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At least 775 records · Page 43

SU-G-201-02: Application of RayStretch in Clinical Cases: A Calculation for Heterogeneity Corrections in LDR Permanent I-125 Prostate Brachytherapy

Purpose: Tissue heterogeneities and calcifications have significant impact on the dosimetry of low energy brachytherapy (BT). RayStretch is an analytical algorithm developed in our institution to incorporate heterogeneity corrections in LDR prostate brachytherapy. The aim of this work is to study its application in clinical cases by comparing its predictions with the results obtained with TG-43 and Monte Carlo (MC) simulations. Methods: A clinical implant (71 I-125 seeds, 15 needles) from a real patient was considered. On this patient, different volumes with calcifications were considered. Its properties were evaluated in three ways by i) the Treatment planning system (TPS) (TG-43), ii) a MC study using the Penelope2009 code, and iii) RayStretch. To analyse the performance of RayStretch, calcifications located in the prostate lobules covering 11% of the total prostate volume and larger calcifications located in the lobules and underneath the urethra for a total occupied volume of 30% were considered. Three mass densities (1.05, 1.20, and 1.35 g/cm3) were explored for the calcifications. Therefore, 6 different scenarios ranging from small low density calcifications to large high density ones have been discussed. Results: DVH and D90 results given by RayStretch agree within 1% with the full MC simulations. Although no effort has been done to improve RayStretch numerical performance, its present implementation is able to evaluate a clinical implant in a few seconds to the same level of accuracy as a detailed MC calculation. Conclusion: RayStretch is a robust method for heterogeneity corrections in prostate BT supported on TG-43 data. Its compatibility with commercial TPSs and its high calculation speed makes it feasible for use in clinical settings for improving treatment quality. It will allow in a second phase of this project, its use during intraoperative ultrasound planning. This study was partly supported by a fellowship grant from the Spanish Ministry of Education, by the Generalitat Valenciana under Project PROMETEOII/2013/010, by the Spanish Government under Project No. FIS2013-42156 and by the European Commission within the SeventhFramework Program through ENTERVISION (grant agreement number 264552).

60 APPLIED LIFE SCIENCES↗

Reaction of amorphous/crystalline SiOC/Fe interfaces by thermal annealing

The development of revolutionary new alloys and composites is crucial to meeting materials requirements for next generation nuclear reactors. The newly developed amorphous silicon oxycarbide (SiOC) and crystalline Fe composite system has shown radiation tolerance over a wide range of temperatures. To advance understanding of this new composite, we investigate the structure and thermal stability of the interface between amorphous SiOC and crystalline Fe by combining various experimental techniques and simulation methods. We show that the SiOC/Fe interface is thermally stable up to at least 400 °C. When the annealing temperature reaches 600 °C, an intermixed region forms at this interface. This region appears to be a crystalline phase that forms an incoherent interface with the Fe layer. Density functional theory (DFT) Molecular dynamics (MD) is performed on the homogeneous SiFeOC phase to study the early stages of 2 formation of the intermixed layer. Both experimental and simulation results suggest this phase has the fayalite crystal structure. As a result, the physical processes involved in the formation of the intermixed region are discussed.

36 MATERIALS SCIENCE↗

Excavator Design Validation

The Excavator Design Validation tool verifies excavator designs by automatically generating control systems and modeling their performance in an accurate simulation of their expected environment. Part of this software design includes interfacing with human operations that can be included in simulation-based studies and validation. This is essential for assessing productivity, versatility, and reliability. This software combines automatic control system generation from CAD (computer-aided design) models, rapid validation of complex mechanism designs, and detailed models of the environment including soil, dust, temperature, remote supervision, and communication latency to create a system of high value. Unique algorithms have been created for controlling and simulating complex robotic mechanisms automatically from just a CAD description. These algorithms are implemented as a commercial cross-platform C++ software toolkit that is configurable using the Extensible Markup Language (XML). The algorithms work with virtually any mobile robotic mechanisms using module descriptions that adhere to the XML standard. In addition, high-fidelity, real-time physics-based simulation algorithms have also been developed that include models of internal forces and the forces produced when a mechanism interacts with the outside world. This capability is combined with an innovative organization for simulation algorithms, new regolith simulation methods, and a unique control and study architecture to make powerful tools with the potential to transform the way NASA verifies and compares excavator designs. Energid's Actin software has been leveraged for this design validation. The architecture includes parametric and Monte Carlo studies tailored for validation of excavator designs and their control by remote human operators. It also includes the ability to interface with third-party software and human-input devices. Two types of simulation models have been adapted: high-fidelity discrete element models and fast analytical models. By using the first to establish parameters for the second, a system has been created that can be executed in real time, or faster than real time, on a desktop PC. This allows Monte Carlo simulations to be performed on a computer platform available to all researchers, and it allows human interaction to be included in a real-time simulation process. Metrics on excavator performance are established that work with the simulation architecture. Both static and dynamic metrics are included.

