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All sources NASA NTRS DOE OSTI

DOE OSTI2016

Scalable shear-exfoliation of high-quality phosphorene nanoflakes with reliable electrochemical cycleability in nano batteries

Atomically thin black phosphorus (called phosphorene) holds great promise as an alternative to graphene and other two-dimensional transition-metal dichalcogenides as an anode material for lithium-ion batteries (LIBs). But, bulk black phosphorus (BP) suffers from rapid capacity fading and poor rechargeable performance. This work reports for the first time the use of in situ transmission electron microscopy (TEM) to construct nanoscale phosphorene LIBs. This enables direct visualization of the mechanisms underlying capacity fading in thick multilayer phosphorene through real-time capture of delithiation-induced structural decomposition, which serves to reduce electrical conductivity thus causing irreversibility of the lithiated phases. Furthermore, we demonstrate that few-layer-thick phosphorene successfully circumvents the structural decomposition and holds superior structural restorability, even when subject to multi-cycle lithiation/delithiation processes and concomitant huge volume expansion. This finding provides breakthrough insights into thickness-dependent lithium diffusion kinetics in phosphorene. More importantly, a scalable liquid-phase shear exfoliation route has been developed to produce high-quality ultrathin phosphorene using simple means such as a high-speed shear mixer or even a household kitchen blender with the shear rate threshold of ~1.25 × 10 4 s -1 . Our results reported here will pave the way for industrial-scale applications of rechargeable phosphorene LIBs.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

MultiTaskDeltaNet: change detection-based image segmentation for operando ETEM with application to carbon gasification kinetics

Transforming in situ transmission electron microscopy (TEM) imaging into a tool for spatially-resolved operando characterization of solid-state reactions requires automated, high-precision semantic segmentation of dynamically evolving features. However, traditional deep learning methods for semantic segmentation often face limitations due to the scarcity of labeled data, visually ambiguous features of interest, and scenarios involving small objects. To tackle these challenges, we introduce MultiTaskDeltaNet (MTDN), a novel deep learning architecture that creatively reconceptualizes the segmentation task as a change detection problem. By implementing a unique Siamese network with a U-Net backbone and using paired images to capture feature changes, MTDN effectively leverages minimal data to produce high-quality segmentations. Furthermore, MTDN utilizes a multi-task learning strategy to exploit correlations between physical features of interest. In an evaluation using data from in situ environmental TEM (ETEM) videos of filamentous carbon gasification, MTDN demonstrated a significant advantage over conventional segmentation models, particularly in accurately delineating fine structural features. Notably, MTDN achieved a 10.22% performance improvement over conventional segmentation models in predicting small and visually ambiguous physical features. This work bridges key gaps between deep learning and practical TEM image analysis, advancing automated characterization of nanomaterials in complex experimental settings.

08 HYDROGEN↗

DOE OSTI2021

Measurement and analysis of GPU-accelerated applications with HPCToolkit

To address the challenge of performance analysis on the US DOE’s forthcoming exascale supercomputers, Rice University has been extending its HPCToolkit performance tools to support measurement and analysis of GPU-accelerated applications. To help developers understand the performance of accelerated applications as a whole, HPCToolkit’s measurement and analysis tools attribute metrics to calling contexts that span both CPUs and GPUs. To measure GPU-accelerated applications efficiently, HPCToolkit employs a novel wait-free data structure to coordinate monitoring and attribution of GPU performance. To help developers understand the performance of complex GPU code generated from high-level programming models, HPCToolkit constructs sophisticated approximations of call path profiles for GPU computations. To support fine-grained analysis and tuning, HPCToolkit uses PC sampling and instrumentation to measure and attribute GPU performance metrics to source lines, loops, and inlined code. To supplement fine-grained measurements, HPCToolkit can measure GPU kernel executions using hardware performance counters. To provide a view of how an execution evolves over time, HPCToolkit can collect, analyze, and visualize call path traces within and across nodes. Finally, on NVIDIA GPUs, HPCToolkit can derive and attribute a collection of useful performance metrics based on measurements using GPU PC samples. Here, we illustrate HPCToolkit’s new capabilities for analyzing GPU-accelerated applications with several codes developed as part of the Exascale Computing Project.

