Phases Of Structure Factors Research Articles

Solving the crystallographic phase problem using dynamical scattering in electron diffraction

Solving crystal structures from kinematical X-ray or electron diffraction patterns of single crystals requires many more diffracted beams to be recorded than there are atoms in the structure, since the phases of the structure factors can only be retrieved from such data if the atoms can be resolved as sharply peaked objects. Here a method is presented by which the fact that multiple scattering encodes structure factor phases in the diffracted intensities is being used for solving the crystallographic phase problem. The retrieval of both amplitudes and phases of electron structure factors from diffraction patterns recorded with varying angle of incidence will be demonstrated. No assumption about the scattering potential itself is being made. In particular, the resolution in the diffraction data does not need to be sufficient to resolve atoms, making this method potentially interesting for electron crystallography of 2-dimensional protein crystals and other beam-sensitive complex structures.

Mask-based approach to phasing of single-particle diffraction data. II. Likelihood-based selection criteria.

A new type of mask-selection criterion is suggested for mask-based phasing. In this phasing approach, a large number of connected molecular masks are randomly generated. Structure-factor phases corresponding to a trial mask are accepted as an admissible solution of the phase problem if the mask satisfies some specified selection rules that are key to success. The admissible phase sets are aligned and averaged to give a preliminary solution of the phase problem. The new selection rule is based on the likelihood of the generated mask. It is defined as the probability of reproducing the observed structure-factor magnitudes by placing atoms randomly into the mask. While the result of the direct comparison of mask structure-factor magnitudes with observed ones using a correlation coefficient is highly dominated by a few very strong low-resolution reflections, a new method gives higher weight to relatively weak high-resolution reflections that allows them to be phased accurately. This mask-based phasing procedure with likelihood-based selection has been applied to simulated single-particle diffraction data of the photosystem II monomer. The phase set obtained resulted in a 16 Å resolution Fourier synthesis (more than 4000 reflections) with 98% correlation with the exact phase set and 69% correlation for about 2000 reflections in the highest resolution shell (20-16 Å). This work also addresses another essential problem of phasing methods, namely adequate estimation of the resolution achieved. A model-trapping analysis of the phase sets obtained by the mask-based phasing procedure suggests that the widely used `50% shell correlation' criterion may be too optimistic in some cases.

Three-beam convergent-beam electron diffraction for measuring crystallographic phases.

Under almost all circumstances, electron diffraction patterns contain information about the phases of structure factors, a consequence of the short wavelength of an electron and its strong Coulombic interaction with matter. However, extracting this information remains a challenge and no generic method exists. In this work, a set of simple analytical expressions is derived for the intensity distribution in convergent-beam electron diffraction (CBED) patterns recorded under three-beam conditions. It is shown that these expressions can be used to identify features in three-beam CBED patterns from which three-phase invariants can be extracted directly, without any iterative refinement processes. The octant, in which the three-phase invariant lies, can be determined simply by inspection of the indexed CBED patterns (i.e. the uncertainty of the phase measurement is ±22.5°). This approach is demonstrated with the experimental measurement of three-phase invariants in two simple test cases: centrosymmetric Si and non-centrosymmetric GaAs. This method may complement existing structure determination methods by providing direct measurements of three-phase invariants to replace 'guessed' invariants in ab initio phasing methods and hence provide more stringent constraints to the structure solution.

Electron density map evaluation functions for determining the quality of protein crystal structures

Finding methods to judge the quality of X-ray crystallographic information is an active research topic. The quality of electron density maps reconstructed by Fourier transform is always limited by the finite resolution, the amplitude/phase error and the completeness of diffraction data. At present, the R value and effective resolution are common ways of evaluating the quality of electron density maps. Unfortunately, the current evaluation methods are only dependent on diffraction amplitude, without any phase information. Advanced evaluation functions in real space are designed to estimate the electron density map quality. The electron density map definition evaluation function relies on the atomicity of the electron density distribution. We use the power spectrum electron density entropy in protein crystallography for the first time. These two functions include both structure factor amplitudes and phases via the Fourier transform of electron density map. We carry out tests on synthetic data sets of known structures, varying the resolution and error, and draw the quality curves of electron density maps with theoretical, noisy and experimental diffraction data by two evaluation functions at different resolutions. The curves reveal the optimum structure and resolution of proteins clearly. The work presented here offers new methods to evaluate the qualities of the electron density maps of proteins with slight differences, and brand new indicators to select the optimum diffraction resolution of protein structures.

