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At least 1,171 records · Page 65
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Decimeter-scale mapping of carbonate-controlled trace element distribution in Neoarchean cuspate stromatolites

We present that ancient stromatolites can provide key insights into the early evolution of life on Earth. Neoarchean fenestrate stromatolites from the ~2520 Ma Upper Nauga Formation (Transvaal Supergroup, South Africa) preserve cuspate morphologies. They possess clearly delineated support and drape structures interpreted as dolomitized microbial mat material. Petrographic observations show that the biogenic structures are composed of planar-s to non-planar ferroan dolomite, encased in ferroan calcite, including herringbone calcite textures. The cuspate stromatolites were analyzed using Synchrotron Rapid Scanning X-Ray Fluorescence (SRS-XRF) and more conventional techniques to determine: (i) whether element distributions could be distinguished in ancient stromatolites at both cm to dm scales, (ii) whether element distributions show variation between biogenic and abiogenic textures, and (iii) the sample’s paragenesis. The distributions of Ca, Fe, Mn, Pb, Cu, As, Br, Al, Si, P, and S directly correspond to dolomitized stromatolitic structures and show trace element distributions are principally controlled by calcite and dolomite occurrence. Dolomite formation was mainly driven by seawater-derived fluids given the high concentrations of Fe and retention of marine shale-normalized rare earth element and yttrium (REY SN ) patterns, however the spatial association of dolomite to stromatolite structures may reflect microbially-influenced mineral nucleation. Lastly, given the complexity of this sample’s paragenetic evolution, trace metal distributions cannot be conclusively tied to specific metabolic pathways, bioaccumulation or passive binding, however, the results show SRS-XRF can be used for quantifiable, spatial, in-situ investigation of ancient microbialites.

58 GEOSCIENCES↗
DOE OSTI2019

Free-free matrix-elements for two-photon opacity

Recent opacity measurements have inspired a close study of the two-photon contributions to the opacity of hot plasmas. The absorption and emission of radiation is controlled by dipole matrix-elements of electrons in an atom or ion. This paper describes two independent methods to calculate matrix-elements needed for the two-photon opacity and tests the results by the f-sum rule. The usual f-sum rule is extended to a matrix f-sum that offers a rigorous test for bound-bound, bound-free and free-free transitions. An additional higher-order sum-rule for the two-photon transition amplitudes is described. In this work, we obtain a simple parametric representation of a key plasma density effect on the matrix-elements. The perturbation theory calculation of two-photon cross-sections is compared to an independent method based on the solution of the time-dependent Schroedinger equation for an atom or ion in a high-frequency electromagnetic field. This is described as a high frequency Stark effect or AC Stark effect. Two-photon cross sections calculated with the AC Stark code agree with perturbation theory to within about 5%. In addition to this cross check, the AC Stark code is well suited to evaluating important questions such as the variation of two-photon opacity for different elements.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Exploration of ICP platforms for measuring elemental impurities in uranium ore concentrates

Presented here is a look at different inductively coupled plasma (ICP) platforms including sector field, triple quadrupole mass spectrometric configurations, and optical emission spectroscopy. The goal of this paper was to convey the feasibility of trace elemental analysis of a uranium ore concentrate with varying ICP-based platforms. It was predicted that low mass elements would be better determined through optical means, or the mass collision/reaction technology of the triple quadrupole mass spectrometer, in comparison to high resolution measurements of magnetic sector-based configurations. Ultimately, it is difficult to achieve an “optimal” analysis of 20+ elements, for numerous samples, on a single platform, especially when time, measurement reproducibility, sensitivity, and robustness should be considered. The work presented here highlights the advantages and disadvantages of the respective trace elemental analysis platforms.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Effect of alloying elements on the $γ$’ antiphase boundary energy in Ni-base superalloys

