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Estimating Bidirectional Reflectance and Monitoring Stability of SNPP-VIIRS Reflective Solar Bands Using A Deep Neural Network

The NASA Clouds and the Earth's Radiant Energy System project provides the scientific community with observed top-of-atmosphere shortwave and longwave fluxes for climate monitoring and climate model validation. To provide consistent VIIRS cloud retrievals, the CERES Imager and Geostationary Calibration Group (IGCG) must understand and quantify the stability of the VIIRS instruments. To achieve this, the IGCG utilizes tropical deep convective clouds (DCCs) as invariant targets. Proper seasonal characterization of the DCC bidirectional reflectance distribution function (BRDF) is key to the success of DCC-based calibration methods, particularly for shortwave infrared (SWIR) bands. This article proposes the use of a deep neural network (DNN) to characterize VIIRS solar reflective band BRDF reflectance, with which individual channel trends are isolated by manipulating the DNN time input. Initial results show that the DNN method can extract statistically significant SNPP-VIIRS band trends, using only SNPP-VIIRS inputs, that are correlative to and match the magnitude of significant trends determined using methods that rely on an external angular distribution model. It may be possible to apply this approach to actively monitor the stability of new instruments without the need for predetermined seasonal BRDF corrections.

Benjamin Scarino↗

DOE OSTI2022

Determination of Molecular Structure and Dynamics of Molten Salts by Advanced Neutron and X-ray Scattering Measurements and Computer Modeling

The design and development of fully functional Molten-Salt Reactors (MSR) require detailed knowledge of the molten salt properties in order to understand and predict the salt’s behavior. Fundamental properties of interest include molecular structure, speciation, and dynamics (such as diffusion coefficients) of salt components and dissolved corrosion and fission products. Computer modeling is necessary to predict changes in physical and chemical properties due to irradiation, burning of dissolved fuel, and corrosion. The modeling requires experimental data, and advanced neutron and x-ray scattering and spectroscopy provide the most reliable and direct determination of the structure (Pair-Distribution Functions, PDF), and dynamics of ions in the melt. This project dealt with both fluoride and chloride salts. The PDFs have been measured by a combination of neutron and x-ray diffraction. We utilized the techniques of isotope substitutions, a very powerful tool available for neutron-scattering, to extract the details of the liquid structure. Although similar measurements have been done before, modern advanced neutron and x-ray-scattering techniques allow collecting the data at much higher resolution and in a wider range of temperatures. Importantly, we were among the first to study fluoride salts by neutron scattering. The importance of impurities and their effects on salt properties have become apparent recently and so new methods of salt purification were developed. We took advantage of these developments to produce reliable data, which have been used for computer simulations of both clean salts and those with added fission and corrosion products most relevant for MSRs. Ab initio molecular dynamics simulations have been performed to understand the multi-component liquid solution, in particular solubility of impurities and thermodynamic interactions in relation to the ionic-cluster structure of the fluid. We applied machine learning to regress from the simulation and experimental data in order to develop a fast-acting model that can handle molten salt with an arbitrary (≥ 10) number of chemical elements and be able to predict chemical potential as a function of composition and temperature. This project resulted in a number of experimental and computer-simulation publications, a patent application, and numerous conference presentations (American Physical Society, American Chemical Society, and The Electrochemical Society among others). Multiple students and postdocs participated and collaborated on aspects of this project. This project seeded new collaborations between MIT and other institutions, such as the University of Massachusetts Lowell, the University of Illinois Urbana-Champaign, the University of California Berkeley, and Oak Ridge and Los Alamos National Labs. As such, this project has had a broad and lasting impact beyond its original scientific scope.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS1991

The joint far-infrared-optical luminosity function for spiral galaxies and data for the Abell 400 and Cancer clusters

Visual and IRAS data for an optically selected sample of 183 late-type galaxies are compiled in tables and graphs and analyzed in detail to determine the joint FIR-optical luminosity function Psi from the FIR/blue luminosity ratio, r = L(FIR)/L(B). It is found that Psi can be approximated by a function of a single variable psi(r-prime), where r-prime is defined as r times L(B)/L(asterisk) exp -delta, with L(asterisk) a constant and delta = about 0.08. A lognormal curve peaking at r-prime = 0.35 and with dispersion of 0.28 is shown to give a good fit to psi(r-prime). From a lack of galaxies with very low r-prime in the present sample it is inferred that there are few spiral galaxies with low interstellar-dust abundances. Also included are data on the distribution function of r-prime for the more distant clusters Abell 400 and Cancer.