97 MATHEMATICS AND COMPUTING↗

NASA NTRSDate not supplied

ExMC Systems Engineering Developments

The Exploration Medical Capability (ExMC) Element within the Human Research Program (HRP) applies systems engineering (SE) principles along with the use of Model-Based Systems Engineering (MBSE) tools to identify and communicate the requirements for medical and crew health and performance (CHP) systems. In the past fiscal year, the MBSE approach sought to advance the digital engineering toolset for medical and CHP system representation. These digital artifacts provide enhanced views of the relationships among requirements, standards, functions, and capabilities, to name a few, that is best suited for a user’s objectives. The MBSE tools and SE practices were applied to the development of the revised Earth Independent Medical Operations (EIMO) medical system and the Artemis III and IV CHP System models. In addition, a System of Systems concept was integrated into the EIMO model to facilitate the identification of system interfaces that interact with the medical system. Finally, the ExMC SE team has initiated several efforts to bring operationally relevant digital engineering practices to the Human Health and Performance Directorate (HHPD). This included the development of a pilot program within the directorate to help foster utilization of tools such as MagicDraw for system modeling and Power BI for visualizing extracted data in an easily accessible dashboard format. Additionally, with the increase in complexity of the integration effort for Artemis missions, ExMC has endeavored to bring digital engineering strategies to potentially increase efficiency in review processes. This talk will provide a high-level overview of the ExMC SE team accomplishments since the last Investigators’ Workshop, an introduction to upcoming SE talks, and the ongoing systems engineering work.

systems engineering↗

DOE OSTI2020

Multimodal Nanoscale Tomographic Imaging for Battery Electrodes

Accurate representations of the 3D structure within a lithium-ion battery are key to understanding performance limitations. However, obtaining exact reconstructions of electrodes, where the active particles, the carbon black and polymeric binder domain, and the pore space are visualized is challenging. In this work, it is shown that multimodal imaging can be used to overcome this challenge. High-resolution ptychographic X-ray computed tomography are combined with lower resolution but higher contrast transmission X-ray tomographic microscopy to obtain 3D reconstructions of pristine and cycled graphite-silicon composite electrodes. This cross-correlation enables quantitative analysis of the surface of active particles, including the heterogeneity of carbon-black and binder domain and solid-electrolyte interphase coverage. Capturing the active particles as well as the carbon black-binder domain allows using these segmented structures for electrochemical simulations to highlight the influence of the particle embedding on local state of charge heterogeneities.

25 ENERGY STORAGE↗

DOE OSTI2017

On the transcritical mixing of fuels at diesel engine conditions

Whilst the physics of both classical evaporation and supercritical fluid mixing are reasonably well characterized and understood in isolation, little is known about the transition from one to the other in the context of liquid fuel systems. The lack of experimental data for microscopic droplets at realistic operating conditions impedes the development of phenomenological and numerical models. To address this issue we performed systematic measurements using high-speed long-distance microscopy, for three single-component fuels (n-heptane, n-dodecane, n-hexadecane), into gas at elevated temperatures (700–1200 K) and pressures (2–11 MPa). We describe these high-speed visualizations and the time evolution of the transition from liquid droplet to fuel vapour at the microscopic level. The measurements show that the classical atomization and vaporisation processes do shift to one where surface tension forces diminish with increasing pressure and temperature, but the transition to diffusive mixing does not occur instantaneously when the fuel enters the chamber. Rather, subcritical liquid structures exhibit surface tension in the near-nozzle region and then, after time surrounded by the hot ambient gas and fuel vapour, undergo a transition to a dense miscible fluid. Although there was clear evidence of surface tension and primary atomization for n-dodecane and n-hexadecane for a period of time at all the above conditions, n-heptane appeared to produce a supercritical fluid from the nozzle outlet when injected at the most elevated conditions (1200 K, 10 MPa). This demonstrates that the time taken by a droplet to transition to diffusive mixing depends on the pressure and temperature of the gas surrounding the droplet as well as the fuel properties. We summarise our observations into a phenomenological model which describes the morphological evolution and transition of microscopic droplets from classical evaporation through a transitional mixing regime and towards diffusive mixing, as a function of operating conditions. We provide criteria for these regime transitions as reduced pressure–temperature correlations, revealing the conditions where transcritical mixing is important to diesel fuel spray mixing.