Pholsiri, Chalongrath↗

Imaging Wavepackets in Real-Space & Disentangling Ultrafast Solvation Dynamics with High-Energy Ultrafast X-ray Scattering

The microscopic information of solute and solute-solvent motions can be measured using ultrafast diffuse x-ray scattering, and compared to molecular dynamics simulations, as demonstrated in studies of solvated molecular systems done in X-ray free-electron lasers. However, for the typical photon energies used in these experiments, the photon momentum transfer range was limited to lower Q values where the signals arising from different parts of the studied system overlap significantly. Recently, this limitation was greatly alleviated by extending the photon energy at the Linac Coherent Light-Source (LCLS), significantly increasing the accessible momentum transfer range. This improvement is transformative for ultrafast diffuse x-ray scattering measurements, enabling for the first time to disentangle the solute scattering difference signal, that persists to higher Q values, from the bulk solvent and solute-solvent cross-terms difference signals. In addition, it enhances the prediction ability of modeling and simulation methods such as the hybrid QM/MM approach. The extended Q range also opens the way to resolve in real-space details regarding coherent wavepackets motions beyond their center-of-mass positions. Here, we present the first results on high-energy (18keV) diffuse x-ray scattering in solution, demonstrating high fidelity time-resolved scattering and analysis of the photoexcited model photocatalysts PtPOP ([Pt2(POP)4]4-), and IrDimen ([Ir2(dimen)4]2+) in several solvents. These complexes provide ideal systems for demonstrating the ability of high-Q ultrafast scattering and QM/MM simulations to decompose ultrafast solvation dynamics into specific changes in the solute-solvent pair distribution function with a particular focus on how electronically excited states change the interaction between the solvent and photo-catalytically active metal sites. We introduce an approach to further utilize the high-energy capability and demonstrate a single-shot ultra-wide-angle X-ray Scattering modality using two perpendicular detectors, spanning a scattering angle range of more than 100 degrees, allowing to extend the accessible momentum transfer range up to Q~14 Å-1. We develop a model-free method to invert the x-ray scattering signals and enable the recovery of multiple pair-density motions that happen simultaneously, allowing us to directly measure in real-space nuclear wavepacket motions. . [1] Natan, Adi. "Real-Space Inversion and Super-Resolution of Ultrafast X-ray Scattering using Natural Scattering Kernels." arXiv preprint arXiv:2107.05576 (2021)

Natan, Adi↗

Study of vibrational modes in protonic conductor LaAlO{sub 3}

In this work, we are presenting the results of our theoretical investigation on the vibrational properties of LaAlO{sub 3} in its cubic phase by using lattice dynamical simulation method based on de Launey angular force (DAF) constant model to understand the role of phonon in this system. The calculated zone centre frequencies agree well with available existing results. The phonon dispersion curves of LaAlO{sub 3} in cubic phase are also drawn.

36 MATERIALS SCIENCE↗

3D Simulation of External Flooding Events for the RISMC Pathway

Incorporating 3D simulations as part of the Risk-Informed Safety Margins Characterization (RISMIC) Toolkit allows analysts to obtain a more complete picture of complex system behavior for events including external plant hazards. External events such as flooding have become more important recently – however these can be analyzed with existing and validated simulated physics toolkits. In this report, we describe these approaches specific to flooding-based analysis using an approach called Smoothed Particle Hydrodynamics. The theory, validation, and example applications of the 3D flooding simulation are described. Integrating these 3D simulation methods into computational risk analysis provides a spatial/visual aspect to the design, improves the realism of results, and can prove visual understanding to validate the analysis of flooding.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Review: Pre-Darcy flows in low-permeability porous media