Corbelli, Edvige↗

DOE OSTI2023

Temperature-Dependent Optical and Structural Properties of Chiral Two-Dimensional Hybrid Lead-Iodide Perovskites

Layered hybrid halide perovskites gain chirality via the incorporation of chiral organic cations in the spacer layer. These so-called chiral two-dimensional (2D) halide perovskites have attracted considerable interest recently for chiral optoelectronic, spintronic, and ferroelectric applications. However, the effect of temperature on these materials, especially how the structure changes with temperature and its impact on the chiral optoelectronic properties, remains an open question. Here, we study the effect of temperature change on chiral optoelectronic and structural properties through temperature-dependent circularly polarized photoluminescence (CPPL) microscopy and synchrotron powder X-ray diffraction as well as pair distribution function analysis to elucidate the intrinsic chiral optoelectronic and structural variations for R-, S-, and racemic-methylbenzylamine lead iodide. Here our results show that the temperature-induced band gap changes indicate a strong electron–phonon coupling compared to the thermal-induced lattice expansion in chiral 2D perovskites. From powder diffraction measurements, a monotonic lattice expansion is observed on heating with no structural phase transitions over 90–360 K detected for the three samples studied herein. Locally, a strongly anisotropic and even negative expansion in some components of the instantaneous Pb–I pair-distance distribution is suggestive of coupling between dynamical intralayer distortions and lattice expansion. This work provides insights into the fundamental understanding of the temperature effect on chiral optoelectronic and structural properties of chiral 2D perovskites, which can lead to new chiral materials design strategies for future chiral optoelectronic applications.

36 MATERIALS SCIENCE↗

NASA NTRS2022

Local Scale (3-M) Soil Moisture Mapping Using SMAP and Planet Superdove

A capability for mapping meter-level resolution soil moisture with frequent temporal sampling over large regions is essential for quantifying local-scale environmental heterogeneity and eco-hydrologic behavior. However, available surface soil moisture (SSM) products generally involve much coarser grain sizes ranging from 30 m to several 10s of kilometers. Hence a new method is proposed to estimate 3-m resolution SSM using a combination of multi-sensor fusion, machine- learning (ML) and Cumulative Distribution Function (CDF) matching approaches. This method established favorable SSM correspondence between 3-m pixels and overlying 9-km grid cells from overlapping Planet SuperDove (PSD) observations and NASA Soil Moisture Active-Passive (SMAP) mission products. The resulting 3-m SSM predictions showed improved accuracy by reducing ab- solute bias and RMSE by ~0.01 cm3/cm3 over the original SMAP data in relation to in-situ soil moisture measurements for the Australian Yanco region, while preserving the high sampling frequency (1-3 day global revisit) and sensitivity to surface wetness (R 0.865) from SMAP. Heterogeneous soil moisture distributions varying with vegetation biomass gradients and irrigation regimes were generally captured within a selected study area. Further algorithm refinement and implementation for regional applications will allow for improvement in water resources management, precision agriculture, and disaster forecasts and responses.

soil moisture↗

DOE OSTI2024

Simulating energetic ions and enhanced fusion rates from ion-cyclotron resonance heating with a full-wave/Fokker–Planck model

Reproducing fast-ion enhanced fusion rates from ion-cyclotron resonance heating (ICRH) in tokamaks requires the self-consistent coupling of a full-wave solver and a Fokker–Planck solver, which evolves multiple simultaneously resonant ion species. We introduce a new self-consistent model that iterates the TORIC full-wave solver with the CQL3D Fokker–Planck solver using the integrated plasma simulator (IPS). This model evolves the bounce-averaged ion distribution functions in both parallel and perpendicular velocity-space with a quasilinear radio frequency (RF) diffusion operator valid in the ion finite Larmor radius (FLR) limit and the RF electric fields with the resultant non-Maxwellian FLR dielectric tensor. This produces non-Maxwellian ICRH simulations that are fully self-consistent, fast, and interoperable with integrated modeling frameworks, such as TRANSP/GACODE/IPS-FASTRAN. We demonstrate our model's capabilities by validating it against experimental data in Alcator C-Mod. We then perform the first RF heating simulations of SPARC using self-consistent non-Maxwellian ion distributions to investigate the potential to enhance fusion rates using ion cyclotron resonance heating generated fast ions.