02 PETROLEUM↗

DOE OSTI2016

SU-G-IeP4-09: Method of Human Eye Aberration Measurement Using Plenoptic Camera Over Large Field of View

Purpose: The measurement based on Shack-Hartmann wave-front sensor(WFS), obtaining both the high and low order wave-front aberrations simultaneously and accurately, has been applied in the detection of human eyes aberration in recent years. However, Its application is limited by the small field of view (FOV), slight eye movement leads the optical bacon image exceeds the lenslet array which result in uncertain detection error. To overcome difficulties of precise eye location, the capacity of detecting eye wave-front aberration over FOV much larger than simply a single conjugate Hartmann WFS accurately and simultaneously is demanded. Methods: Plenoptic camera’s lenslet array subdivides the aperture light-field in spatial frequency domain, capture the 4-D light-field information. Data recorded by plenoptic cameras can be used to extract the wave-front phases associated to the eyes aberration. The corresponding theoretical model and simulation system is built up in this article to discuss wave-front measurement performance when utilizing plenoptic camera as wave-front sensor. Results: The simulation results indicate that the plenoptic wave-front method can obtain both the high and low order eyes wave-front aberration with the same accuracy as conventional system in single visual angle detectionand over FOV much larger than simply a single conjugate Hartmann systems. Meanwhile, simulation results show that detection of eye aberrations wave-front in different visual angle can be achieved effectively and simultaneously by plenoptic method, by both point and extended optical beacon from the eye. Conclusion: Plenoptic wave-front method possesses the feasibility in eye aberrations wave-front detection. With larger FOV, the method can effectively reduce the detection error brought by imprecise eye location and simplify the eye aberrations wave-front detection system comparing with which based on Shack-Hartmann WFS. Unique advantage of the plenoptic method lies in obtaining wave-front in different visual angle simultaneously, which provides an approach in building up 3-D model of eye refractor tomographically. Funded by the key Laboratory of High Power Laser and Physics, CAS Research Project of National University of Defense Technology No. JC13-07-01; National Natural Science Foundation of China No. 61205144.

60 APPLIED LIFE SCIENCES↗

NASA NTRS1987

Integration of altitude and airspeed information into a primary flight display via moving-tape formats

A ground-based aircraft simulation study was conducted to determine the effect on pilot performance of replacing the electromechanical altimeter and airspeed indicators with electronically generated representations integrated into the primary flight display via moving-tape (linear moving scale) formats. Several key factors relating to moving-tape formats were examined during the study: tape centering, secondary (trend) information, and tape orientation. The factor of centering refers to whether the tape was centered about the actual airspeed or altitude or about some defined reference value. Tape orientation refers to whether the values represented are arranged in either descending or ascending order. Six pilots participated in this study, with each subject performing 18 runs along a single, known flight profile. Subjective results indicated that the moving-tape formats were generally better than that of the conventional instruments. They also indicated that an actual-centered fixed pointer was preferred to a reference-centered pointer. Performance data for a visual secondary task showed that formats not containing trend information produced better performance; however, no difference was noted in airspeed tracking or altitude tracking performance. Regarding tape orientation, subjective comments indicated that there was lower work load and better performance when the airspeed tape had the high numbers at the top.

Abbott, Terence S.↗

NASA NTRS1998

Luminescent Paints Used for Rotating Temperature and Pressure Measurements on Scale-Model High-Bypass-Ratio Fans

NASA Lewis Research Center is a leader in the application of temperature- and pressuresensitive paints (TSP and PSP) in rotating environments. Tests were recently completed on several scale model, high-bypass-ratio turbofans in Lewis' 9- by 15-Foot Low-Speed Wind Tunnel. Two of the test objectives were to determine the aerodynamic and acoustic performance of the fan designs. Using TSP and PSP, researchers successfully achieved fullfield aerodynamic loading profiles. The visualized loading profiles may help researchers identify factors contributing to the fans' performance and to the acoustic characteristics associated with the flow physics on the surface of the blades.

