Simulated Diffraction Data Research Articles

Quantum tomography of molecules using ultrafast electron diffraction.

We propose a quantum tomography (QT) approach to retrieve the temporally evolving reduced density matrix in electronic state basis, where the populations and coherence between the ground state and excited state are reconstructed from the ultrafast electron diffraction signal. In order to showcase the capability of the proposed QT approach, we simulate the nuclear wavepacket dynamics and ultrafast electron diffraction of photoexcited pyrrole molecules using the abinitio quantum chemical CASSCF method. From the simulated time-resolved diffraction data, we retrieve the evolving density matrix in a crude diabatic representation basis and reveal the symmetry of the excited pyrrole wavepacket. Our QT approach opens the route to make a quantum version of "molecular movie" that covers the electronic degree of freedom and equips ultrafast electron diffraction with the power to reveal the coherence between electronic states, relaxation, and dynamics of population transfer.

Accuracy Limits of Pair Distribution Function Analysis in Structural Characterization of Nanocrystalline Powders by X-ray Diffraction

We report the minimum errors of structural parameters, namely lattice parameter, crystallite size, and atomic displacement parameters, expected from Pair Distribution Function (PDF) analysis of nanocrystalline gold powders for the first time by a self-consistent computational methodology. Although PDF analysis has been increasingly used to characterize nanocrystalline powders by X-rays, the current literature includes no established error bounds to be expected from the resulting structural parameters. For accurate interpretation of X-ray diffraction data, these error bounds must be determined, and the obtained structural parameters must be cleared from them. Our novel methodology includes: 1) simulation of ideal powder diffraction experiments with the use of the Debye scattering equation, 2) pair distribution function analysis of the diffraction data with the Diffpy-CMI analysis software, and 3) determination of the errors from PDF analysis of the simulated diffraction data by comparing them with real-space analysis of spherical gold nanocrystals that are 30 nm size and smaller. Our results show that except for the lattice parameters and even with an ideal crystalline powder sample and ideal diffraction data, the extracted structural parameters from PDF analysis diverge from their true values for the studied nanopowder. These deviations are dependent on the average size of the nanocrystals and the energy of the X-rays selected for the diffraction experiments, where lower X-ray energies and small-sized nanocrystalline powders lead to greater errors.

Coarse-Grained Diffraction Template Matching Model to Retrieve Multiconformational Models for Biomolecule Structures from Noisy Diffraction Patterns.

Biomolecular imaging using X-ray free-electron lasers (XFELs) has been successfully applied to serial femtosecond crystallography. However, the application of single-particle analysis for structure determination using XFELs with 100 nm or smaller biomolecules has two practical problems: the incomplete diffraction data sets for reconstructing 3D assembled structures and the heterogeneous conformational states of samples. A new diffraction template matching method is thus presented here to retrieve a plausible 3D structural model based on single noisy target diffraction patterns, assuming candidate structures. Two concepts are introduced here: prompt candidate diffraction, generated by enhanced sampled coarse-grain (CG) candidate structures, and efficient molecular orientation searching for matching based on Bayesian optimization. A CG model-based diffraction-matching protocol is proposed that achieves a 100-fold speed increase compared to exhaustive diffraction matching using an all-atom model. The conditions that enable multiconformational analysis were also investigated by simulated diffraction data for various conformational states of chromatin and ribosomes. The proposed method can enable multiconformational analysis, with a structural resolution of at least 20 Å for 270-800 Å flexible biomolecules, in experimental single-particle structure analyses that employ XFELs.

Supercell refinement: a cautionary tale.

Theoretically, crystals with supercells exist at a unique crossroads where they can be considered as either a large unit cell with closely spaced reflections in reciprocal space or a higher dimensional superspace with a modulation that is commensurate with the supercell. In the latter case, the structure would be defined as an average structure with functions representing a modulation to determine the atomic location in 3D space. Here, a model protein structure and simulated diffraction data were used to investigate the possibility of solving a real incommensurately modulated protein crystal using a supercell approximation. In this way, the answer was known and the refinement method could be tested. Firstly, an average structure was solved by using the `main' reflections, which represent the subset of the reflections that belong to the subcell and in general are more intense than the `satellite' reflections. The average structure was then expanded to create a supercell and refined using all of the reflections. Surprisingly, the refined solution did not match the expected solution, even though the statistics were excellent. Interestingly, the corresponding superspace group had multiple 3D daughter supercell space groups as possibilities, and it was one of the alternate daughter space groups that the refinement locked in on. The lessons learned here will be applied to a real incommensurately modulated profilin-actin crystal that has the same superspace group.

Structure of a Superhydrophilic Surface: Wet Chemically Prepared Rutile-TiO2(110)(1 × 1).