Embedded Isoreticular Zeolites: Concept and Beyond

The structure solution, prediction, and targeted synthesis of a family of embedded isoreticular zeolites (EIZs) with expanding structural complexity, denoted the RHO family, were reported recently. Here, the naming and building rules of body-centered cubic EIZs with the lta cage as the lattice point are presented. The rearrangement of a pair of pau and d8r cages between two lta cages and its repetitive insertion, combined with the strong reflections and fragment methods, allows the creation of three other new zeolite families, designated the HPO, RHO(b), and KFI families. Among them, the KFI family is found to be the only EIZ family, on the basis of the similarity of structure factor amplitudes and phases of strong reflections, that is, structural coding, within its family members. The structural credibility of this family is confirmed by both local interatomic distances and T-T-T angle analyses. The existence of tetragonal EIZ families is also demonstrated. The overall results provide useful insights into the prediction of unprecedented EIZ families.

Stochastic and Deterministic Crystal Structure Solution Methods in GSAS-II: Monte Carlo/Simulated Annealing Versus Charge Flipping

Crystallographic studies frequently involve the determination of a previously unknown crystal structure; General Structure Analysis System (GSAS)-II provides two methods for this purpose. The Monte Carlo/simulated annealing method is fundamentally stochastic in nature; random trials are tested for suitability by comparing calculated structure factors with a suite of observed ones. In contrast, the charge flipping method may begin with a suite of random structure factor phases, but the subsequent mathematical steps are entirely deterministic even though they appear to display chaotic behavior. This paper will briefly describe these methods as implemented in GSAS-II, illustrating their use with examples.

Level crossing, spin structure factor and quantum phases of the frustrated spin-1/2 chain with first and second neighbor exchange

The spin-1/2 chain with isotropic Heisenberg exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed J1/J2 between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor S(q) that diverges at wave vector qm in QLRO(qm) phases with periodicity 2π/qm but remains finite in gapped phases. S(qm) is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of S(qm) are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO(π/2) phase between −1.2 < J1/|J2| < 0.45. Numerical results and field theory agree well for quantum critical points at small frustration J2 but disagree in the sector of weak exchange J1 between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.

How phase measurements can change X-ray crystallography

"From inorganic crystals to macromolecular crystals, structure determination with atomic resolution is based on diffraction techniques (electrons, X-rays and neutrons). However, since the coherent scattering cross-section for X-rays by atoms have intermediate values between those for electrons and neutrons, experimental measurements of the phases of structure factors are feasible only for X-rays. Unprecedented results [1-3] have show how phase measurements can reveal important information of the crystalline structure of doped crystals, which are inaccessible by other techniques: ""differently from any other method in X-ray crystallography based on structure refinement of intensity data, the presented method pinpoint a specific feature of the structure and directly prove its existence beyond of any reliability parameter or of goodness-of-fitting value."" Although there are instrumental and computational challenges, the method can be extended to protein crystallography for improving resolution of poorly solved features of the structures. As depicted in Fig. 1, the structure factor of a given reflection has contributions of distinct groups of atoms. Low resolution at one group can result in phase deviation of Fickle, without significantly affecting its amplitude. For instance, small differences in the symmetry of charge balance at group B can cause great difference in the phase angle of this group, which increases the phase angle of Fickle, Fig. 1 (b). On the other hand, reduction in the scattering amplitude of group E due to atomic disordering decreases the phase angle, Fig. 1 (c). In this presentation, we discuss the current understanding of this method, its perspectives, and importance of providing a tool for structural analysis of macromolecules that is able to go beyond the resolution achieved by the techniques actually in use."

Local atomic structure determination using focused electron beams

This talk will give an overview of methods for solving the atomic structure of nanostructured materials using focused electron beams. It will illustrate these methods with a range of applications, such as the determination of the atomic structure and stability of nanoparticle facets [1]; the local atomic structure of "chessboard' nanostructures in lithium-based titanate perovskites; and the measurement of local polarity, dopant concentration and atomic-scale morphology in semiconducting nanowire quantum wells. These methods take advantage of the fact that electron wavefields can be brought to a focal point smaller than an Ångström in diameter, enabling small volumes of matter to be probed and characterized. The wealth of information contained in the resulting diffraction patterns can be interrogated selectively to isolate and `image' specific structural information. Several methods using small focused electron beams will be described in this talk, including; (i) An approach for the determination of centrosymmetric structures from the direct observation of structure factor phases by inspection of features in convergent beam electron diffraction patterns [2]. The method can achieve high resolution from just a few phase observations and no intensity measurements or iterative refinements are required; (ii) Methods for the quantitative interpretation of the intensity in atomic resolution imaging and diffraction data for the measurement of local atomic and electronic structure; (iii) Pseudo-confocal scanning transmission electron microscopy methods for obtaining depth and chemical information which record the scattered intensity in a plane conjugate to the specimen (as opposed to the diffraction plane) [3].

