Showing papers in "Acta Crystallographica Section A in 2015"

SHELXT - Integrated space-group and crystal- structure determination

Abstract: 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.

The anatomy of a comprehensive constrained, restrained refinement program for the modern computing environment – Olex2 dissected

Abstract: 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.

Complex modeling: a strategy and software program for combining multiple information sources to solve ill posed structure and nanostructure inverse problems

Abstract: 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).

Structure refinement using precession electron diffraction tomography and dynamical diffraction: theory and implementation.

Abstract: 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.

MicroED data collection and processing

Abstract: 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.

Analysis of rapidly synthesized guest‐filled porous complexes with synchrotron radiation: practical guidelines for the crystalline sponge method

Abstract: 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.

High-symmetry embeddings of interpenetrating periodic nets. Essential rings and patterns of catenation

Abstract: Symmetrical embeddings are given for multiply intergrown sets of some commonly occurring nets such as dia (diamond), qtz (quartz), pcu (net of primitive cubic lattice) and srs (labyrinth net of the G minimal surface). Data are also given for all known pairs of nets which have edge-transitive self-dual tilings. Examples are given for symmetrical polycatenation of the 2-periodic nets sql (square lattice) and hcb (honeycomb). The idea that the rings that are the faces of natural tilings form a complete basis set (essential rings) is explored and patterns of catenation of such rings described.

More statistics on intermetallic compounds - ternary phases.

Abstract: How many different intermetallic compounds are known so far, and in how many different structure types do they crystallize? What are their chemical compositions, the most abundant ones and the rarest ones? These are some of the questions we are trying to find answers for in our statistical analysis of the structures of the 20 829 intermetallic phases included in the database Pearson's Crystal Data, with the goal of gaining insight into some of their ordering principles. In the present paper, we focus on the subset of 13 026 ternary intermetallics, which crystallize in 1391 different structure types; remarkably, 667 of them have just one representative. What makes these 667 structures so unique that they are not adopted by any other of the known intermetallic compounds? Notably, ternary compounds are known in only 5109 of the 85 320 theoretically possible ternary intermetallic systems so far. In order to get an overview of their chemical compositions we use structure maps with Mendeleev numbers as ordering parameters.

Magnetic structure determination from the magnetic pair distribution function (mPDF): ground state of MnO.

Abstract: An experimental determination of the magnetic pair distribution function (mPDF) defined in an earlier paper [Frandsen et al. (2014). Acta Cryst. A70, 3–11] is presented for the first time. The mPDF was determined from neutron powder diffraction data from a reactor and a neutron time-of-flight total scattering source on a powder sample of the antiferromagnetic oxide MnO. A description of the data treatment that allowed the measured mPDF to be extracted and then modelled is provided and utilized to investigate the low-temperature structure of MnO. Atomic and magnetic co-refinements support the scenario of a locally monoclinic ground-state atomic structure, despite the average structure being rhombohedral, with the mPDF analysis successfully recovering the known antiferromagnetic spin configuration. The total scattering data suggest a preference for the spin axis to lie along the pseudocubic [10{\overline 1}] direction. Finally, r-dependent PDF refinements indicate that the local monoclinic structure tends toward the average rhombohedral R{\overline 3}m symmetry over a length scale of approximately 100 A.

Direct phasing of protein crystals with high solvent content

Abstract: An iterative transform method is proposed for solving the phase problem in protein crystallography. In each iteration, a weighted average electron-density map is constructed to define an estimated protein mask. Solvent flattening is then imposed through the hybrid input–output algorithm [Fienup (1982). Appl. Opt. 21, 2758–2769]. Starting from random initial phases, after thousands of iterations the mask evolves into the correct shape and the phases converge to the correct values with an average error of 30–40° for high-resolution data for several protein crystals with high solvent content. With the use of non-crystallographic symmetry, the method could potentially be extended to phase protein crystals with less than 50% solvent fraction. The new phasing algorithm can supplement and enhance the traditional refinement tools.

Uniqueness of the macromolecular crystallographic phase problem.

Abstract: Uniqueness of the phase problem in macromolecular crystallography, and its relationship to the case of single particle imaging, is considered. The crystallographic problem is characterized by a constraint ratio that depends only on the size and symmetry of the molecule and the unit cell. The results are used to evaluate the effect of various real-space constraints. The case of an unknown molecular envelope is considered in detail. The results indicate the quite wide circumstances under which ab initio phasing should be possible.

Unique atom hyper-kagome order in Na4Ir3O8 and in low-symmetry spinel modifications.