Mechanical and fatigue performance of γ-γ' Ni-base superalloys are strongly affected by the antiphase boundary energy (APBE) of the γ' precipitates which, in turn, is dictated by the alloy's composition. Due to the multicomponent character of these alloys, establishing composition-APBE relationships are challenging, even though the qualitative effect of individual solutes on the APBE may be known. This work attempts to utilize density functional theory-based cluster expansion calculations to systematically assess the effect of composition on the APBE of the γ' phases in Ni-base superalloys. We aim to elucidate the influence of not only one single element but also multiple coexisting alloying elements on the γ' APBE. By explicit consideration of configurational disorder via Monte Carlo sampling, the effect of temperature on the APBE has also been analyzed. Overall, this work reveals that (1) effects of individual solute element M on the APBE energy obtained in an isolated, ternary condition (i.e. in Ni 3-x Al 1-y M x+y ) does not directly translate to a multi-solute case and that (2) the mutual synergistic interactions among different solute elements are not negligible. Based on the present results, an empirical master equation that predicts the APBE based on the composition of the γ' phases has been obtained.

36 MATERIALS SCIENCE↗
DOE OSTI2018

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructs high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss–Lobatto–Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

NaroNet: Discovery of tumor microenvironment elements from highly multiplexed images

Understanding the spatial interactions between the elements of the tumor microenvironment -i.e. tumor cells. fibroblasts, immune cells- and how these interactions relate to the diagnosis or prognosis of a tumor is one of the goals of computational pathology. We present NaroNet, a deep learning framework that models the multi-scale tumor microenvironment from multiplex-stained cancer tissue images and provides patient-level interpretable predictions using a seamless end-to-end learning pipeline. Trained only with multiplex-stained tissue images and their corresponding patient-level clinical labels, NaroNet unsupervisedly learns which cell phenotypes, cell neighborhoods, and neighborhood interactions have the highest influence to predict the correct label. To this end, NaroNet incorporates several novel and state-of-the-art deep learning techniques, such as patch-level contrastive learning, multi-level graph embeddings, a novel max-sum pooling operation, or a metric that quantifies the relevance that each microenvironment element has in the individual predictions. We validate NaroNet using synthetic data simulating multiplex-immunostained images where a patient label is artificially associated to the -adjustable- probabilistic incidence of different microenvironment elements. We then apply our model to two sets of images of human cancer tissues: 336 seven-color multiplex-immunostained images from 12 high-grade endometrial cancer patients; and 382 35-plex mass cytometry images from 215 breast cancer patients. In both synthetic and real datasets, NaroNet provides outstanding predictions of relevant clinical information while associating those predictions to the presence of specific microenvironment elements.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2026

First measurement of polarized spin-density matrix elements and differential cross sections dσ/dt in ω photoproduction off the proton for 2.7 < Eγ < 5.2 GeV using CLAS at Jefferson Lab

We report on the differential cross sections dσ/dt, the unpolarized spin-density matrix elements ρ 00 0 , ρ 1 − 1 0 , Re ρ 10 0 , and the first extraction of the polarized elements Im ρ 10 3 , Im ρ 1 − 1 3 for the reaction γp → pω using the CLAS spectrometer at Jefferson Laboratory. The ω mesons were detected in their dominant charged decay mode, ω → π + π − π 0 , and all t-dependent results are presented in a fine binning for incident photon energies between 2.73 and 5.16 GeV (corresponding to the center-of-mass energy range W ∈ [ 2.45, 3.25 ] GeV). All matrix elements are first measurements for − t > 0.6 GeV2. Moreover, differential cross sections dσ/d(cos Θ c . m . ω ) and the corresponding angle-dependent unpolarized spin-density matrix elements in the Adair frame are presented for the incident photon energy range 1.56–3.80 GeV (corresponding to W ∈ [ 1.95, 2.83 ] GeV). These new ω photoproduction data are consistent with earlier CLAS results but extend the energy range well beyond the nucleon resonance region into the Regge regime. The comparison with Regge-theory-based model predictions shows that the new data impose more stringent constraints on our understanding of ω photoproduction.