Precession Electron Diffraction

The strong Coulombic interaction between a high energy electron and a thin crystal film gives rise to electron diffraction patterns encoded with information that is remarkably sensitive to the crystal potential. That exquisite sensitivity can be advantageous, for example in the determination of local symmetry and bonding, but can also be problematic in that in general the dynamical scattering inherent in electron diffraction prohibits the use of conventional crystallographic methods to recover structure factor phase information and solve unknown structures. One way to reduce this problem is to use precession electron diffraction (PED), introduced 20 years ago [1] as the electron analogue of Buerger's X-ray technique, in which the electron beam is first rocked in a hollow cone above the sample and then de-rocked below, the net effect of which is equivalent to precessing the sample about a stationary electron beam. PED is now used almost routinely as a starting point to solve crystal structures that cannot be solved for a variety of reasons using x-ray or neutron methods. In this keynote lecture we explore why the PED technique has been successful for structure determination, focussing on the PED geometry, the variation of intensities with precession angle and specimen thickness, and how this `mimics' kinematic behaviour, and the use of unconventional structure solution and refinement approaches [2]. New acquisition geometries will be discussed that rely on tilt series of PED patterns to yield a more complete 3D data set. The lecture will focus on how PED has been used also as a method for nanoscale orientation mapping [3], providing more information than conventional electron diffraction and a robust method with which to determine local crystallographic orientation. By scanning the beam, accurate orientation images can be derived from series of PED patterns and, by combining with tomographic methods, sub-volume orientation information is also available.

Coherent convergent-beam time-resolved X-ray diffraction.

The use of coherent X-ray lasers for structural biology allows the use of nanometre diameter X-ray beams with large beam divergence. Their application to the structure analysis of protein nanocrystals and single particles raises new challenges and opportunities. We discuss the form of these coherent convergent-beam (CCB) hard X-ray diffraction patterns and their potential use for time-resolved crystallography, normally achieved by Laue (polychromatic) diffraction, for which the monochromatic laser radiation of a free-electron X-ray laser is unsuitable. We discuss the possibility of obtaining single-shot, angle-integrated rocking curves from CCB patterns, and the dependence of the resulting patterns on the focused beam coordinate when the beam diameter is larger or smaller than a nanocrystal, or smaller than one unit cell. We show how structure factor phase information is provided at overlapping interfering orders and how a common phase origin between different shots may be obtained. Their use in refinement of the phase-sensitive intensity between overlapping orders is suggested.

Approaches to time-resolved diffraction using an XFEL.

We describe several schemes for time-resolved imaging of molecular motion using a free-electron laser (XFEL), in response to the many challenges and opportunities which XFEL radiation has created for accurate time-resolved measurement of structure. For pump-probe experiments using crystals, the problem of recording full Bragg reflections (not partials) in each shot arises. Two solutions, the use of the large bandwith which necesarily results from using attosecond pulses, and the use the coherent convergent beam mode are suggested. We also show that with attosecond recording times shorter than the temporal coherence time, Bragg reflections excited by different wavelengths from different reflections can interfere, providing structure factor phase information. For slower processes, a mixing jet sample-delivery device is described to allow snapshot solution scattering during molecular reactions on the microsecond scale. For optically excited membrane proteins, we suggest the use of the lipid cubic phase sample delivery device operating at atmospheric pressure. The use of two-color and split-and-delay schemes is suggested for improved accuracy in the Monte-Carlo method of serial femtosecond crystallography (SFX).

Analyzing structure factor phases in pure and doped single crystals by synchrotron X-ray Renninger scanning

X-ray multiple diffraction has been applied to study the substitutional incorporation of Mg2+ions into NSH crystals (nickel sulfate hexahydrate, NiSO4·6H2O). Intensity profiles provide information on invariant phases, while angular positions of the multiple diffractions allow accurate determination of lattice parameters. By increasing the atomic disordering only of O2−sites in model structures of doped NSH, the sense and magnitude of induced phase shifts match those necessary to justify the observed changes in the intensity profiles. Causes of disordering and lattice parameter variation are discussed. Although the amount of extra oxygen disordering is relatively large with respect to the small difference in the ionic radii of the metallic ions, this disordering is beyond the resolution power achievable by analyzing diffracted intensities of isolated reflections, such as in standard crystallographic techniques.