Bencic, Timothy J.↗

NASA NTRS1987

Adhesive evaluation of water-soluble LARC-TPI

The water-soluble polyimide, identified as TPI(MTC)/H2O, was evaluated as a high temperature thermoplastic adhesive for bonding Ti-6Al-4V and comparing those results primarily with results reported in earlier work with the polyamic-acid/diglyme material. The lap shear strength test was the primary test performed to evaluate the adhesive before (controls) and after thermal exposure in air at 204 C for up to 5000 hours and after a 72 hour water-boil exposure. Lap shear strengths were determined at RT, 177, 204, and 232 C. The adhesive was also characterized after fracture by determining the glass transition temperature as well as defining the mode of failure by visual observation. In general, the results indicate that the TPI(MTC)/H2O retains high lap shear strengths after thermal exposure but had reduced strengths after the water-boil exposure. All failures were cohesive. The TPI(MTC)/H2O compared very well with previous data reported for the standard polyamic-acid/diglyme LARC-TPI results, and therefore, shows promise as a water-soluble adhesive for use in various applications.

Progar, Donald J.↗

NASA NTRS1988

Adhesive evaluation of water-soluble LARC-TPI

Progar, D. J.↗

DOE OSTI2021

SIR Glass Test Vehicle Designed to Characterize Process Materials

The exposure to harsh environments can result in reliability issues on highly dense electronic packages and circuit assemblies. One area of concern is electrochemical failure mechanisms caused by electrolytic corrosion, electrochemical migration (dendrites), and leakage currents under humid conditions. These no-fault and no-trouble found failure points are challenging to detect and to perform process control during the assembly process. The focus of this research centers on metallizing glass surface insulation resistance test vehicles to characterize materials on electronic packages and components that exhibit the highest risk of failure due to material interactions and process residues. Glass test vehicles provide a visual representation of materials when they are exposed to moisture content in the air. Harsh environments can cause material interactions resulting in morphology changes, opening, and release of active components. A transparent test vehicle is highly functional for environmental stress testing under temperature, humidity, and bias. The designed experiment will evaluate the effects of material and process residues that can undergo changes in behavior during application, which leads to local failures.

Capen, Bill↗

DOE OSTI2015

Diverging polygon-based modeling (DPBM) of concentrated solar flux distributions

This paper presents an efficient and robust methodology for modeling concentrated solar flux distributions. Compared to ray tracing methods, which provide high accuracy but can be computationally intensive, this approach makes a number of simplifying assumptions in order to reduce complexity by modeling incident and reflected flux as a series of simple geometric diverging polygons, then applying shading and blocking effects. A reduction in processing time (as compared to ray tracing) allows for evaluating and visualizing numerous combinations of engineering and operational variables (easily exceeding 106 unique iterations) to ascertain instantaneous, transient, and annual system performance. The method is demonstrated on a linear Fresnel reflector array and a number of variable iteration examples presented. While some precision is sacrificed for computational speed, flux distributions were compared to ray tracing (SolTrace) and average concentration ratio generally found to agree within ~3%. Furthermore, this method presents a quick and very flexible coarse adjust method for concentrated solar power (CSP) field design, and can be used to both rapidly gain an understanding of system performance as well as to narrow variable constraint windows for follow-on high accuracy system optimization.