Physics↗

DOE OSTI2024

Tracking copper nanofiller evolution in polysiloxane during processing into SiOC ceramic

Polymer-derived ceramics (PDCs) remain at the forefront of research for a variety of applications including ultra-high-temperature ceramics, energy storage and functional coatings. Despite their wide use, questions remain about the complex structural transition from polymer to ceramic and how local structure influences the final microstructure and resulting properties. This is further complicated when nanofillers are introduced to tailor structural and functional properties, as nanoparticle surfaces can interact with the matrix and influence the resulting structure. The inclusion of crystalline nanofiller produces a mixed crystalline–amorphous composite, which poses characterization challenges. With this study, we aim to address these challenges with a local-scale structural study that probes changes in a polysiloxane matrix with incorporated copper nanofiller. Composites were processed at three unique temperatures to capture mixing, pyrolysis and initial crystallization stages for the pre-ceramic polymer. We observed the evolution of the nanofiller with electron microscopy and applied synchrotron X-ray diffraction with differential pair distribution function (d-PDF) analysis to monitor changes in the matrix's local structure and interactions with the nanofiller. The application of the d-PDF to PDC materials is novel and informs future studies to understand interfacial interactions between nanofiller and matrix throughout PDC processing.

Chemistry↗

DOE OSTI2022

Quantifying radial migration in the Milky Way: inefficient over short time-scales but essential to the very outer disc beyond ∼15 kpc

ABSTRACT Stellar radial migration plays an important role in reshaping a galaxy’s structure and the radial distribution of stellar population properties. In this work, we revisit reported observational evidence for radial migration and quantify its strength using the age–[Fe/H] distribution of stars across the Milky Way with APOGEE data. We find a broken age–[Fe/H] relation in the Galactic disc at r > 6 kpc, with a more pronounced break at larger radii. To quantify the strength of radial migration, we assume stars born at each radius have a unique age and metallicity, and then decompose the metallicity distribution function (MDF) of mono-age young populations into different Gaussian components that originated from various birth radii at rbirth < 13 kpc. We find that, at ages of 2 and 3 Gyr, roughly half the stars were formed within 1 kpc of their present radius, and very few stars (<5 per cent) were formed more than 4 kpc away from their present radius. These results suggest limited short-distance radial migration and inefficient long-distance migration in the Milky Way during the last 3 Gyr. In the very outer disc beyond 15 kpc, the observed age–[Fe/H] distribution is consistent with the prediction of pure radial migration from smaller radii, suggesting a migration origin of the very outer disc. We also estimate intrinsic metallicity gradients at ages of 2 and 3 Gyr of −0.061 and −0.063 dex kpc−1, respectively.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2012

Extending the International Space Station Life and Operability

The International Space Station (ISS) is in an operational configuration with final assembly complete. To fully utilize ISS and extend the operational life, it became necessary to upgrade and extend the onboard systems with the Obsolescence Driven Avionics Redesign (ODAR) project. ODAR enabled a joint project between the Johnson Space Center (JSC) and Marshall Space Flight Center (MSFC) focused on upgrading the onboard payload and Ku-Band systems, expanding the voice and video capabilities, and including more modern protocols allowing unprecedented access for payload investigators to their on-orbit payloads. The MSFC Huntsville Operations Support Center (HOSC) was tasked with developing a high-rate enhanced Functionally Distributed Processor (eFDP) to handle 300Mbps Return Link data, double the legacy rate, and incorporate a Line Outage Recorder (LOR). The eFDP also provides a 25Mbps uplink transmission rate with a Space Link Extension (SLE) interface. HOSC also updated the Payload Data Services System (PDSS) to incorporate the latest Consultative Committee for Space Data Systems (CCSDS) protocols, most notably the use of the Internet Protocol (IP) Encapsulation, in addition to the legacy capabilities. The Central Command Processor was also updated to interact with the new onboard and ground capabilities of Mission Control Center -- Houston (MCC-H) for the uplink functionality. The architecture, implementation, and lessons learned, including integration and incorporation of Commercial Off The Shelf (COTS) hardware and software into the operational mission of the ISS, is described herein. The applicability of this new technology provides new benefits to ISS payload users and ensures better utilization of the ISS by the science community