Hu, T.↗
DOE OSTI2020

Nuclear matrix elements from lattice QCD for electroweak and beyond-Standard-Model processes

Over the last decade, numerical solutions of Quantum Chromodynamics (QCD) using the technique of lattice QCD have developed to a point where they are beginning to connect fundamental aspects of nuclear physics to the underlying degrees of freedom of the Standard Model. In this review, the progress of lattice QCD studies of nuclear matrix elements of electroweak currents and beyond-Standard-Model operators is summarized, and connections with effective field theories and nuclear models are outlined. Lattice QCD calculations of nuclear matrix elements can provide guidance for low-energy nuclear reactions in astrophysics, dark matter direct detection experiments, and experimental searches for violations of the symmetries of the Standard Model, including searches for additional CP violation in the hadronic and leptonic sectors, baryon-number violation, and lepton-number or flavor violation. Similarly, important inputs to neutrino experiments seeking to determine the neutrino-mass hierarchy and oscillation parameters, as well as other electroweak and beyond-Standard-Model processes can be determined. As a result, the phenomenological implications of existing studies of electroweak and beyond-Standard-Model matrix elements in light nuclear systems are discussed, and future prospects for the field toward precision studies of these matrix elements are outlined.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Elemental source attribution signatures for calcium ammonium nitrate (CAN) fertilizers used in homemade explosives

Calcium ammonium nitrate (CAN) is a widely available fertilizer composed of ammonium nitrate mixed with some form of calcium carbonate such as limestone or dolomite. CAN is also frequently used to make homemade explosives. The potential of using elemental profiling and chemometrics to match both pristine and reprocessed CAN fertilizers to their factories for use in future forensic investigations was examined. Inductively coupled plasma-mass spectrometry (ICP-MS) analysis was performed on 64 elements in 125 samples from 11 CAN stocks from 6 different CAN factories. Fisher ratio, degree-of-class-separation, and partial least squares discriminant analysis (PLSDA) were used to develop a model using the concentrations of Na, V, Mn, Cu, Ga, Sr, Ba and U to classify a validation set of CAN samples into 5 factory groups; one group was two factories from the same fertilizer company. In terms of the pristine CAN samples, i.e., unadulterated prills, 64% of the test samples were matched to their correct factory group with zero false positives. The same PLSDA model was used to correctly match 100% of the CAN samples that were reprocessed by crushing and mixing the CAN with powdered sugar. In the case of crushed CAN samples mixed with aluminum powder, correct matches were made for zero to 100% of the samples depending on the factory the CAN originated. Remarkably, for one factory, 100% of the ammonium nitrate samples that were extracted from CAN using tap or bottled water were matched to the correct CAN factory group. Lastly, the water-insoluble (calcium carbonate) portions of CAN provided a greater degree of discrimination between factories than the water-soluble portions of CAN. In summary, this work illustrates that sourcing unadulterated CAN fertilizer can potentially be done with high frequency and high confidence using elemental profiling and chemometrics while the sourcing of reprocessed CAN is dependent on how much an adulterant alters the recovered elemental profile of CAN.

chemical forensics↗
DOE OSTI2017

Tracking Radionuclide Fractionation in the First Atomic Explosion Using Stable Elements