Surface X-ray diffraction has been employed to quantitatively determine the geometric structure of an X-ray-induced superhydrophilic rutile-TiO2(110)(1 × 1) surface. A scatterer, assumed to be oxygen, is found at a distance of 1.90 ± 0.02 Å above the five-fold-coordinated surface Ti atom, indicating surface hydroxylation. Two more oxygen atoms, situated further from the substrate, are also included to achieve the optimal agreement between experimental and simulated diffraction data. It is concluded that these latter scatterers are from water molecules, surface-localized through hydrogen bonding. Comparing this interfacial structure with previous studies suggests that the superhydophilicity of titania is most likely to be a result of the depletion of surface carbon contamination coupled to extensive surface hydroxylation.

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.

Ab initio structure determination and quantitative disorder analysis on nanoparticles by electron diffraction tomography.

Nanoscaled porous materials such as zeolites have attracted substantial attention in industry due to their catalytic activity, and their performance in sorption and separation processes. In order to understand the properties of such materials, current research focuses increasingly on the determination of structural features beyond the averaged crystal structure. Small particle sizes, various types of disorder and intergrown structures render the description of structures at atomic level by standard crystallographic methods difficult. This paper reports the characterization of a strongly disordered zeolite structure, using a combination of electron exit-wave reconstruction, automated diffraction tomography (ADT), crystal disorder modelling and electron diffraction simulations. Zeolite beta was chosen for a proof-of-principle study of the techniques, because it consists of two different intergrown polymorphs that are built from identical layer types but with different stacking sequences. Imaging of the projected inner Coulomb potential of zeolite beta crystals shows the intergrowth of the polymorphs BEA and BEB. The structures of BEA as well as BEB could be extracted from one single ADT data set using direct methods. A ratio for BEA/BEB = 48:52 was determined by comparison of the reconstructed reciprocal space based on ADT data with simulated electron diffraction data for virtual nanocrystals, built with different ratios of BEA/BEB. In this way, it is demonstrated that this smart interplay of the above-mentioned techniques allows the elaboration of the real structures of functional materials in detail - even if they possess a severely disordered structure.

Coherence-resolution relationship in holographic and coherent diffractive imaging.

We study by numerical simulation how spatial coherence affects the reconstruction quality of images in coherent diffractive x-ray imaging. Using a conceptually simple, but computationally demanding approach, we have simulated diffraction data recorded under partial coherence, and then use the data for iterative reconstruction algorithms using a support constraint. By comparison of experimental regimes and parameters, we observe a significantly higher robustness against partially coherent illumination in the near-field compared to the far-field setting.

Evaluating the electron density model by applying an imaginary modification

A function has been proposed to evaluate the electron density model constructed by inverse Fourier transform using the observed structure amplitudes and trial phase set. The strategy of this function is applying an imaginary electron density modification to the model, and then measuring how well the calculated structure amplitudes of the modified model matches the expected structure amplitudes for the modified correct model. Since the correct model is not available in advance, a method has been developed to estimate the structure amplitudes of the modified correct model. With the estimated structure amplitudes of the modified correct model, the evaluation function can be calculated approximately. Limited tests on simulated diffraction data indicate that this evaluation function may be valid at the data resolution better than 2.5 Å.

Coexistence of multiple metastable polytypes in rhombohedral bismuth.

Derivative structural polytypes coexisting with the rhombohedral A7 structure of elemental bismuth (Bi) have been discovered at ambient condition, based on microstructure analyses of pure Bi samples treated under high pressure and high temperature conditions. Three structures with atomic positions close to those of the A7 structure have been identified through first-principles calculations, showing these polytypes energetically comparable to the A7 structure under ambient condition. Simulated diffraction data are in excellent agreement with the experimental observations. We argue that previously reported some variations of physical properties (e.g., density, electrical conductivity, and magnetism) in bismuth could be due to the formation of these polytypes. The coexistence of metastable derivative structural polytypes may be a widely occurring phenomenon in other elemental materials.

Reliability of twin-dependent triple junction distributions measured from a section plane

Numerous studies indicate polycrystalline triple junctions are independent microstructural features with distinct properties from their constituent grain boundaries. Despite the influence of triple junctions on material properties, it is impractical to characterize triple junctions on a large scale using current three-dimensional methods. This work demonstrates the ability to characterize twin-dependent triple junction distributions from a section plane by adopting a grain boundary plane stereology. The technique is validated through simulated distributions and simulated electron back-scatter diffraction (EBSD) data. Measures of validation and convergence are adopted to demonstrate the quantitative reliability of the technique as well as the convergence behavior of twin-dependent triple junction distributions. This technique expands the characterization power of EBSD and prepares the way for characterizing general triple junction distributions from a section plane.

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

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 Ba(8)Ga(16)Sn(30). 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, Ba(8)Ga(16)Sn(30), 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.