Abstract: Group-theoretical and thermodynamic methods of the Landau theory of phase transitions are used to investigate the hyper-kagome atomic order in structures of ordered spinels and a spinel-like Na4Ir3O8 crystal. The formation of an atom hyper-kagome sublattice in Na4Ir3O8 is described theoretically on the basis of the archetype (hypothetical parent structure/phase) concept. The archetype structure of Na4Ir3O8 has a spinel-like structure (space group Fd\bar 3m) and composition [Na1/2Ir3/2]16d[Na3/2]16cO32e4. The critical order parameter which induces hypothetical phase transition has been stated. It is shown that the derived structure of Na4Ir3O8 is formed as a result of the displacements of Na, Ir and O atoms, and ordering of Na, Ir and O atoms, ordering dxy, dxz, dyz orbitals as well. Ordering of all atoms takes place according to the type 1:3. Ir and Na atoms form an intriguing atom order: a network of corner-shared Ir triangles called a hyper-kagome lattice. The Ir atoms form nanoclusters which are named decagons. The existence of hyper-kagome lattices in six types of ordered spinel structures is predicted theoretically. The structure mechanisms of the formation of the predicted hyper-kagome atom order in some ordered spinel phases are established. For a number of cases typical diagrams of possible crystal phase states are built in the framework of the Landau theory of phase transitions. Thermodynamical conditions of hyper-kagome order formation are discussed by means of these diagrams. The proposed theory is in accordance with experimental data.

Diffuse multiple scattering

Abstract: A new form of diffraction lines has been identified, similar to Rutherford, Kikuchi and Kossel lines. This paper highlights some of the properties of these lines and shows how they can be used to eliminate the need for sample/source matching in Lonsdale's triple convergent line method in lattice-parameter determination.

Pairwise correlations in layered close-packed structures.

Abstract: Given a description of the stacking statistics of layered close-packed structures in the form of a hidden Markov model, analytical expressions are developed for the pairwise correlation functions between the layers. These may be calculated analytically as explicit functions of model parameters or the expressions may be used as a fast, accurate and efficient way to obtain numerical values. Several examples are presented, finding agreement with previous work as well as deriving new relations.

Interpretation of angular symmetries in electron nanodiffraction patterns from thin amorphous specimens.

Abstract: The interpretation of angular symmetries in electron nanodiffraction patterns from thin amorphous specimens is examined. It is found that in general there are odd symmetries in experimental electron nanodiffraction patterns. Using simulation, it is demonstrated that this effect can be attributed to dynamical scattering, rather than other divergences from the ideal experimental conditions such as probe-forming lens aberrations and camera noise. The departure of opposing diffracted intensities from Friedel's law in the phase grating formalism is calculated using a general structure factor for disordered materials. On the basis of this, a simple correction procedure is suggested to recover the kinematical angular symmetries, and thus readily interpretable information that reflects the symmetries of the original projected object. This correction is numerically tested using both the phase object and multislice calculations, and is demonstrated to fully recover all the kinematical diffracted symmetries from a simulated atomic model of a metallic glass.

Nuclear-weighted X-ray maximum entropy method - NXMEM.

Abstract: Subtle structural features such as disorder and anharmonic motion may be accurately characterized from nuclear density distributions (NDDs). As a viable alternative to neutron diffraction, this paper introduces a new approach named the nuclear-weighted X-ray maximum entropy method (NXMEM) for reconstructing pseudo NDDs. It calculates an electron-weighted nuclear density distribution (eNDD), exploiting that X-ray diffraction delivers data of superior quality, requires smaller sample volumes and has higher availability. NXMEM is tested on two widely different systems: PbTe and Ba8Ga16Sn30. The first compound, PbTe, possesses a deceptively simple crystal structure on the macroscopic level that is unable to account for its excellent thermoelectric properties. The key mechanism involves local distortions, and the capability of NXMEM to probe this intriguing feature is established with simulated powder diffraction data. In the second compound, Ba8Ga16Sn30, disorder among the Ba guest atoms is analysed with both experimental and simulated single-crystal diffraction data. In all cases, NXMEM outperforms the maximum entropy method by substantially enhancing the nuclear resolution. The induced improvements correlate with the amount of available data, rendering NXMEM especially powerful for powder and low-resolution single-crystal diffraction. The NXMEM procedure can be implemented in existing software and facilitates widespread characterization of disorder in functional materials.

Algorithm for systematic peak extraction from atomic pair distribution functions.