Compositional analysis of postdetonation fallout is a tool for forensic identification of nuclear devices. However, the relationship between device composition and fallout composition is difficult to interpret because of the complex combination of physical mixing, nuclear reactions, and chemical fractionations that occur in the chaotic nuclear fireball. By using a combination of in situ microanalytical techniques (electron microprobe analysis and secondary ion mass spectrometry), we show that some heavy stable elements (Rb, Sr, Zr, Ba, Cs, Ba, La, Ce, Nd, Sm, Dy, Lu, U, Th) in glassy fallout from the first nuclear test, Trinity, are reliable chemical proxies for radionuclides generated during the explosion. Stable-element proxies show that radionuclides from the Trinity device were chemically, but not isotopically, fractionated by condensation. Moreover, stable-element proxies delineate chemical fractionation trends that can be used to connect present-day fallout composition to past fireball composition. Stable-element proxies therefore offer a novel approach for elucidating the phenomenology of the nuclear fireball as it relates to the formation of debris and the fixation of device materials within debris.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Crystal Growth and Elemental Homogeneity of the Multicomponent Rare-Earth Garnet (Lu 1/6 Y 1/6 Ho 1/6 Dy 1/6 Tb 1/6 Gd 1/6 ) 3 Al 5 O 12

We report high-entropy aluminum garnets were grown as bulk single crystals using the micro-pulling-down method, taking the synthesis of complex ceramics a step further from the conventional preparation of polycrystalline materials. We studied the effects of growth parameters on the elemental distribution in high optical quality crystals of (Lu 1/6 Y 1/6 Ho 1/6 Dy 1/6 Tb 1/6 Gd 1/6 ) 3 Al 5 O 12 containing six cations (yttrium and rare-earths) taken in equimolar amounts. A single garnet structure was confirmed by powder X-ray diffraction. Electron microprobe measurements were obtained to correlate the radial distribution of rare-earth elements with pulling rates and molten zone height. The nature of the elemental distribution in the radial direction was associated with ionic radius: smaller rare-earths concentrated in the center of the crystal, while larger rare-earths segregated toward the outer edge of the cylindrical crystal. Faster pulling rates led to a flattening of the concentration profiles toward the nominal concentration, promoting a more homogeneous radial elemental distribution, while varying the molten zone height did not have a significant effect. The demonstrated success with crystal growth enables the practical availability of single crystals of multicomponent aluminum garnets for further discovery of new phenomena and applications.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Controlling Extraction of Rare Earth Elements Using Functionalized Aryl-vinyl Phosphonic Acid Esters

Ligands that can discriminate between individual rare earth elements are important for production of these critical elements. A set of aryl-vinyl phosphonic acid ligands for extracting rare earth elements were designed and synthesized under the hypothesis that the strength of the rare earth-ligand interactions could be tuned by changing the dipole moment of the ligand. The ligands were synthesized via a two-step reaction procedure using a Heck coupling reaction to functionalize vinyl phosphonic acid, followed by Steglich esterification to obtain high-purity styryl phosphonic acid monoesters with varying dipole moments along the P-C bond. The metal binding strength and composition of the rare earth complexes formed with these styryl phosphonic acid monoesters were experimentally studied by liquid-liquid extraction techniques, while DFT calculations were performed to determine the dipole moments of the free and complexed ligands and the electronic structure of the complexes formed. All three prepared ligands were much stronger extracting agents for europium(III) than the dialkylphosphonic acids usually used for this separation. However, the order of increasing extraction strength was found to match the order of the decreasing calculated dipole moment along the P-C bond of the three styryl-based ligands, rather than correlating with increasing ligand basicity, as reflected by the pK a of the ligands. Finally, these findings suggest that this approach can be used to systematically alter the extraction strength of aromatic phosphonic monoesters for rare earth element purification.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Spin–Orbit Coupling Constants in Atoms and Ions of Transition Elements: Comparison of Effective Core Potentials, Model Core Potentials, and All-Electron Methods