Cecil, Andrew J.↗

DOE OSTI2021

Confronting lattice parton distributions with global QCD analysis

We present the first Monte Carlo based global QCD analysis of spin-averaged and spin-dependent parton distribution functions (PDFs) that includes nucleon isovector matrix elements in coordinate space from lattice QCD. We investigate the degree of universality of the extracted PDFs when the lattice and experimental data are treated under the same conditions within the Bayesian likelihood analysis. For the unpolarized sector, we find rather weak constraints from the current lattice data on the phenomenological PDFs, and difficulties in describing the lattice matrix elements at large spatial distances. In contrast, for the polarized PDFs we find good agreement between experiment and lattice data, with the latter providing significant constraints on the spin-dependent isovector quark and antiquark distributions.

Sato, Nobuo↗

DOE OSTI2020

The chemical properties of the Milky Way’s on-bar and off-bar regions: evidence for inhomogeneous star formation history in the bulge

ABSTRACT Numerous studies of integrated starlight, stellar counts, and kinematics have confirmed that the Milky Way is a barred galaxy. However, far fewer studies have investigated the bar’s stellar population properties, which carry valuable independent information regarding the bar’s formation history. Here, we conduct a detailed analysis of chemical abundance distributions ([Fe/H] and [Mg/Fe]) in the on-bar and off-bar regions to study the azimuthal variation of star formation history (SFH) in the inner Galaxy. We find that the on-bar and off-bar stars at Galactocentric radii 3 kpc < rGC < 5 kpc have remarkably consistent [Fe/H] and [Mg/Fe] distribution functions and [Mg/Fe]–[Fe/H] relation, suggesting a common SFH shared by the long bar and the disc. In contrast, the bar and disc at smaller radii (2 kpc < rGC < 3 kpc) show noticeable differences, with relatively more very metal-rich ($\rm [Fe/H] \sim 0.4$) stars but fewer solar abundance stars in the bar. Given the three-phase star formation history proposed for the inner Galaxy in Lian et al., these differences could be explained by the off-bar disc having experienced either a faster early quenching process or recent metal-poor gas accretion. Vertical variations of the abundance distributions at small rGC suggest a wider vertical distribution of low-α stars in the bar, which may serve as chemical evidence for vertical heating through the bar buckling process. The lack of such vertical variations outside the bulge may then suggest a lack of vertical heating in the long bar.

Lian, Jianhui↗

DOE OSTI2021

Modelling the stellar halo with RR-Lyrae stars

ABSTRACT A seven-parameter distribution function (DF) is fitted to $20\, 000$ RR-Lyrae stars for which only astrometric data are available. The observational data are predicted by the DF in conjunction with the gravitational potential of a self-consistent model Galaxy defined by DFs for the dark halo, the bulge, and a four-component disc. Tests of the technique developed to deal with missing line-of-sight velocities show that adding such velocities tightens constraints on the DF only slightly. The recovered model of the RR-Lyrae population confirms that the population is flattened and has a strongly radially biased velocity distribution. At large radii, its density profile tends to ρ ∼ r−4.5 but no power law provides a good fit inside the solar sphere. The model is shown to provide an excellent fit to the data for stars brighter than r = 16.5 but at certain longitudes it predicts too few faint stars at Galactocentric radii $\sim 20\, \mathrm{kpc}$, possibly signalling that the halo is not axisymmetric. The DF is used to predict the velocity distribution of BHB stars for which space velocities are available. The z components are predicted successfully but too much anisotropy in the vRvϕ plane is expected.