Abstract: The study presents an algorithm, ParSCAPE, for model-independent extraction of peak positions and intensities from atomic pair distribution functions (PDFs). It provides a statistically motivated method for determining parsimony of extracted peak models using the information-theoretic Akaike information criterion (AIC) applied to plausible models generated within an iterative framework of clustering and chi-square fitting. All parameters the algorithm uses are in principle known or estimable from experiment, though careful judgment must be applied when estimating the PDF baseline of nanostructured materials. ParSCAPE has been implemented in the Python program SrMise. Algorithm performance is examined on synchrotron X-ray PDFs of 16 bulk crystals and two nanoparticles using AIC-based multimodeling techniques, and particularly the impact of experimental uncertainties on extracted models. It is quite resistant to misidentification of spurious peaks coming from noise and termination effects, even in the absence of a constraining structural model. Structure solution from automatically extracted peaks using the Liga algorithm is demonstrated for 14 crystals and for C60. Special attention is given to the information content of the PDF, theory and practice of the AIC, as well as the algorithm's limitations.

Approximation of virus structure by icosahedral tilings.

Abstract: Viruses are remarkable examples of order at the nanoscale, exhibiting protein containers that in the vast majority of cases are organized with icosahedral symmetry. Janner used lattice theory to provide blueprints for the organization of material in viruses. An alternative approach is provided here in terms of icosahedral tilings, motivated by the fact that icosahedral symmetry is non-crystallographic in three dimensions. In particular, a numerical procedure is developed to approximate the capsid of icosahedral viruses by icosahedral tiles via projection of high-dimensional tiles based on the cut-and-project scheme for the construction of three-dimensional quasicrystals. The goodness of fit of our approximation is assessed using techniques related to the theory of polygonal approximation of curves. The approach is applied to a number of viral capsids and it is shown that detailed features of the capsid surface can indeed be satisfactorily described by icosahedral tilings. This work complements previous studies in which the geometry of the capsid is described by point sets generated as orbits of extensions of the icosahedral group, as such point sets are by construction related to the vertex sets of icosahedral tilings. The approximations of virus geometry derived here can serve as coarse-grained models of viral capsids as a basis for the study of virus assembly and structural transitions of viral capsids, and also provide a new perspective on the design of protein containers for nanotechnology applications.

Group‐theoretical analysis of aperiodic tilings from projections of higher‐dimensional lattices Bn

Abstract: A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group W(a)(B(n)) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup D(h) of W(B(n)) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A(3)), W(H(2)) × W(A(1)) and W(H(3)) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B(4) onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B(5) lattice is used to describe both fivefold and tenfold symmetries. The lattice B(6) can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

Powder diffraction in Bragg-Brentano geometry with straight linear detectors

Abstract: A common way of speeding up powder diffraction measurements is the use of one- or two-dimensional detectors. This usually goes hand in hand with worse resolution and asymmetric peak profiles. In this work the influence of a straight linear detector on the resolution function in the Bragg-Brentano focusing geometry is discussed. Because of the straight nature of most modern detectors geometrical defocusing occurs, which heavily influences the line shape of diffraction lines at low angles. An easy approach to limit the resolution-degrading effects is presented. The presented algorithm selects an adaptive range of channels of the linear detector at low angles, resulting in increased resolution. At higher angles the whole linear detector is used and the data collection remains fast. Using this algorithm a well behaved resolution function is obtained in the full angular range, whereas using the full linear detector the resolution function varies within one pattern, which hinders line-shape and Rietveld analysis.

Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests.

Abstract: Entanglements of two-dimensional honeycomb nets are constructed from free tilings of the hyperbolic plane (H2) on triply periodic minimal surfaces. The 2-periodic nets that comprise the structures are guaranteed by considering regular, rare free tilings in H2. This paper catalogues an array of entanglements that are both beautiful and challenging for current classification techniques, including examples that are realized in metal-organic materials. The compactification of these structures to the genus-3 torus is considered as a preliminary method for generating entanglements of finite θ-graphs, potentially useful for gaining insight into the entanglement of the periodic structure. This work builds on previous structural enumerations given in Periodic entanglement Parts I and II [Evans et al. (2013). Acta Cryst. A69, 241-261; Evans et al. (2013). Acta Cryst. A69, 262-275].

Structure factor for an icosahedral quasicrystal within a statistical approach.

Abstract: This paper describes a detailed derivation of a structural model for an icosahedral quasicrystal based on a primitive icosahedral tiling (three-dimensional Penrose tiling) within a statistical approach. The average unit cell concept, where all calculations are performed in three-dimensional physical space, is used as an alternative to higher-dimensional analysis. Comprehensive analytical derivation of the structure factor for a primitive icosahedral lattice with monoatomic decoration (atoms placed in the nodes of the lattice only) presents in detail the idea of the statistical approach to icosahedral quasicrystal structure modelling and confirms its full agreement with the higher-dimensional description. The arbitrary decoration scheme is also discussed. The complete structure-factor formula for arbitrarily decorated icosahedral tiling is derived and its correctness is proved. This paper shows in detail the concept of a statistical approach applied to the problem of icosahedral quasicrystal modelling.