The widely used Darcy’s law specifies a linear relation between the Darcy velocity of fluid flow and the pressure gradient that drives the flow. However, studies have shown that Darcy velocity can exhibit a nonlinear dependence on the pressure gradient in low-permeability porous media such as clay and shale when the pressure gradient is adequately low. This phenomenon is referred to as low-velocity non-Darcian flow or pre-Darcy flow. This paper provides a comprehensive review of the theories, experimental data, and modeling methods for pre-Darcy flow in low-permeability porous media. The review begins by outlining the fundamental mechanisms underlying pre-Darcy flow that regulate the unique characteristics such as nonlinear dependence of the Darcy velocity on the pressure gradient and its relevance to fluid–rock interactions. The review then proceeds to present a thorough compilation of experimental investigations performed in various low-permeability geomaterials including tight sandstones, shales, and clays. Next, empirical and theoretical models and simulation methods that have been developed to fit and interpret experimental data are reviewed. Finally, the review underscores the challenges encountered in conducting and interpreting pre-Darcy flow experiments and suggests future research directions. By analyzing previous experimental investigations, this review aims to offer a valuable resource for researchers and practitioners seeking to enhance their understanding of fluid dynamics in low-permeability geomaterials. This provides insights into the application of pre-Darcy flow in numerous natural and engineered processes such as shale oil and gas recovery, contaminant transport in low-permeability aquifers, and geological disposal of nuclear waste.

58 GEOSCIENCES↗

Two-Phase Fluid Flow Properties of Rough Fractures With Heterogeneous Wettability: Analysis With Lattice Boltzmann Simulations

Fractures are conduits for fluid flow in low-permeability geological formations. Multiphase flow properties of fractures are important in natural processes and in engineering applications such as the evaluation of the sealing capacity of caprocks and productivity of hydrocarbon-bearing tight rocks. Investigations of flow and transport through fractures typically focus on the effects of fracture geometric and mechanical factors such as aperture, roughness, and compressibility. The wettability of the fracture surfaces and its influence on microscale interfacial phenomena and macroscale effective transport properties are seldom studied. In this study, we investigated the effect of heterogeneous wetting properties on the displacement of water by supercritical CO2 through a series of lattice Boltzmann method simulations. The results show the evolution of the CO2 plume within a fracture is controlled by both the roughness of the aperture field and the wetting distribution. We combined these factors into a capillary pressure map that can be related to the macroscopic flow behavior of the fracture. We observed that heterogeneous wetting distributions promote the residual trapping of water where lower capillary pressures allowed for isolated water pockets in higher capillary pressure zones. Analysis of fracture unsteady relative permeability shows the effect of wetting on permeability evolution and provides support for the viscous-coupling relative permeability model. Finally, analysis of the steady-state relative permeability and saturation demonstrates a strong correlation between permeability and the standard deviation of the capillary pressure field. Thus, characterizing the distribution of wetting properties of fractures is crucial to understanding multiphase fracture flow and transport properties.

58 GEOSCIENCES↗

Laser-excited optical emission response of CdTe quantum dot/polymer nanocomposite under shock compression

Laser-driven shock compression experiments and corresponding finite element method simulations are carried out to investigate the blueshift in the optical emission spectra under continuous laser excitation of a dilute composite consisting of 0.15% CdTe quantum dots by weight embedded in polyvinyl alcohol polymer. This material is a potential candidate for use as internal stress sensors. The analyses focus on the time histories of the wavelength blue-shift for shock loading with pressures up to 7.3 GPa. The combined measurements and calculations allow a relation between the wavelength blueshift and pressure for the loading conditions to be extracted. It is found that the blueshift first increases with pressure to a maximum and subsequently decreases with pressure. This trend is different from the monotonic increase of blueshift with pressure observed under conditions of quasistatic hydrostatic compression. Additionally, the blueshift in the shock experiments is much smaller than that in hydrostatic experiments at the same pressure levels. The differences in responses are attributed to the different stress states achieved in the shock and hydrostatic experiments and the time dependence of the mechanical response of the polymer in the composite. The findings offer a potential guide for the design and development of materials for internal stress sensors for shock conditions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Accelerating dissipative particle dynamics simulations for soft matter systems

