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Results for “mathematical crystallography”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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31 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Deep Bayesian local crystallography

Abstract The advent of high-resolution electron and scanning probe microscopy imaging has opened the floodgates for acquiring atomically resolved images of bulk materials, 2D materials, and surfaces. This plethora of data contains an immense volume of information on materials structures, structural distortions, and physical functionalities. Harnessing this knowledge regarding local physical phenomena necessitates the development of the mathematical frameworks for extraction of relevant information. However, the analysis of atomically resolved images is often based on the adaptation of concepts from macroscopic physics, notably translational and point group symmetries and symmetry lowering phenomena. Here, we explore the bottom-up definition of structural units and symmetry in atomically resolved data using a Bayesian framework. We demonstrate the need for a Bayesian definition of symmetry using a simple toy model and demonstrate how this definition can be extended to the experimental data using deep learning networks in a Bayesian setting, namely rotationally invariant variational autoencoders.

36 MATERIALS SCIENCE↗
DOE OSTI2022

The Euler characteristic as a basis for teaching topology concepts to crystallographers

The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V – E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χ m ) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χ m has an elegant topological interpretation through the concept of orbifolds. Alternatively, χ m can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Inversion of Many-Beam Bragg Intensities for Phasing by Iterated Projections: Removal of Multiple Scattering Artifacts from Diffraction Data

An iterated projection algorithm (N-Phaser) is developed that reconstructs a scattering potential from N-beam multiple Bragg scattered intensities. The method may be used to eliminate multiple scattering artifacts from electron diffraction data, solving the phase problem and increasing the thicknesses of samples used in materials science, solid-state chemistry, and small molecule crystallography. For high-energy transmission electron diffraction, we show that the algorithm recovers accurate complex structure factors from a wide range of thicknesses, orientations, and relativistic beam energies, and does not require known thickness or atomic-resolution data if sufficient multiple scattering occurs. Furthermore, extensions to Cryo-electron microscopy and Micro-electron diffraction are suggested.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Growth functions of periodic space tessellations

This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.

Chemistry↗
DOE OSTI2023

Decahedra and Icosahedra Everywhere: The Anomalous Crystallization of Au and Other Metals at the Nanoscale

Throughout history, objects with fivefold symmetry have been a popular topic of interest for artists, philosophers, and scientists. This may be because fivefold symmetry is very conspicuous in nature. In the case of crystals, fivefold rotational symmetry is mathematically forbidden, and macroscopic crystals exhibiting fivefold symmetry have never been shown to exist. Nevertheless, in the nanoworld, nanoparticles are often found with decahedral and icosahedral shapes that have an overall pseudo–fivefold symmetry. These structures are observed at many length scales, from 1 nm up to 1 µm. In this review, several reasons for the stability of fivefold nanoparticles are discussed. Finally, these include the formation of twin boundaries, surface reconstruction faceting, and other factors.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Small-wedge synchrotron and serial XFEL datasets for Cysteinyl leukotriene GPCRs

Structural studies of challenging targets such as G protein-coupled receptors (GPCRs) have accelerated during the last several years due to the development of new approaches, including small-wedge and serial crystallography. Here, we describe the deposition of seven datasets consisting of X-ray diffraction images acquired from lipidic cubic phase (LCP) grown microcrystals of two human GPCRs, Cysteinyl leukotriene receptors 1 and 2 (CysLT 1 R and CysLT 2 R), in complex with various antagonists. Five datasets were collected using small-wedge synchrotron crystallography (SWSX) at the European Synchrotron Radiation Facility with multiple crystals under cryo-conditions. Two datasets were collected using X-ray free electron laser (XFEL) serial femtosecond crystallography (SFX) at the Linac Coherent Light Source, with microcrystals delivered at room temperature into the beam within LCP matrix by a viscous media microextrusion injector. All seven datasets have been deposited in the open-access databases Zenodo and CXIDB. Here, we describe sample preparation and annotate crystallization conditions for each partial and full datasets. We also document full processing pipelines and provide wrapper scripts for SWSX and SFX data processing. A Correction to this paper has been published: https://doi.org/10.1038/s41597-020-00759-w

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Structure analysis of the telomere resolvase from the Lyme disease spirochete Borrelia garinii reveals functional divergence of its C-terminal domain