Iterative projection algorithms in protein crystallography. II. Application.

Abstract: Iterative projection algorithms (IPAs) are a promising tool for protein crystallographic phase determination. Although related to traditional density-modification algorithms, IPAs have better convergence properties, and, as a result, can effectively overcome the phase problem given modest levels of structural redundancy. This is illustrated by applying IPAs to determine the electron densities of two protein crystals with fourfold non-crystallographic symmetry, starting with only the experimental diffraction amplitudes, a low-resolution molecular envelope and the position of the non-crystallographic axes. The algorithm returns electron densities that are sufficiently accurate for model building, allowing automated recovery of the known structures. This study indicates that IPAs should find routine application in protein crystallography, being capable of reconstructing electron densities starting with very little initial phase information.

Generalized Penrose tiling as a quasilattice for decagonal quasicrystal structure analysis.

Abstract: The generalized Penrose tiling is, in fact, an infinite set of decagonal tilings. It is constructed with the same rhombs (thick and thin) as the conventional Penrose tiling, but its long-range order depends on the so-called shift parameter (s ∈ 〈0; 1)). The structure factor is derived for the arbitrarily decorated generalized Penrose tiling within the average unit cell approach. The final formula works in physical space only and is directly dependent on the s parameter. It allows one to straightforwardly change the long-range order of the refined structure just by changing the s parameter and keeping the tile decoration unchanged. This gives a great advantage over the higher-dimensional method, where every change of the tiling (change in the s parameter) requires the structure model to be built from scratch, i.e. the fine division of the atomic surfaces has to be redone.

Celebrating the past, looking to the future.

Abstract: Copyright c © International Union of Crystallography Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr. For further information see http://journals.iucr.org/services/authorrights.html

Statistical tests against systematic errors in data sets based on the equality of residual means and variances from control samples: theory and applications

Abstract: Statistical tests are applied for the detection of systematic errors in data sets from least-squares refinements or other residual-based reconstruction processes. Samples of the residuals of the data are tested against the hypothesis that they belong to the same distribution. For this it is necessary that they show the same mean values and variances within the limits given by statistical fluctuations. When the samples differ significantly from each other, they are not from the same distribution within the limits set by the significance level. Therefore they cannot originate from a single Gaussian function in this case. It is shown that a significance cutoff results in exactly this case. Significance cutoffs are still frequently used in charge-density studies. The tests are applied to artificial data with and without systematic errors and to experimental data from the literature.

Partial order among the 14 Bravais types of lattices: basics and applications

Abstract: Neither International Tables for Crystallography (ITC) nor available crystallography textbooks state explicitly which of the 14 Bravais types of lattices are special cases of others, although ITC contains the information necessary to derive the result in two ways, considering either the symmetry or metric properties of the lattices. The first approach is presented here for the first time, the second has been given by Michael Klemm in 1982. Metric relations between conventional bases of special and general lattice types are tabulated and applied to continuous equi-translation phase transitions.

Solution of the phase problem at non-atomic resolution by the phantom derivative method.

Abstract: For a given unknown crystal structure (the target), n random structures, arbitrarily designed without any care for their chemical consistency and usually uncorrelated with the target, are sheltered in the same unit cell as the target structure and submitted to the same space-group symmetry. (These are called ancil structures.) The composite structures, whose electron densities are the sum of the target and of the ancil electron densities, are denoted derivatives. No observed diffraction amplitudes are available for them: in order to emphasize their unreal nature, the term phantom is added. The paper describes the theoretical basis by which the phantom derivative method may be used to phase the target structure. It may be guessed that 100-300 ancil structures may be sufficient for phasing a target structure, so that the phasing technique may be denoted as the multiple phantom derivative method. Ancil phases and amplitudes may be initially combined with observed target magnitudes to estimate amplitudes and phases of the corresponding phantom derivative. From them suitable algorithms allow one to obtain poor target phase estimates, which are often improved by combining the indications arising from each derivative. Probabilistic criteria are described to recognize the most reliable target phase estimates. The method is cyclic: the target phase estimates just obtained are used to improve amplitudes and phases of each derivative, which, in their turn, are employed to provide better target phase estimates. The method is a fully ab initio method, because it needs only the experimental data of the target structure. The term derivative is maintained with reference to SIR-MIR (single isomorphous replacement-multiple isomorphous replacement) techniques, even if its meaning is different: therefore the reader should think of the phantom derivative method more as a new method than as a variant of SIR-MIR techniques. The differences are much greater than the analogies. The paper also describes how phantom derivatives may be used for improving structure models obtained via other ab initio or non-ab initio techniques. The method is expected to be insensitive to the structural complexity of the target and to the target experimental data resolution, provided it is better than 4-6 A.