Dissipative particle dynamics (DPD) is a coarse-grained particle-based simulation method that offers microscopic-scale insights into soft matter systems. We present an efficient implementation of a DPD model for graphical processing units (GPUs). As implemented in the LAMMPS molecular dynamics package, it can run effectively on current-generation supercomputers which often have hybrid nodes containing multi-core CPUs and (one or more) GPUs. Using efficient communication of information between the CPUs and GPUs, DPD interactions can be computed on the GPU while other portions of a full simulation model (boundary conditions, constraints, bonded interactions, diagnostic calculations, etc.) can be performed on the CPU. Our GPU-enhanced runs show a speedup of up to 9.5× versus many-core CPU simulations, and can run scalably across thousands of compute nodes. We briefly discuss how the new GPU implementation was validated against the CPU version for thermodynamics, diffusion, and hydrodynamic behavior. We also highlight large-scale models which the faster DPD implementation has enabled, for studies of monolayer self-assembly and thin-film instabilities.

36 MATERIALS SCIENCE↗

Strain relaxation of GaAs/Ge crystals on patterned Si substrates

We report on the mask-less integration of GaAs crystals several microns in size on patterned Si substrates by metal organic vapor phase epitaxy. The lattice parameter mismatch is bridged by first growing 2-μm-tall intermediate Ge mesas on 8-μm-tall Si pillars by low-energy plasma enhanced chemical vapor deposition. We investigate the morphological evolution of the GaAs crystals towards full pyramids exhibiting energetically stable (111) facets with decreasing Si pillar size. The release of the strain induced by the mismatch of thermal expansion coefficients in the GaAs crystals has been studied by X-ray diffraction and photoluminescence measurements. The strain release mechanism is discussed within the framework of linear elasticity theory by Finite Element Method simulations, based on realistic geometries extracted from scanning electron microscopy images.

36 MATERIALS SCIENCE↗

Study of Air Ingress Across the Duct During the Accident Conditions

The goal of this project is to study the fundamental physical phenomena associated with air ingress in very high temperature reactors (VHTRs). Air ingress may occur due to a rupture of primary piping and a subsequent breach in the primary pressure boundary in helium-cooled and graphite-moderated VHTRs. Significant air ingress is a concern because it introduces potential to expose the fuel, graphite support rods, and core to a risk of severe graphite oxidation. Two of the most probable air ingress scenarios involve rupture of a control rod or fuel access standpipe, and rupture in the main coolant pipe on the lower part of the reactor pressure vessel. Therefore, establishing a fundamental understanding of air ingress phenomena is critical in order to rationally evaluate safety of existing VHTRs and develop new designs that minimize these risks. But despite this importance, progress toward development these predictive capabilities has been slowed by the complex nature of the underlying phenomena. The combination of inter-diffusion among multiple species, molecular diffusion, natural convection, and complex geometries, as well as the multiple chemical reactions involved, impose significant roadblocks to both modeling and experiment design. The project team will employ a coordinated experimental and computational effort that will help gain a deeper understanding of multiphased air ingress phenomena. This project will enhance advanced modeling and simulation methods, enabling calculation of nuclear power plant transients and accident scenarios with a high degree of confidence. The following are the project tasks: Perform particle image velocimetry measurement of multiphase air ingresses; and, Perform computational fluid dynamics analysis of air ingress phenomena.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Chemical transformations and transport phenomena at interfaces

Abstract Interfaces, the boundary that separates two or more chemical compositions and/or phases of matter, alters basic chemical and physical properties including the thermodynamics of selectivity, transition states, and pathways of chemical reactions, nucleation events and phase growth, and kinetic barriers and mechanisms for mass transport and heat transport. While progress has been made in advancing more interface‐sensitive experimental approaches, their interpretation requires new theoretical methods and models that in turn can further elaborate on the microscopic physics that make interfacial chemistry so unique compared to the bulk phase. In this review, we describe some of the most recent theoretical efforts in modeling interfaces, and what has been learned about the transport and chemical transformations that occur at the air–liquid and solid–liquid interfaces. This article is categorized under: Structure and Mechanism > Reaction Mechanisms and Catalysis Structure and Mechanism > Computational Materials Science Software > Quantum Chemistry Software > Simulation Methods