14 SOLAR ENERGY↗

DOE OSTI2022

The discovery of a radio galaxy of at least 5 Mpc

Context. Giant radio galaxies (GRGs, or colloquially ‘giants’) are the Universe’s largest structures generated by individual galaxies. They comprise synchrotron-radiating active galactic nucleus ejecta and attain cosmological (megaparsec-scale) lengths. However, the main mechanisms that drive their exceptional growth remain poorly understood. Aims. To deduce the main mechanisms that drive a phenomenon, it is usually instructive to study extreme examples. If there exist host galaxy characteristics that are an important cause for GRG growth, then the hosts of the largest GRGs are likely to possess them. Similarly, if there exist particular large-scale environments that are highly conducive to GRG growth, then the largest GRGs are likely to reside in them. For these reasons, we aim to perform a case study of the largest GRG available. Methods. We reprocessed the LOFAR Two-Metre Sky Survey DR2 by subtracting compact sources and performing multi-scale CLEAN de-convolutions at 60" and 90" resolution. The resulting images constitute the most sensitive survey yet for radio galaxy lobes, whose diffuse nature and steep synchrotron spectra have allowed them to evade previous detection attempts at higher resolution and shorter wavelengths. We visually searched these images for GRGs. Results. We have discovered Alcyoneus, a low-excitation radio galaxy with a projected proper length l p = 4.99 ± 0.04 Mpc. Both its jets and lobes are detected at very high significance, and the SDSS-based identification of the host, at spectroscopic redshift z spec = 0.24674 ± 6 × 10 -5 , is unambiguous. The total luminosity density at ν = 144 MHz is L ν = 8 ± 1 × 10 25 W Hz -1 , which is below average, though near median (percentile 45 ± 3%) for GRGs. The host is an elliptical galaxy with a stellar mass M* = 2.4 ± 0.4 × 10 11 M ⊙ and a super-massive black hole mass M• = 4 ± 2 × 10 8 M ⊙ , both of which tend towards the lower end of their respective GRG distributions (percentiles 25 ± 9% and 23 ± 11%). The host resides in a filament of the Cosmic Web. Through a new Bayesian model for radio galaxy lobes in three dimensions, we estimate the pressures in the megaparsec-cubed-scale northern and southern lobes to be P min,1 = 4.8 ± 0.3 × 10 -16 Pa and P min,2 = 4.9 ± 0.6 × 10 -16 Pa, respectively. The corresponding magnetic field strengths are B min,1 = 46 ± 1 pT and B min,2 = 46 ± 3 pT. Conclusions. We have discovered what is in projection the largest known structure made by a single galaxy – a GRG with a projected proper length l p = 4.99 ± 0.04 Mpc. The true proper length is at least l min = 5.04 ± 0.05 Mpc. Beyond geometry, Alcyoneus and its host are suspiciously ordinary: the total low-frequency luminosity density, stellar mass, and super-massive black hole mass are all lower than, though similar to, those of the medial GRG. Thus, very massive galaxies or central black holes are not necessary to grow large giants, and, if the observed state is representative of the source over its lifetime, neither is high radio power. A low-density environment remains a possible explanation. The source resides in a filament of the Cosmic Web, with which it might have significant thermodynamic interaction. The pressures in the lobes are the lowest hitherto found, and Alcyoneus therefore represents the most promising radio galaxy yet to probe the warm–hot inter-galactic medium.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2005

Ocular examination for trauma; clinical ultrasound aboard the International Space Station

BACKGROUND: Ultrasound imaging is a successful modality in a broad variety of diagnostic applications including trauma. Ultrasound has been shown to be accurate when performed by non-radiologist physicians; recent reports have suggested that non-physicians can perform limited ultrasound examinations. A multipurpose ultrasound system is installed on the International Space Station (ISS) as a component of the Human Research Facility (HRF). This report documents the first ocular ultrasound examination conducted in space, which demonstrated the capability to assess physiologic alterations or pathology including trauma during long-duration space flight. METHODS: An ISS crewmember with minimal sonography training was remotely guided by an imaging expert from Mission Control Center (MCC) through a comprehensive ultrasound examination of the eye. A multipurpose ultrasound imager was used in conjunction with a space-to-ground video downlink and two-way audio. Reference cards with topological reference points, hardware controls, and target images were used to facilitate the examination. Multiple views of the eye structures were obtained through a closed eyelid. Pupillary response to light was demonstrated by modifying the light exposure of the contralateral eye. RESULTS: A crewmember on the ISS was able to complete a comprehensive ocular examination using B- and M-mode ultrasonography with remote guidance from an expert in the MCC. Multiple anteroposterior, oblique, and coronal views of the eye clearly demonstrated the anatomic structures of both segments of the globe. The iris and pupil were readily visualized with probe manipulation. Pupillary diameter was assessed in real time in B- and M-mode displays. The anatomic detail and fidelity of ultrasound video were excellent and could be used to answer a variety of clinical and space physiologic questions. CONCLUSIONS: A comprehensive, high-quality ultrasound examination of the eye was performed with a multipurpose imager aboard the ISS by a non-expert operator using remote guidance. Ocular ultrasound images were of diagnostic quality despite the 2-second communication latency and the unconventional setting of a weightless spacecraft environment. The remote guidance techniques developed to facilitate this successful NASA research experiment will support wider applications of ultrasound for remote medicine on Earth including the assessment of pupillary reactions in patients with severe craniofacial trauma and swelling.