The spin–orbit coupling constants (SOCC) in atoms and ions of the first- through third-row transition elements were calculated for the low-lying atomic states whose main electron configuration is [ n d] q ( q = 1–4 and 6–9, n = the principal quantum number), using four different approaches: (1) a nonrelativistic Hamiltonian used to construct multiconfiguration self-consistent field (MCSCF) wave functions utilizing effective core potentials and their associated basis sets within the framework of second-order configuration interaction (SOCI) to calculate spin–orbit couplings (SOC) using one-electron Breit–Pauli Hamiltonian (BPH), (2) a nonrelativistic Hamiltonian used to construct MCSCF wave functions utilizing model core potentials and their associated basis sets within the framework of SOCI to calculate SOC using the full BPH, (3) nonrelativistic and spin-independent relativistic Hamiltonians used to construct MCSCF wave functions utilizing all-electron (AE) basis sets within the framework of SOCI to calculate SOC using the full BPH, and (4) a relativistic Hamiltonian given by the exact two-component (X2C) transformation for construction of Kramers-restricted relativistic configuration interaction wave functions. In this investigation, these four approaches are referred to as ECP, MCP, AE, and X2C methods, respectively. The ECP, MCP, and AE methods are so-called two-step approaches (TSA), while the X2C method is a one-step approach (OSA). In the AE method, three different calculations—relativistic elimination of small components (RESC), third-order Douglas–Kroll–Hess (DKH3), and infinite-order two-component (IOTC) relativistic correction—were performed for the estimation of the scalar relativistic components in addition to those of the nonscalar relativistic (NSR) contributions. The calculated SOCC are compared to the available experimental data via the Landé interval rule. Although there are several exceptions, including states whose main configuration is [ n d] 5 , the average differences between the ECP and AE (IOTC) SOCC and between the ECP and the X2C SOCC are mostly less than 20%. The differences between the ECP and the experimental SOCC are even smaller. No serious discrepancy was found between the TSA and OSA predictions of SOCC for the first- and second-row transition elements in comparison to experiment. For atoms and ions of the third-row transition elements, the SOCC calculated through the Landé interval rule are not reliable. The low-energy spin-mixed (SM) states originating from a [5d]q configuration ( q = 2–4) have a larger energy lowering due to the SOC effects, in comparison with those for atoms and ions of the first- and second-row transition elements. For the spin-mixed (SM) states originating from a [5d]q configuration ( q = 6–8), the energy lowering of all 4 F 7/2 , 5 D 1 , and 5 D 3 states due to the SOC effects is smaller than those of the other SM states. Furthermore this difficulty, which also arises for the MCP, AE, and X2C (OSA) approaches, suggests that the LS-coupling scheme is inappropriate.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2023

3D Quantification of Elemental Gradients within Heterostructured Particles of Battery Cathodes

Heterogenous architectures with elemental gradients tailored within particles have been pursued to combat the instabilities limiting Ni-rich cathode materials for lithium-ion batteries. The growth of different compositional layers is accomplished during the synthesis of hydroxide precursors. However, the extent to which these concentration gradients are modified during high-temperature reactions is difficult to establish in their intact, spherical form. Here, we show the entire three-dimensional structure of a secondary particle can be resolved nondestructively with differential X-ray absorption spectroscopy (XAS) through transmission X-ray microscopy (TXM). The relationship between particle location and elemental content was fully quantified, with high statistical significance, for heterostructures possessing different compositional gradients in the precursors with 90:5:5 Ni:Mn:Co core compositions. Reduced elemental heterogeneity was observed after high-temperature synthesis, but gradients remained. In conclusion, the methodology presented should be used to guide synthesis while assuring that gains in electrochemical performance are linked to precise elemental distributions at the nanoscale.

25 ENERGY STORAGE↗
DOE OSTI2013

β-Rhombohedral Boron: At the Crossroads of the Chemistry of Boron and the Physics of Frustration [Boron: a frustrated element]

In the periodic table boron occupies a peculiar, crossover position: on the first row, it is surrounded by metal forming elements on the left and by non-metals on the right. In addition, it is the only non-metal of the third column. Therefore it is perhaps not surprising that the crystallographic structure and topology of its stable allotrope at room temperature (β-boron) are not shared by any other element, and are extremely complex. The formidable intricacy of β- boron, with interconnecting icosahedra, partially occupied sites, and an unusually large number of atoms per unit cell (more than 300) has been known for more than 40 years. Nevertheless boron remains the only element purified in significant quantities whose ground state geometry has not been completely determined by experiments. However theoretical progress reported in the last decade has shed light on numerous properties of elemental boron, leading to a thorough characterization of its structure at ambient conditions, as well as of its electronic and thermodynamic properties. This review discusses in detail the properties of β-boron, as inferred from experiments and the ab-initio theories developed in the last decade.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Real-Time Elemental and Isotopic Measurements of Molten Salt Systems through Laser-Induced Breakdown Spectroscopy