Hao, Hongxia↗

GaAs/Ge crystals grown on Si substrates patterned down to the micron scale

Monolithic integration of III-V compounds into high density Si integrated circuits is a key technological challenge for the next generation of optoelectronic devices. In this work, we report on the metal organic vapor phase epitaxy growth of strain-free GaAs crystals on Si substrates patterned down to the micron scale. The differences in thermal expansion coefficient and lattice parameter are adapted by a 2-μm-thick intermediate Ge layer grown by low-energy plasma enhanced chemical vapor deposition. The GaAs crystals evolve during growth towards a pyramidal shape, with lateral facets composed of (111) planes and an apex formed by (137) and (001) surfaces. The influence of the anisotropic GaAs growth kinetics on the final morphology is highlighted by means of scanning and transmission electron microscopy measurements. The effect of the Si pattern geometry, substrate orientation, and crystal aspect ratio on the GaAs structural properties was investigated by means of high resolution X-ray diffraction. The thermal strain relaxation process of GaAs crystals with different aspect ratio is discussed within the framework of linear elasticity theory by Finite Element Method simulations based on realistic geometries extracted from cross-sectional scanning electron microscopy images.

36 MATERIALS SCIENCE↗

The influence of kinematics of deformation on polycrystalline halite dynamic recrystallization: Full-field simulation of simple shear versus pure shear

Rock salt, composed mainly of halite, flows viscoplastically over a wide range of geological conditions, strongly impacting the dynamic evolution of sedimentary basins and orogens. Understanding how dislocation creep, which involves dislocation glide, intracrystalline recovery and dynamic recrystallization, influences the microstructure and rheology of halite under various deformation kinematics and temperatures is crucial for enhancing knowledge of salt flow dynamics. Here, this study employs a full-field numerical simulation method to compare the viscoplastic deformation of polycrystalline halite under simple shear and pure shear conditions up to a natural strain of ε = 1.5 at temperatures ranging from 100 °C to 300 °C. The results are presented in terms of crystallographic preferred orientation (CPO), grain shape preferred orientation (SPO), subgrain boundary direction, grain size and strain rate distribution. The results indicate that the crystallographic anisotropy of individual halite crystals is transferred to the polycrystalline scale, resulting in strain localization, particularly in simple shear simulations. The kinematics of deformation affect the evolution and distribution of high strain-rate bands, determining the direction of intragranular substructures and the morphology of strain-induced grain boundaries, with minimal impact on grain size. The intensity of grain boundary migration increases with temperature, significantly influencing grain morphology and size, thereby obscuring strain localization, while having little effect on CPOs. At low strain (ε < 1.0), CPOs relative to both the maximum shortening direction and the grain SPO are similar regardless of the deformation kinematics. At high strain (ε > 1.0), simple shear CPOs exhibit three stronger {100} maxima with a monoclinic symmetry relative to the grain SPO compared to the six {100} maxima with an orthotropic symmetry relative to the grain SPO generated under pure shear. Therefore, microstructures and CPOs can serve as indicators of the strain path in polycrystalline halite under various conditions, aiding in determining the shear sense and elucidating the deformation kinematics of salt structures.

58 GEOSCIENCES↗

Monte Carlo simulation based study of a proposed multileaf collimator for a telecobalt machine

