small when neat crystals are used, but are higher when photoactive species are embedded in an inert framework in supramolecular crystals. With polychromatic techniques and increasing source brightness, smaller samples down to tenths of a micrometre or less can be used, increasing homogeneity of exposure and the fractional population of the excited species. Experiments described include a series of transition metal complexes and a fully organic example involving excimer formation. In the final section, experimental findings are compared with those from theoretical calculations on the isolated species. Qualitative agreement is generally obtained, but the theoretical results are strongly dependent on the details of the calculation, indicating the need for further systematic analysis.

Algorithms and geometrical properties are described for the automated building of nucleic acids in experimental electron density.

Medium- to high-resolution X-ray structures of DNA and RNA molecules were investigated to find geometric properties useful for automated model building in crystallographic electron-density maps. We describe a simple method, starting from a list of electron-density ‘blobs’, for identifying backbone phosphates and nucleic acid bases based on properties of the local electron-density distribution. This knowledge should be useful for the automated building of nucleic acid models into electron-density maps. We show that the distances and angles involving C1′ and the P atoms, using the pseudo-torsion angles and that describe the …P—C1′—P—C1′… chain, provide a promising basis for building the nucleic acid polymer. These quantities show reasonably narrow distributions with asymmetry that should allow the direction of the phosphate backbone to be established.

A procedure for obtaining depth-specific scattering profiles of laterally defined nanostructure using grazing-incidence small-angle X-ray scattering is described.

A model-free method of reconstructing depth-specific lateral scattering from incident-angle-resolved grazing-incidence small-angle X-ray scattering (GISAXS) data is proposed. The information on the material which is available through variation of the X-ray penetration depth with incident angle is accessed through reference to the reflected branch of the GISAXS process. Reconstruction of the scattering from lateral density fluctuations is achieved by solving the resulting Fredholm integral equation with minimal a priori information about the experimental system. Results from simulated data generated for hypothetical multilayer polymer systems with constant absorption coefficient are used to verify that the method can be applied to cases with large X-ray penetration depths, as typically seen with polymer materials. Experimental tests on a spin-coated thick film of a blend of diblock copolymers demonstrate that the approach is capable of reconstruction of the scattering from a multilayer structure with the identification of lateral scattering profiles as a function of sample depth.

Time-resolved structural studies of proteins have undergone several significant developments during the last decade. Recent developments using time-resolved X-ray methods, such as time-resolved Laue diffraction, low-temperature intermediate trapping, time-resolved wide-angle X-ray scattering and time-resolved X-ray absorption spectroscopy, are reviewed.

Proteins undergo conformational changes during their biological function. As such, a high-resolution structure of a protein’s resting conformation provides a starting point for elucidating its reaction mechanism, but provides no direct information concerning the protein’s conformational dynamics. Several X-ray methods have been developed to elucidate those conformational changes that occur during a protein’s reaction, including time-resolved Laue diffraction and intermediate trapping studies on three-dimensional protein crystals, and time-resolved wide-angle X-ray scattering and X-ray absorption studies on proteins in the solution phase. This review emphasizes the scope and limitations of these complementary experimental approaches when seeking to understand protein conformational dynamics. These methods are illustrated using a limited set of examples including myoglobin and haemoglobin in complex with carbon monoxide, the simple light-driven proton pump bacteriorhodopsin, and the superoxide scavenger superoxide reductase. In conclusion, likely future developments of these methods at synchrotron X-ray sources and the potential impact of emerging X-ray free-electron laser facilities are speculated upon.

Here it is demonstrated how five-dimensional crystallography can be used to determine a comprehensive chemical kinetic mechanism in concert with the atomic structures of transient intermediates that form and decay during the course of the reaction.

A method for determining a comprehensive chemical kinetic mechanism in macromolecular reactions is presented. The method is based on five-dimensional crystallography, where, in addition to space and time, temperature is also taken into consideration and an analysis based on singular value decomposition is applied. First results of such a time-resolved crystallographic study are presented. Temperature-dependent time-resolved X-ray diffraction measurements were conducted on the newly upgraded BioCARS 14-ID-B beamline at the Advanced Photon Source and aimed at elucidating a comprehensive kinetic mechanism of the photoactive yellow protein photocycle. Extensive time series of crystallographic data were collected at two temperatures, 293 K and 303 K. Relaxation times of the reaction extracted from these time series exhibit measurable differences for the two temperatures, hence demonstrating that five-dimensional crystallography is feasible.

The approach used and the progress made in the assignment of structure types to the crystal structures contained in the ICSD database are reported.

Both the approach used and the progress made in the assignment of structure types to the crystal structures contained in the ICSD database are reported. Extending earlier work, an hierarchical set of criteria for the separation of isopointal structures into isoconfigurational structure types is used. It is shown how these criteria, which include the space group (number), Wyckoff sequence and Pearson symbol, c/a ratio, β ranges, ANX formulae and, in certain cases, the necessary elements and forbidden elements, may be used to uniquely identify the representative structure types of the compounds contained in the ICSD database.

Erratum to Livet [Acta Cryst. (2007), A63, 63–87].

In the paper by Livet [Acta Cryst. (2007), A63, 63–87], equation (15) is incorrect. The correct equation is

Techniques for coherent X-ray scattering measurements are detailed. Applications in the study of the dynamics of fluctuations and in lensless high-resolution imaging are described.

Methods for carrying out coherent X-ray scattering experiments are reviewed. The brilliance of the available synchrotron sources, the characteristics of the existing optics, the various ways of obtaining a beam of controlled coherence properties and the detectors used are summarized. Applications in the study of the dynamics of speckle patterns are described. In the case of soft condensed matter, the movement of inclusions like fillers in polymers or colloidal particles can be observed and these can reflect polymer or liquid-crystal fluctuations. In hard condensed-matter problems, like phase transitions, charge-density waves or phasons in quasicrystals, the study of speckle fluctuations provides new time-resolved methods. In the domain of lensless imaging, the coherent beam gives the modulus of the sample Fourier transform. If oversampling conditions are fulfilled, the phase can be obtained and the image in the direct space can be reconstructed. The forthcoming improvements of all these techniques are discussed.

In icosahedral carbon nanoparticles, the diamond-like core can undergo a reversible topological transition into and coexist coherently with the onion shells.

The general approach for describing and designing complex hierarchical icosahedral structures is discussed. Structural models of icosahedral carbon nanoparticles in which the local arrangement of atoms is virtually identical to that in diamond are derived. It is shown that icosahedral diamond-like particles can be transformed into onion-like shell structures (and vice versa) by the consecutive smoothing (puckering) of atomic networks without disturbance of their topological integrity. The possibility of coherent coexistence of icosahedral diamond-like core with onion shells is shown.

Multiple twinning in cubic crystals is represented geometrically by three-dimensional fractals and algebraically by groupoids. The groupoid composition table can be used to identify the Σ3n grain boundaries in EBSD maps.

Multiple twinning in cubic crystals is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the Σ3n operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of variants and the number of Σ3n operators for any twinning order. Different substructures of this groupoid (free group, semigroup) can be equivalently introduced to encode the operations with strings. For any coding substructure, the operators are expressed by sets of equivalent strings. The composition of two operators is determined without any matrix calculation by string concatenations. It is multivalued due to the groupoid structure. The composition table of the operators is used to identify the Σ3n grain boundaries and to reconstruct the twin related domains in the electron back-scattered diffraction maps.