Applicability study of the structure-factor phase method for determining the polarity of binary semiconductors

The structure-factor phase method of convergent-beam electron diffraction (CBED) has been widely applied as an effective tool in determining the polarity of binary compound materials, for example, the typical sphalerite material, GaAs. However, its validity on other polar materials is still unknown. In this paper we extensively investigated its potential applicability onto 11 AB-type semiconductors by dynamical simulations of CBED. Two key factors during the simulation, the difference between A and B atomic numbers and the sample thickness, are discussed in detail. It was found that this method is efficient to determine the polarity for a sphalerite structure under certain conditions, and, reversely, limited to determine the polarity for a wurtzite structure even though it is very similar to the sphalerite structure.

Structural Determination of Ordered Porous Solids by Electron Crystallography

Knowing the structure of porous materials is essential for understanding their properties and exploiting them for applications. Electron crystallography has two main advantages compared to X‐ray diffraction for structure determination. Crystals too small or too complicated to be studied by X‐ray diffraction can be studied by electron crystallography. The crystallographic structure factor phase information, which is lost in X‐ray diffraction, can be obtained from high‐resolution transmission electron microscopy (HRTEM) images. Here, different electron microscopic techniques and their applications for structure determination of porous materials are reviewed. The recently developed automated diffraction tomography (ADT), the rotation electron diffraction (RED), and the through‐focus structure projection reconstruction (QFcous) methods make the structure determination by electron crystallography more feasible for non‐TEM experts and as efficient as that by X‐ray diffraction. How the various electron crystallographic methods are chosen are demonstrated and these methods used for solving different structural problems in porous materials. The benefits of combining electron crystallography and X‐ray diffraction for studying complex zeolite structures are also shown. A large number of examples are given to demonstrate the use of various electron crystallographic techniques for structure determination of zeolites, metal‐organic frameworks and ordered mesoporous materials. These electron crystallographic methods are general and can also be used for structural studies of other functional materials.

High-resolution noise substitution to measure overfitting and validate resolution in 3D structure determination by single particle electron cryomicroscopy

Three-dimensional (3D) structure determination by single particle electron cryomicroscopy (cryoEM) involves the calculation of an initial 3D model, followed by extensive iterative improvement of the orientation determination of the individual particle images and the resulting 3D map. Because there is much more noise than signal at high resolution in the images, this creates the possibility of noise reinforcement in the 3D map, which can give a false impression of the resolution attained. The balance between signal and noise in the final map at its limiting resolution depends on the image processing procedure and is not easily predicted. There is a growing awareness in the cryoEM community of how to avoid such over-fitting and over-estimation of resolution. Equally, there has been a reluctance to use the two principal methods of avoidance because they give lower resolution estimates, which some people believe are too pessimistic. Here we describe a simple test that is compatible with any image processing protocol. The test allows measurement of the amount of signal and the amount of noise from overfitting that is present in the final 3D map. We have applied the method to two different sets of cryoEM images of the enzyme beta-galactosidase using several image processing packages. Our procedure involves substituting the Fourier components of the initial particle image stack beyond a chosen resolution by either the Fourier components from an adjacent area of background, or by simple randomisation of the phases of the particle structure factors. This substituted noise thus has the same spectral power distribution as the original data. Comparison of the Fourier Shell Correlation (FSC) plots from the 3D map obtained using the experimental data with that from the same data with high-resolution noise (HR-noise) substituted allows an unambiguous measurement of the amount of overfitting and an accompanying resolution assessment. A simple formula can be used to calculate an unbiased FSC from the two curves, even when a substantial amount of overfitting is present. The approach is software independent. The user is therefore completely free to use any established method or novel combination of methods, provided the HR-noise test is carried out in parallel. Applying this procedure to cryoEM images of beta-galactosidase shows how overfitting varies greatly depending on the procedure, but in the best case shows no overfitting and a resolution of ~6 Å. (382 words)

Generation and applications of structure envelopes for porous metal–organic frameworks