Approximations to the full description of stacking disorder in ice I for powder diffraction

Abstract Different descriptions of the stacking disorder of the so-called “cubic” phase “ice Ic” and stacking-faulted hexagonal ice Ih exist. We present an overview of the effect of different stacking disorder interaction ranges s from s = 2 to s = 4 after Jagodzinski on the (neutron) powder diffraction patterns of stacking-disordered ice I, which we propose to name ice Ich. We fit in a systematic approach simulated diffraction data of ice Ich for s up to 4 with a multi-peak approach. In this way we allow for estimating the relative proportion of cubic sequences in the stacking sequences by using readily accessible observables of a diffraction pattern.

Nuclear Enhanced MEM Used to Analyze Local Distortions in Lead Chalcogenides

We introduce a novel method for reconstructing nuclear density distributions (NDDs): Nuclear Enhanced X-ray Maximum Entropy Method (NEXMEM). NEXMEM offers an alternative route to experimental NDDs, exploiting the superior quality of synchrotron X-ray data compared to neutron data. The method was conceived to analyse local distortions in the thermoelectric lead chalcogenides, PbX (X = S, Se, Te). Thermoelectric materials are functional materials with the unique ability to interconvert heat and electricity, holding much promise for green energy solutions such as waste heat recovery. The extraordinary thermoelectric performance of binary lead chalcogenides has caused huge research activity, but the mechanisms governing their unexpected low thermal conductivity still remain a controversial topic. It has been proposed to result from giant anharmonic phonon scattering or from local fluctuating dipoles on the Pb site.[1,2] No macroscopic symmetry change are associated with these effects, rendering them invisible to conventional crystallographic techniques. For this reason PbX was until recently believed to adopt the ideal, undistorted rock-salt structure. In the present study, we investigate PbX using multi-temperature synchrotron powder X-ray diffraction data in combination with the maximum entropy method (MEM) and NEXMEM. In addition NEXMEM has been validated by testing against simulated powder diffraction data of PbTe with known displacements of Pb. The increased resolution of NEXMEM proved essential for resolving Pb-displacement of 0.2 Å in simulated data. The figure below shows Pb in the (100) plane for MEM, NEXMEM and the actual NDD of the test structure. Our findings outline the extent of disorder in lead chalcogenides, promoting our understanding of this class of high-performance thermoelectric materials. Furthermore we introduce NEXMEM which can be used for widespread characterization of subtle atomic features in crystals with unusual properties.

Breaking the structure complexity limits for powder based structure solution

There exist theoretical predictions trying to determine how much complex crystal structure can be solved from perfect powder diffraction data [1]. The theoretical limit for perfect synchrotron data is about 300 DOF (degree of freedom) while one of the current record solves only 42 DOF problem simplified by heavy atom presence [2]. We have tried to determine a realistic DOF limit based on perfect simulated powder diffraction data. For the simulation we have chosen structures from CSD with 1 single peptide molecule in asymmetric unit cell, 2-8 amino acids, 10-39 DOF. The parameters of the simulated powder diffractogram used were close to typical perfect measurement on ID31 of ESRF - wavelength 0.5 Å, range 0.5°-15°, step 0.002°, FWHM 0.01°. The structure solution tests were done by SA (simulated annealing) in DASH 3.2 software [3]. To speed up the computation we have used parallel processing obtained by MDASH extension. Influence of Mogul CSD based torsion angles bias on the calculation effectiveness was investigated as well. The results demonstrate the required number of SA steps depends exponentially on the problems DOF. This requires for problems close to 30 DOF about 10E+10 SA steps and years of single CPU computational time. The Mogul based bias can significantly help for compounds like peptides - e.g. for simulation based on compound CSD code AHAREH (4 peptides, DOF 24) the Mogul based calculation gives 50 times more often correct result than non-restricted SA run. We believe the 40 DOF structures can be solved routinely on 16-32 CPU clusters from perfect data not influenced by preferred orientation when the Mogul CSD torsion angles bias will be used. Without developing a more efficient algorithm than SA solution we do not see a way how to get really close to the 300 DOF theoretical limit. Acknowledgement: This work was supported by the Grant Agency of Czech Republic, Grant No. 106/14/03636S. Fig. caption: Dependence of required simulated annealing steps required to get one solution on DOF and the use of Mogul based bias.

Complete crystal structure of decafluorocyclohex-1-ene at 4.2 K from original neutron diffraction data

The crystal structure model of decafluorocyclohex-1-ene at 4.2 K derived from simulated powder diffraction data and solid-state energy minimization [Smrčok et al. (2013). Acta Cryst. B69, 395-404] is found to be incomplete. In this study it is completed by an additional alternative molecular orientation revealed from the difference density analysis and direct space search. The structure is refined by the derivative difference method in the rigid-body approximation leading to perfect agreement between observed and calculated neutron powder patterns.