Li, Chengdong↗

NASA NTRS1976

Lossy radial diffusion of relativistic Jovian electrons

The radial diffusion equation with synchrotron losses is solved by the Laplace-transform method for near equatorially mirroring relativistic electrons. The evolution of a power-law distribution function is found, and the characteristics of synchrotron burnoff are stated in terms of explicit parameters for an arbitrary diffusion coefficient of a specific form. The peaking of the 10.4-cm volume emissivity from Jupiter at an L shell of about 1.8 provides an estimate of the diffusion coefficient in the radiation belts; one value is suggested as the appropriate modification, for an equatorial field strength of 4.2 G, of the Birmingham et al. (1974) result. Nonsynchrotron losses are included phenomenologically; from the phase-space densities reported by McIlwain and Fillius (1975), the particle lifetime is estimated. Asymptotic forms for the distribution in the strong synchrotron loss regime are provided.

Barbosa, D. D.↗

NASA NTRS1993

An alternative representation of the ice canopy for calculating microwave brightness temperatures over a thunderstorm

Passive microwave brightness temperatures (TBs) at 92 and 183 GHz from an aircraft thunderstorm overflight are compared with values calculated from radar-derived hydrometeor profiles and a modified proximity sounding. Two methods for modeling particles in the ice canopy are contrasted. The first is a 'traditional' approach employing Marshall-Palmer ice spheres. The second, or 'alternative', method partitions 20 percent of the ice water content into a Marshall-Palmer component for graupel and hail, and 80 percent into a modified gamma spherical particle size distribution function representing ice crystals. Results from the alternative approach are superior to those from the traditional method in the anvil and mature convective core. In the decaying convective region, the traditional approach yields better agreement with observed magnitudes. Neither method, however, matches the geometry of the observed TB depression associated with the decaying convective core. This is likely due to the presence of graupel, which is not detected as a special signature in radar reflectivity, but does diminish TBs through scattering. Brightness temperatures at the relatively high microwave frequencies considered are shown to be very sensitive to the ice-particle size distribution.

Muller, Bradley M.↗

DOE OSTI2023

Spatiotemporal Studies of Soluble Inorganic Nanostructures with X‐rays and Neutrons

Abstract This Review addresses the use of X‐ray and neutron scattering as well as X‐ray absorption to describe how inorganic nanostructured materials assemble, evolve, and function in solution. We first provide an overview of techniques and instrumentation (both large user facilities and benchtop). We review recent studies of soluble inorganic nanostructure assembly, covering the disciplines of materials synthesis, processes in nature, nuclear materials, and the widely applicable fundamental processes of hydrophobic interactions and ion pairing. Reviewed studies cover size regimes and length scales ranging from sub‐Ångström (coordination chemistry and ion pairing) to several nanometers (molecular clusters, i.e. polyoxometalates, polyoxocations, and metal‐organic polyhedra), to the mesoscale (supramolecular assembly processes). Reviewed studies predominantly exploit 1) SAXS/WAXS/SANS (small‐ and wide‐angle X‐ray or neutron scattering), 2) PDF (pair‐distribution function analysis of X‐ray total scattering), and 3) XANES and EXAFS (X‐ray absorption near‐edge structure and extended X‐ray absorption fine structure, respectively). While the scattering techniques provide structural information, X‐ray absorption yields the oxidation state in addition to the local coordination. Our goal for this Review is to provide information and inspiration for the inorganic/materials science communities that may benefit from elucidating the role of solution speciation in natural and synthetic processes.

Yin, Jia‐Fu↗

DOE OSTI2020

Measurement of single-diffractive dijet production in proton-proton collisions at $\sqrt{s} =$ 8 TeV with the CMS and TOTEM experiments