Borrelia spirochetes are the causative agents of Lyme disease and relapsing fever, two of the most common tick-borne illnesses. A characteristic feature of these spirochetes is their highly segmented genomes which consists of a linear chromosome and a mixture of up to approximately 24 linear and circular extrachromosomal plasmids. The complexity of this genomic arrangement requires multiple strategies for efficient replication and partitioning during cell division, including the generation of hairpin ends found on linear replicons mediated by the essential enzyme ResT, a telomere resolvase. Using an integrative structural biology approach employing advanced modelling, circular dichroism, X-ray crystallography and small-angle X-ray scattering, we have generated high resolution structural data on ResT from B. garinii. Our data provides the first high-resolution structures of ResT from Borrelia spirochetes and revealed active site positioning in the catalytic domain. We also demonstrate that the C-terminal domain of ResT is required for both transesterification steps of telomere resolution, and is a requirement for DNA binding, distinguishing ResT from other telomere resolvases from phage and bacteria. These results advance our understanding of the molecular function of this essential enzyme involved in genome maintenance in Borrelia pathogens.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2025

Robust error calibration for serial crystallography

Serial crystallography is an important technique with unique abilities to resolve enzymatic transition states, minimize radiation damage to sensitive metalloenzymes and perform de novo structure determination from micrometre-sized crystals. This technique requires the merging of data from thousands of crystals, making manual identification of errant crystals unfeasible. cctbx.xfel.merge uses filtering to remove problematic data. However, this process is imperfect, and data reduction must be robust to outliers. We add robustness to cctbx.xfel.merge at the step of uncertainty determination for reflection intensities. This step is a critical point for robustness because it is the first step where the data sets are considered as a whole, as opposed to individual lattices. Robustness is conferred by reformulating the error-calibration procedure to have fewer and less stringent statistical assumptions and incorporating the ability to down-weight low-quality lattices. We then apply this method to five macromolecular XFEL data sets and observe the improvements to each. The appropriateness of the intensity uncertainties is demonstrated through internal consistency. This is performed through theoretical CC 1/2 and I /σ relationships and by weighted second moments, which use Wilson's prior to connect intensity uncertainties with their expected distribution. This work presents new mathematical tools to analyze intensity statistics and demonstrates their effectiveness through the often underappreciated process of uncertainty analysis.

Mittan-Moreau, David W.↗
DOE OSTI2024

Towards Autonomous Experiments by Connecting High Performance Microscopy with High Performance Computing

The digitization of controls, data, and analysis in microscopy is bringing the idea of autonomous microscopes closer to reality than ever before. Automated transmission electron microscopy (TEM) is already fairly routine for some experiments the only require simple repetitive tasks such as imaging biological macromolecules for single particle cryoEM [1], tilt series for electron tomography [2], and movies for crystallography [3]. The vast majority of TEM experiments are conducted completely by human operators who choose the regions of interest, optimize experimental parameters, and make decisions about data quality visually during an experiment. The field is still a long way from having completely autonomous TEMs that can adapt to sample difficulties and tune experimental parameters based on data quality and desired experimental outcomes. Part of the issue is the lack of capability for feeding information learned from on-line, live data analysis back into the on-going experiment [4]. Furthermore, this presentation will discuss current capabilities for large scale data reduction and analysis using high performance computing (i.e. supercomputing) and progress towards developing a true feed-back loop that places data analysis and theory in the experimental loop.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A method for crystallographic mapping of an alpha-beta titanium alloy with nanometre resolution using scanning precession electron diffraction and open-source software libraries

An approach for the crystallographic mapping of two-phase alloys on the nanoscale using a combination of scanned precession electron diffraction and open-source python libraries is introduced in this paper. This method is demonstrated using the example of a two-phase α/β titanium alloy. The data were recorded using a direct electron detector to collect the patterns, and recently developed algorithms to perform automated indexing and analyse the crystallography from the results. Very high-quality mapping is achieved at a 3 nm step size. The results show the expected Burgers orientation relationships between the α laths and β matrix, as well as the expected misorientations between α laths. A minor issue was found that one area was affected by 180° ambiguities in indexing occur due to this area being aligned too close to a zone axis of the α with twofold projection symmetry (not present in 3D) in the zero-order Laue Zone, and this should be avoided in data acquisition in the future. Nevertheless, this study demonstrates a good workflow for the analysis of nanocrystalline two- or multi-phase materials, which will be of widespread use in analysing two-phase titanium and other systems and how they evolve as a function of thermomechanical treatments.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Combining Reactive Quantum-Mechanical Molecular-Dynamics Simulations with Mutagenesis, Crystallography, and Enzyme Kinetics to Reveal Plausible Steps of Isocyanide Hydratase Catalysis