Clinical Trial↗

DOE OSTI2023

Strictly Enforcing Invertibility and Conservation in CNN-Based Super Resolution for Scientific Datasets

Abstract Recently, deep convolutional neural networks (CNNs) have revolutionized image “super resolution” (SR), dramatically outperforming past methods for enhancing image resolution. They could be a boon for the many scientific fields that involve imaging or any regularly gridded datasets: satellite remote sensing, radar meteorology, medical imaging, numerical modeling, and so on. Unfortunately, while SR-CNNs produce visually compelling results, they do not necessarily conserve physical quantities between their low-resolution inputs and high-resolution outputs when applied to scientific datasets. Here, a method for “downsampling enforcement” in SR-CNNs is proposed. A differentiable operator is derived that, when applied as the final transfer function of a CNN, ensures the high-resolution outputs exactly reproduce the low-resolution inputs under 2D-average downsampling while improving performance of the SR schemes. The method is demonstrated across seven modern CNN-based SR schemes on several benchmark image datasets, and applications to weather radar, satellite imager, and climate model data are shown. The approach improves training time and performance while ensuring physical consistency between the super-resolved and low-resolution data. Significance Statement Recent advancements in using deep learning to increase the resolution of images have substantial potential across the many scientific fields that use images and image-like data. Most image super-resolution research has focused on the visual quality of outputs, however, and is not necessarily well suited for use with scientific data where known physics constraints may need to be enforced. Here, we introduce a method to modify existing deep neural network architectures so that they strictly conserve physical quantities in the input field when “super resolving” scientific data and find that the method can improve performance across a wide range of datasets and neural networks. Integration of known physics and adherence to established physical constraints into deep neural networks will be a critical step before their potential can be fully realized in the physical sciences.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

High-speed characterization of two-phase flow and bubble dynamics in titanium felt porous media for hydrogen production

Bubble dynamics and two-phase flow phenomena are closely related to the performance of proton exchange membrane electrolyzer cells (PEMECs). This paper reports an in-situ study of the oxygen bubble behavior and associated multiphase evolutions in the anode side of PEMECs with titanium (Ti) felt liquid gas diffusion layers (LGDLs) by a high-speed visualization system. The micro oxygen bubble dynamics was captured and analyzed at different locations and virous operating conditions. The results show that the bubble detachment frequency and detachment diameter greatly increase with the operating current density. Additionally, they are significantly impacted by the local pore structure and morphology of Ti felt LGDLs. In the flow channels, there exist only several discrete micro bubbles at a low current density (0.04 A/cm 2 ) and a large flow velocity (133 mm/s). At a current density (0.2 A/cm 2 ) and a flow velocity (67 mm/s), a number of gas slugs are formed in the follow channels, in addition to discrete micro bubbles. At a high current density (1 A/cm 2 ) and a flow velocity (67 mm/s), more bubbles appear in the channel, and the flow field is dominated by slug or annular flows. These investigations can help to better understand the two-phase flow and bubble detachment mechanism, and provide a foundation for electrochemical reaction, multiphase flow studies and optimize the design of gas diffusion layers and flow fields for PEMECs in the future.

08 HYDROGEN↗

DOE OSTI2015

Structural study of growth, orientation and defects characteristics in the functional microelectromechanical system material aluminium nitride

The real structure and morphology of piezoelectric aluminum nitride (AlN) thin films as essential components of magnetoelectric sensors are investigated via advanced transmission electron microscopy methods. State of the art electron diffraction techniques, including precession electron diffraction and automated crystal orientation mapping (ACOM), indicate a columnar growth of the AlN grains optimized for piezoelectric application with a (0 0 0 1) texture. Comparing ACOM with piezoresponse force microscopy measurements, a visual correlation of the structure and the piezoelectric properties is enabled. With a quantitative analysis of the ACOM measurements, a statistical evaluation of grain rotations is performed, indicating the presence of coincidence site lattices with Σ7, Σ13a, Σ13b, Σ25. Using a geometric phase analysis on high resolution micrographs, the occurrence of strain is detected almost exclusively at the grain boundaries. Moreover, high resolution imaging was applied for solving the atomic structure at stacking mismatch boundaries with a displacement vector of 1/2 〈1 0 -1 1〉. All real structural features can be interpreted via simulations based on crystallographic computing in terms of a supercell approach.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