The synthesis of polycrystalline, as opposed to single-crystalline, porous materials, such as zeolites and metal–organic frameworks (MOFs), is usually beneficial because the former have shorter synthesis times and higher yields. However, the structural determination of these materials using powder X-ray diffraction (PXRD) data is usually complicated. Recently, several methods for the structural investigation of zeolite polycrystalline materials have been developed, taking advantage of the structural characteristics of zeolites. Nevertheless, these techniques have rarely been applied in the structure determination of a MOF even though, with the electron-density contrast between the metal-containing units and pore regions, the construction of a structure envelope, the surface between high- and low-electron-density regions, should be straightforward for a MOF. Herein an example of such structure solution of MOFs based on PXRD data is presented. To start, a Patterson map was generated from powder diffraction intensities. From this map, structure factor phases for several of the strongest reflections were extracted and a structure envelope (SE) of a MOF was subsequently constructed. This envelope, together with all extracted reflection intensities, was used as input to theSUPERFLIPsoftware and a charge-flipping (CF) structure solution was performed. This structure solution method has been tested on the PXRD data of both activated (solvent removed from the pores;dmin= 0.78 Å) and as-synthesized (dmin= 1.20 Å) samples of HKUST-1. In both cases, our method has led to structure solutions. In fact, charge-flipping calculations using SE provided correct solutions in minutes (6 min for activated and 3 min for as-synthesized samples), while regular charge flipping or charge flipping with histogram matching calculation provided meaningful solutions only after several hours. To confirm the applicability of structure envelopes to low-symmetry MOFs, the structure of monoclinic PCN-200 has been solvedviaCF+SE calculations.

Atomic imaging and direct phase retrieval using anomalous surface x-ray diffraction

The application of multi-wavelength anomalous diffraction to thin films, interfaces and surface structures is presented. The method directly determines the amplitudes and phases of the complex surface structure factors from surface x-ray diffraction data, measured at three different energies around the absorption edge of one of the elements present in the film. Thereby, one is able to directly Fourier transform the data, which immediately provides meaningful and unambiguous electron-density distributions. These serve as a starting point for subsequent structural refinement. The robustness of the algorithm was evaluated on simulated data as a proof of principle. The experimental limitations and their effect on the method will be discussed as well as stability tests for the algorithm, such as the positions of the anomalous scatterers and the interfacial roughness. It will be shown that the method can be applied to real structures. The algorithm was tested on real data from a thin film of SrTiO3 grown on NdGaO3(110).

Using phases retrieved from two-dimensional projections to facilitate structure solution from X-ray powder diffraction data

A single-crystal charge-flipping algorithm has been applied to two-dimensional projections derived from X-ray powder diffraction data to retrieve structure-factor phases. These phases proved to be as reliable as those obtained from high-resolution transmission electron microscopy (HRTEM) images or from precession electron diffraction data. In particular, the stronger reflections tend to be correctly phased. The two-dimensional electron-density `images' obtained in this way show the same features as the corresponding HRTEM images, but with higher resolution. Application of the powder charge-flipping algorithm to the full three-dimensional powder diffraction data in conjunction with phases derived from several such (arbitrarily selected) projections was found to have a significant and beneficial effect on the structure solution. The approach was first developed using data collected on the complex zeolite TNU-9, and was then tested further using data for IM-5 and SSZ-74, two similarly complex zeolites. All three of these structures were originally solved by combining X-ray powder diffraction and electron microscopy data, because X-ray diffraction data alone were not sufficient. In all three cases, the phase information derived from two-dimensional subsets of the X-ray powder diffraction data resulted in a significant improvement in the electron-density maps generated by the powder charge-flipping algorithm. The inclusion of this phase information allowed all three structures to be determined from the X-ray data alone. This two-dimensional X-ray powder diffraction approach appears to offer a remarkably simple and powerful method for solving the structures of complex polycrystalline materials.

Structure study of the tri-continuous mesoporous silica IBN-9 by electron crystallography

High resolution electron microscopy (HRTEM) has unique advantages for structural determination of nano-sized porous materials compared to X-ray diffraction, because it provides the important structure factor phase information which is lost in diffraction. Here we demonstrate the structure determination of the first tri-continuous mesoporous silica IBN-9 by electron crystallography. IBN-9 has a hexagonal unit cell with the space group P6 3/ mcm and a = 88.4 Å, c = 84.3 Å. HRTEM images taken along three main directions, [0 0 1], [ 1 1 ¯ 0 ] and [1 0 0] were combined to reconstruct the 3D electrostatic potential map, from which the tri-continuous pore structure of IBN-9 was discovered. The different steps of structure determination of unknown mesoporous structures by electron crystallography are described in details. Similar procedures can also be applied for structure determination of other porous and nonporous crystalline materials.