A complete understanding of enzyme mechanisms requires atomistic details of chemical reactions. Quantum-based molecular dynamics simulations (QMD) are a potential source of this information, but trade-offs between accuracy and computational cost have limited their use. We previously developed extended Lagrangian Born–Oppenheimer molecular dynamics (XL-BOMD) methods that leverage a negligible compromise in accuracy to substantially decrease the cost of QMD simulations. Here, we develop a reactive QMD approach using the latest XL-BOMD formulation, which enables efficient simulations of highly reactive systems, and use it to investigate mechanisms of intermediate formation in isocyanide hydratase (ICH) catalysis. In QMD simulations, molecular analogs of ICH active site residues reacted with para-nitrophenyl isocyanide, forming a thioimidate. Analysis of simulated atomic configurational and charge dynamics revealed a pathway where protonation of the isocyanide carbon occurs prior to thioimidate formation and suggested a possible role of Asp17 as a proton donor in the early phase of ICH catalysis. To test whether the pathway seen using the reactive QMD approach might be relevant to ICH catalysis, we performed X-ray crystallography and pre-steady-state enzyme kinetics studies of wild-type and D17N mutant ICH. Both the structure and kinetics are sensitive to the D17N mutation in a manner that is consistent with the order of the reaction steps seen in the simulations. Mobile protons play essential roles in many enzymes, yet they are difficult to observe experimentally, making the ordering of proton-dependent steps ambiguous in many enzyme mechanisms. The ability to directly simulate model reactions for the design of experiments that provide information about enzyme mechanisms involving mobile protons demonstrates the significance of our reactive QMD approach and motivates further biological applications.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Integrated edge-to-exascale workflow for real-time steering in neutron scattering experiments

We introduce a computational framework that integrates artificial intelligence (AI), machine learning, and high-performance computing to enable real-time steering of neutron scattering experiments using an edge-to-exascale workflow. Focusing on time-of-flight neutron event data at the Spallation Neutron Source, our approach combines temporal processing of four-dimensional neutron event data with predictive modeling for multidimensional crystallography. At the core of this workflow is the Temporal Fusion Transformer model, which provides voxel-level precision in predicting 3D neutron scattering patterns. The system incorporates edge computing for rapid data preprocessing and exascale computing via the Frontier supercomputer for large-scale AI model training, enabling adaptive, data-driven decisions during experiments. This framework optimizes neutron beam time, improves experimental accuracy, and lays the foundation for automation in neutron scattering. Although real-time experiment steering is still in the proof-of-concept stage, the demonstrated potential of this system offers a substantial reduction in data processing time from hours to minutes via distributed training, and significant improvements in model accuracy, setting the stage for widespread adoption across neutron scattering facilities and more efficient exploration of complex material systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

X-ray thermal diffuse scattering as a texture-robust temperature diagnostic for dynamically compressed solids

We present a model of x-ray thermal diffuse scattering (TDS) from a cubic polycrystal with an arbitrary crystallographic texture, based on the classic approach of Warren [B. E. Warren, Acta Crystallogr. 6, 803 (1953)]. We compare the predictions of our model with femtosecond x-ray diffraction patterns gathered from ambient and dynamically compressed rolled copper foils obtained at the High Energy Density instrument of the European X-Ray Free-Electron Laser facility and find that the texture-aware TDS model yields more accurate results than does the conventional powder model owed to Warren. Nevertheless, we further show: with sufficient angular detector coverage, the TDS signal is largely unchanged by sample orientation and in all cases strongly resembles the signal from a perfectly random powder; shot-to-shot fluctuations in the TDS signal resulting from grain-sampling statistics are at the percent level, in stark contrast to the fluctuations in the Bragg-peak intensities (which are over an order of magnitude greater); and TDS is largely unchanged even following texture evolution caused by compression-induced plastic deformation. We conclude that TDS is robust against texture variation, making it a flexible temperature diagnostic applicable just as well to off-the-shelf commercial foils as to ideal powders.

Crystal lattices↗