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.

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.

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.