Purpose: The objective of the present work was to propose a design of a secondary multileaf collimator (MLC) for a telecobalt machine and optimize its design features through Monte Carlo simulation. Methods: The proposed MLC design consists of 72 leaves (36 leaf pairs) with additional jaws perpendicular to leaf motion having the capability of shaping a maximum square field size of 35 Multiplication-Sign 35 cm{sup 2}. The projected widths at isocenter of each of the central 34 leaf pairs and 2 peripheral leaf pairs are 10 and 5 mm, respectively. The ends of the leaves and the x-jaws were optimized to obtain acceptable values of dosimetric and leakage parameters. Monte Carlo N-Particle code was used for generating beam profiles and depth dose curves and estimating the leakage radiation through the MLC. A water phantom of dimension 50 Multiplication-Sign 50 Multiplication-Sign 40 cm{sup 3} with an array of voxels (4 Multiplication-Sign 0.3 Multiplication-Sign 0.6 cm{sup 3}= 0.72 cm{sup 3}) was used for the study of dosimetric and leakage characteristics of the MLC. Output files generated for beam profiles were exported to the PTW radiation field analyzer software through locally developed software for analysis of beam profiles in order to evaluate radiation field width, beam flatness, symmetry, and beam penumbra. Results: The optimized version of the MLC can define radiation fields of up to 35 Multiplication-Sign 35 cm{sup 2} within the prescribed tolerance values of 2 mm. The flatness and symmetry were found to be well within the acceptable tolerance value of 3%. The penumbra for a 10 Multiplication-Sign 10 cm{sup 2} field size is 10.7 mm which is less than the generally acceptable value of 12 mm for a telecobalt machine. The maximum and average radiation leakage through the MLC were found to be 0.74% and 0.41% which are well below the International Electrotechnical Commission recommended tolerance values of 2% and 0.75%, respectively. The maximum leakage through the leaf ends in closed condition was observed to be 8.6% which is less than the values reported for other MLCs designed for medical linear accelerators. Conclusions: It is concluded that dosimetric parameters and the leakage radiation of the optimized secondary MLC design are well below their recommended tolerance values. The optimized design of the proposed MLC can be integrated into a telecobalt machine by replacing the existing adjustable secondary collimator for conformal radiotherapy treatment of cancer patients.

60 APPLIED LIFE SCIENCES↗

Effects of nanoconfinement and surface charge on iron adsorption on mesoporous silica

Here, we present a combined molecular dynamics (MD) simulation and X-ray absorption fine structure (XAFS) spectroscopic investigation of aqueous iron adsorption on nanoconfined amorphous silica surfaces. The simulation models examine the effects of pore size, pH (surface charge), iron valency, and counter-ion (chloride or hydroxide). The simulation methods were validated by comparing the coordination environment of adsorbed iron with coordination numbers and bond lengths derived from XAFS. In the MD models, nanoconfinement effects on local iron coordination were investigated by comparing results for unconfined silica surfaces and in confined domains within 2 nm, 4 nm, and 8 nm pores. Experimentally, coordination environments of iron adsorbed onto mesoporous silica with 4 nm and 8 nm pores at pH 7.5 were investigated. The effect of pH in the MD models was included by simulating Fe(II) adsorption onto negatively charged SiO 2 surfaces and Fe(III) adsorption on neutral surfaces. The simulation results show that iron adsorption depends significantly on silica surface charge, as expected based on electrostatic interactions. Adsorption on a negatively charged surface is an order of magnitude greater than on the neutral surface, and simulated surface coverages are consistent with experimental results. Pore size effects from the MD simulations were most notable in the adsorption of Fe(II) at deprotonated surface sites (SiO - ), but adsorption trends varied with concentration and aqueous Fe speciation. The coordination environment of adsorbed iron varied significantly with the type of anion. Considerable ion pairing with hydroxide anions led to the formation of oligomeric surface complexes and aqueous species, resulting in larger iron hydroxide clusters at higher surface loadings.

54 ENVIRONMENTAL SCIENCES↗

Elucidating size effects on the yield strength of single-crystal Cu via the Richtmyer–Meshkov instability

Capturing the dynamic response of a material under high strain-rate deformation often demands challenging and time consuming experimental effort. While shock hydrodynamic simulation methods can aid in this area, a priori characterizations of the material strength under shock loading and spall failure are needed in order to parameterize constitutive models needed for these computational tools. Moreover, parameterizations of strain-rate-dependent strength models are needed to capture the full suite of Richtmyer–Meshkov instability (RMI) behavior of shock compressed metals, creating an unrealistic demand for these training data solely on experiments. Herein, we sweep a large range of geometric, crystallographic, and shock conditions within molecular dynamics (MD) simulations and demonstrate the breadth of RMI in Cu that can be captured from the atomic scale. In this work, yield strength measurements from jetted and arrested material from a sinusoidal surface perturbation were quantified as Y RMI = 0.787 ± 0.374 GPa, higher than strain-rate-independent models used in experimentally matched hydrodynamic simulations. Defect-free, single-crystal Cu samples used in MD will overestimate Y RMI , but the drastic scale difference between experiment and MD is highlighted by high confidence neighborhood clustering predictions of RMI characterizations, yielding incorrect classifications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