Determination of the composition and thickness of semi-polar and non-polar III-nitride films and quantum wells using X-ray scattering

There is increasing interest in III-nitride films and multiple quantum well structures grown in non-polar or semi-polar orientations for application in light-emitting devices. We describe a method to obtain the compositions and the thicknesses of layers within III-nitride quantum well or superlattice structures grown in non-polar or semi-polar orientations, based on X-ray scattering. For each new crystallographic orientation considered, new axes were obtained and both the lengths and angles of these new axes calculated relative to the original conventional reference axes. These angles provide the coefficients of the matrix to transform the elastic constants published in the conventional setting (as used for polar c-plane oriented III-nitrides) into the appropriate new values. The new characteristic lengths and new elastic constants are then put into the general equation that relates the composition of a fully strained layer to the experimentally measured out-of-plane alloy d-spacing. Thus we have (a) determined the alloy composition from the difference between the experimentally measured alloy d-spacing and that of the substrate and (b) calculated the strained d-spacing for a given alloy composition for input to simple kinematical simulation software. In addition for quantum well structures the thickness ratio of well-to-barrier must be determined. Here we use the minima in the low angle reflectivity data. The repeat thickness and thus the thicknesses of the well and barrier layers, can be obtained from either the low or the high-angle data. We then cross-check by comparing the experimental and the simulated high-angle diffraction data. This method has been applied successfully to heteroepitaxial non-polar and semi-polar GaN/AlGaN and InGaN/GaN multiple quantum well structures and may also be used to find the composition of epilayers. The method works even in the presence of tilt between the superlattice and the GaN “template’, although in this case additional high-angle diffraction data at different settings must be collected.

Common arc method for diffraction pattern orientation

Very short pulses of X-ray free-electron lasers opened the way to obtaining diffraction signal from single particles beyond the radiation dose limit. For three-dimensional structure reconstruction many patterns are recorded in the object's unknown orientation. A method is described for the orientation of continuous diffraction patterns of non-periodic objects, utilizing intensity correlations in the curved intersections of the corresponding Ewald spheres, and hence named the common arc orientation method. The present implementation of the algorithm optionally takes into account Friedel's law, handles missing data and is capable of determining the point group of symmetric objects. Its performance is demonstrated on simulated diffraction data sets and verification of the results indicates a high orientation accuracy even at low signal levels. The common arc method fills a gap in the wide palette of orientation methods.

Far-field high-energy diffraction microscopy: a tool for intergranular orientation and strain analysis

The far-field high-energy diffraction microscopy technique is presented in the context of high-energy synchrotron x-ray diffraction. For each grain in an illuminated polycrystalline volume, the volume-averaged lattice orientations, lattice strain tensors, and centre-of-mass (COM) coordinates may be determined to a high degree of precision: better than 0.05°, 1 × 10−4, and 0.1 pixel, respectively. Because the full lattice strain tensors are available, corresponding mean stress tensors may be calculated unambiguously using single-crystal elastic moduli. A novel formulation for orientation indexing and cell refinement is introduced and demonstrated using two examples: first, sequential indexing and lattice refinement of a single-crystal ruby standard with known COM coordinates; and second, indexing and refinement of simulated diffraction data from an aggregate of 819 individual grains using several sample rotation ranges and including the influence of experimental uncertainties. The speed of acquisition and penetration depth achievable with high-energy (that is, >50 keV) x-rays make this technique ideal for studies of strain/stress evolution in situ, as well as for residual stress analysis.

Ab initiostructure solution by iterative phase-retrieval methods: performance tests on charge flipping and low-density elimination

Comprehensive tests on the density-modification methods charge flipping [Oszlányi & Sütő (2004).Acta Cryst.A60, 134–141] and low-density elimination [Shiono & Woolfson (1992).Acta Cryst.A48, 451–456] for solving crystal structures are performed on simulated diffraction data of periodic structures and quasicrystals. A novel model-independent figure of merit, which characterizes the reliability of the retrieved phase of each reflection, is introduced and tested. The results of the performance tests show that the quality of the phase retrieval highly depends on the presence or absence of an inversion center and on the algorithm used for solving the structure. Charge flipping has a higher success rate for solving structures, while low-density elimination leads to a higher accuracy in phase retrieval. The best results can be obtained by combining the two methods,i.e.by solving a structure with charge flipping followed by a few cycles of low-density elimination. It is shown that these additional cycles dramatically improve the phases not only of the weak reflections but also of the strong ones. The results can be improved further by averaging the results of several runs and by applying a correction term that compensates for a reduction of the structure-factor amplitudes by averaging of inconsistently observed reflections. It is further shown that in most cases the retrieved phases converge to the best solution obtainable with a given method.