Molten salt reactors are an emerging advanced nuclear reactor concept in which the fuel is dissolved into the working fluid in the form of a high-temperature molten salt. The complex, mobile, and corrosive nature of this fluid presents a fundamental challenge for analytical measurements. Tracking species throughout the reactor is important for ensuring proper operation. This article presents laser-induced breakdown spectroscopy (LIBS) used to monitor the elemental composition of a molten salt and corresponding hydrogen isotopic shifts in real-time. A NaNO 3 –KNO 3 eutectic salt was saturated with protium and deuterium gases, then the effluent aerosol stream formed using an argon sparging vessel was monitored with LIBS. This modular LIBS system permitted several spectrometers to be used simultaneously to capture high-resolution isotopic shifts and provide broadband elemental coverage. Further, the results exhibit how LIBS can be used to understand salt–gas chemical and physical interactions such as diffusion. Furthermore, LIBS’ broad elemental coverage can provide greater insight into the chemical reactions within the salt vessel such as the formation of water vapor by monitoring hydrogen and oxygen signatures simultaneously. Ultimately, this study demonstrates the analytical possibilities of LIBS for real-time monitoring of isotopes and elemental composition in molten salt systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Pressurized Gas-Driven Elemental Redistribution Enables Ultrastable PtNi Catalysts for Heavy-Duty Vehicles

Gas-driven element redistribution, characterized by the preferential enrichment of one element at the surface relative to the bulk, is frequently observed in multicomponent alloys. Using L1 0 -ordered PtNi as a model system, we reveal that gas pressure plays a critical role in governing adsorption-driven surface composition during annealing in reducing gases: low pressure favors Pt surface segregation, while high pressure facilitates Ni surface enrichment. In this study, we developed a high-pressure nitriding (HPN) strategy that modulates the surface structure and composition of PtNi catalysts. The resulting HPN-PtNi exhibits enhanced performance and durability in membrane electrode assemblies for heavy-duty fuel cell applications, maintaining a high current density of 1.19 A cm −2 at 0.7 V after 90,000 voltage cycles. Through a combination of experimental and theoretical analyses, we reveal that the HPN process forms additional stabilizing Ni−N bonds and induces elemental redistribution with Ni surface enrichment and a Ni-deficient Pt subsurface. These modifications alter the atomic coordination environment of the ordered PtNi phase. This work presents a generalizable strategy to design robust and high-performing Pt-based catalysts by controlling gas-pressure-driven elemental redistribution and dopant incorporation.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2021

Separating Physics and Dynamics Grids for Improved Computational Efficiency in Spectral Element Earth System Models

Previous studies have shown that atmospheric models with a spectral element grid can benefit from putting physics calculations on a relatively coarse finite volume grid. Here we demonstrate an alternative high-order, element-based mapping approach used to implement a quasi-equal-area, finite volume physics grid in E3SM. Unlike similar methods, the new method in E3SM requires topology data purely local to each spectral element, which trivially allows for regional mesh refinement. Simulations with physics grids defined by 2 × 2, 3 × 3, and 4 × 4 divisions of each element are shown to verify that the alternative physics grid does not qualitatively alter the model solution. The model performance is substantially affected by the reduction of physics columns when using the 2 × 2 grid, which can increase the throughput of physics calculations by roughly 60%–120% depending on whether the computational resources are configured to maximize throughput or efficiency. A pair of regionally refined cases are also shown to highlight the refinement capability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