Measurements are presented of the single-diffractive dijet cross section and the diffractive cross section as a function of the proton fractional momentum loss $\xi $ and the four-momentum transfer squared t. Both processes ${\text{ p }{}{}} {\text{ p }{}{}} \rightarrow {\text{ p }{}{}} {\text{ X }} $ and ${\text{ p }{}{}} {\text{ p }{}{}} \rightarrow {\text{ X }} {\text{ p }{}{}} $, i.e. with the proton scattering to either side of the interaction point, are measured, where ${\text{ X }} $ includes at least two jets; the results of the two processes are averaged. The analyses are based on data collected simultaneously with the CMS and TOTEM detectors at the LHC in proton–proton collisions at $\sqrt{s} = 8\,\text {Te}\text {V} $ during a dedicated run with $\beta ^{*} = 90\,\text {m} $ at low instantaneous luminosity and correspond to an integrated luminosity of $37.5{\,\text {nb}^{-1}} $. The single-diffractive dijet cross section $\sigma ^{{\text{ p }{}{}} {\text{ X }}}_{\mathrm {jj}}$, in the kinematic region $\xi < 0.1$, $0.03< |t | < 1\,\text {Ge}\text {V} ^2$, with at least two jets with transverse momentum $p_{\mathrm {T}} > 40\,\text {Ge}\text {V} $, and pseudorapidity $|\eta | < 4.4$, is $21.7 \pm 0.9\,\text {(stat)} \,^{+3.0}_{-3.3}\,\text {(syst)} \pm 0.9\,\text {(lumi)} \,\text {nb} $. The ratio of the single-diffractive to inclusive dijet yields, normalised per unit of $\xi $, is presented as a function of x, the longitudinal momentum fraction of the proton carried by the struck parton. The ratio in the kinematic region defined above, for x values in the range $-2.9 \le \log _{10} x \le -1.6$, is $R = (\sigma ^{{\text{ p }{}{}} {\text{ X }}}_{\mathrm {jj}}/\Delta \xi )/\sigma _{\mathrm {jj}} = 0.025 \pm 0.001\,\text {(stat)} \pm 0.003\,\text {(syst)} $, where $\sigma ^{{\text{ p }{}{}} {\text{ X }}}_{\mathrm {jj}}$ and $\sigma _{\mathrm {jj}}$ are the single-diffractive and inclusive dijet cross sections, respectively. The results are compared with predictions from models of diffractive and nondiffractive interactions. Monte Carlo predictions based on the HERA diffractive parton distribution functions agree well with the data when corrected for the effect of soft rescattering between the spectator partons.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1983

Interpreting the solar wind ionization state

The ionization state of the solar coronal expansion is frozen within a few solar radii of the solar photosphere, and spacecraft measurements of the solar wind heavy ion charge state can therefore yield information about coronal conditions (e.g., electron temperature). Previous interpretations of the frozen-in ionization state have always assumed that in the coronal freezing-in region, (1) all heavy ions flow at the same bulk speed as protons, (2) the electron velocity distribution function is Maxwellian, and (3) conditions vary in space but not in time. The consequences of relaxing these assumptions for the interpretation of solar wind charge state measurements are examined. It is found that: (1) the temperature inferred by traditional interpretation of the interplanetary ionization state overestimates (underestimate) the actual coronal electron temperature if higher ion charge stages flow systematically faster (slower) than lower stages at the coronal freezing radius; (2) temperatures inferred from relative abundance measurements of ion-charge-stages with high ionization potentials moderately overestimate the actual coronal electron temperature if the high-energy tail of the coronal electron velocity distribution is enhanced relative to a Maxwellian distribution; (3) the propagation of a disturbance, e.g., a shock wave, through the corona can strongly affect the frozen-in charge state, but only over a time (a few times ten minutes) corresponding to the coronal transit time for the disturbance.

Owocki, S. P.↗

DOE OSTI2020

Minimal Experimental Bias on the Hydrogen Bond Greatly Improves Ab Initio Molecular Dynamics Simulations of Water

Experiment Directed Simulations (EDS) is a method within a class of techniques seeking to improve molecular simulations by minimally biasing the system Hamiltonian to reproduce certain experimental observables. In a previous application of EDS to ab initio molecular dynamics (AIMD) simulation based on electronic density functional theory (DFT), the AIMD simulations of water were biased to reproduce its experimentally derived solvation structure. In particular, by solely biasing the O-O pair correlation functions, other structural and dynamical properties that were not biased were improved. In this work, the hypothesis is tested that directly biasing the OH pair correlation (and hence the H-O∙∙∙H hydrogen bonding), will provide an even better improvement of DFT-based water properties in AIMD simulations. The logic behind this hypothesis is that for most electronic DFT descriptions of water the hydrogen bonding is known to be deficient due to anomalous charge transfer and over polarization in the DFT. Using recent advances to the EDS learning algorithm, we thus train a minimal bias on AIMD water that reproduces the O-H radial distribution function derived from the highly accurate MB-pol model of water. Finally, it is then confirmed that biasing the O-H pair correlation alone can lead to improved AIMD water properties, with structural and dynamical properties in even closer to experiment than the previous EDS-AIMD model.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