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

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At least 1,135 records · Page 63
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Coupling solidification kinetics with phase-behavior computations in hydrodynamic simulations of high-pressure, dynamic-compression processes

In this study, we report a numerical scheme to integrate models for the kinetics of solidification processes together with phase-behavior computations in the context of continuum-scale hydrodynamic simulations. The objective of the phase-behavior computations is to determine the pressure and temperature, given the following three sets of inputs: (1) an appropriate equation of state to describe our system, (2) the phase fraction(s) produced by the kinetic models, (3) and the volume and internal energy obtained by solving the conservation equations that govern the hydrodynamic behavior. The kinetics are assumed to be governed by the Kolmogorov–Johnson–Mehl–Avrami equation, and the nucleation and growth rates that enter into that equation are functions of the pressure and temperature produced by the phase-behavior computations. Our formulation allows for the fluid and solid phases to be at different temperatures (thermal nonequilibrium) and pressures (arising from surface-tension-induced Laplace contributions). The formulation is presented in a fairly general setting that is independent of any particular material, although we demonstrate it in some examples related to high-energy-density science applications where materials are rapidly compressed to pressures exceeding several gigapascals in less than a microsecond. We conclude with a critical evaluation of our approach and provide suggestions for future work to improve the predictive capabilities and generality of the models.

42 ENGINEERING↗
NASA NTRS2018

Satellite Observations and Modeling to Understand the Lower Mekong River Basin Streamflow Variability

In this work, we have used the Soil & Water Assessment Tool (SWAT) to examine streamflow variability of the Lower Mekong River Basin (LMRB) associated with changes in the Upper Mekong River Basin (UMRB) inflows. Two hypothetical experiments were formulated and evaluated for the LMRB, where we conducted runoff simulations with multiple inflow changes that include upstream runoff yield increase and decrease scenarios. Streamflow variability of the LMRB was quantified by two streamflow metrics that explain flow variability and predictability, and high flow disturbance. The model experiments were performed for the Lower Mekong River Basin with identical climate, soil, and other watershed characteristics data. Remote sensing precipitation (Tropical Rainfall Measurement Mission, TRMM, and Global Precipitation Measurement mission, GPM), meteorological data as well as spatial data that include a digital elevation model, newly developed soil information (Harmonized World Soil Database, HWSD), and land use and land cover were processed as input to the LMRB model simulations. Observed daily streamflow data along the Lower Mekong River from Chiang Sean, Thailand to Kratie, Cambodia were used for calibration and validation. Our work results suggest that the Lower Mekong River streamflow is highly variable and has a low predictability (Colwell index of about 32%). We found that releasing more water from upstream Mekong during rainfall months by 30% would result in a reduction in the Lower Mekong streamflow predictability by about 21%. This reduction in predictability is mainly attributed to a decrease in the Contingency index. Our work shows that the ability to predict floods/droughts at the Lower Mekong River would be reduced if there is any anticipated change (i.e., increase/decrease) from UMRB releases. Our results also show that releasing more flows from the upstream Mekong would also affect flood duration and the frequency of flood occurrences downstream. The results of this work thus help to quantify the sensitivity of streamflow variability at the Lower Mekong River Basin to upstream anthropogenic changes.

Streamflow Predictability↗
DOE OSTI2018

Toward supplying food, energy, and water demand: Integrated solar desalination process synthesis with power and hydrogen coproduction

The increase in population coupled with rising per capita income and associated change in consumption habits will put unprecedented stress on food, energy and water (FEW) resources. Sustainable and reliable fresh water supply is central for life and also for all sectors that support our existence. Uncertainty on water security prompted interest in investigation of renewable energy driven desalination processes. One particularly promising option is to produce fresh water from the two most abundant resources on earth: solar energy and seawater. In this study, using Solar Electricity, Water, Food and Chemical (SEWFAC) process synthesis concept, we explore and identify synergistic integration alternatives of multi stage flash desalination, solar thermal power, and hydrogen production processes. The promising options have been analyzed by detailed process simulation and optimization using an integrated Aspen Plus and MATLAB modeling environment. The proposed process designs can meet the water and electricity demand with rather high conversion efficiencies. Furthermore, integration of solar hydrogen production and hydrogen-fired power plant can enable continuous production of fresh water and electricity in solar-rich water-poor regions. In addition to other metrics, we have evaluated the performance of the desalination process from power point of view with a new metric, Electricity Equivalent Water (EEW) to demonstrate the marginal energy penalty of desalination. Integration of thermal desalination processes with electricity and hydrogen production is a synergistic alliance and can play a pivotal role in approaching FEW nexus.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2017

Integrated Computational Materials Engineering Development of Advanced High Strength Steel for Lightweight Vehicles

The goal of the ICME 3GAHSS project was to successfully demonstrate the applicability of Integrated Computational Materials Engineering (ICME) for the development and deployment of third generation advanced high strength steels (3GAHSS) for immediate weight reduction in passenger vehicles. The ICME approach integrated results from well-established computational and experimental methodologies to develop a suite of material constitutive models (deformation and failure), manufacturing process and performance simulation modules, a properties database, as well as the computational environment linking them together for both performance prediction and material optimization. This is the Final Report for the ICME 3GAHSS project, which achieved the fol-lowing objectives: 1) Developed a 3GAHSS ICME model, which includes atomistic, crystal plasticity, state variable and forming models. The 3GAHSS model was implemented in commercially available LS-DYNA and a user guide was developed to facilitate use of the model. 2) Developed and produced two 3GAHSS alloys using two different chemistries and manufacturing processes, for use in calibrating and validating the 3GAHSS ICME Model. 3) Optimized the design of an automotive subassembly by substituting 3GAHSS for AHSS yielding a design that met or exceeded all baseline performance requirements with a 30% mass savings. A technical cost model was also developed to estimate the cost per pound of weight saved when substituting 3GAHSS for AHSS. The project demonstrated the potential for 3GAHSS to achieve up to 30% weight savings in an automotive structure at a cost penalty of up to $0.32 to $1.26 per pound of weight saved. The 3GAHSS ICME Model enables the user to design 3GAHSS to desired mechanical properties in terms of strength and ductility.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Quantifying soil carbon accumulation in Alaskan terrestrial ecosystems during the last 15 000 years

Abstract. Northern high latitudes contain large amounts of soil organic carbon (SOC), of which Alaskan terrestrial ecosystems account for a substantial proportion. In this study, the SOC accumulation in Alaskan terrestrial ecosystems over the last 15 000 years was simulated using a process-based biogeochemistry model for both peatland and non-peatland ecosystems. Comparable with the previous estimates of 25–70 Pg C in peatland and 13–22 Pg C in non-peatland soils within 1 m depth in Alaska using peat-core data, our model estimated a total SOC of 36–63 Pg C at present, including 27–48 Pg C in peatland soils and 9–15 Pg C in non-peatland soils. Current vegetation stored 2.5–3.7 Pg C in Alaska, with 0.3–0.6 Pg C in peatlands and 2.2–3.1 Pg C in non-peatlands. The simulated average rate of peat C accumulation was 2.3 Tg C yr−1, with a peak value of 5.1 Tg C yr−1 during the Holocene Thermal Maximum (HTM) in the early Holocene, 4-fold higher than the average rate of 1.4 Tg C yr−1 over the rest of the Holocene. The SOC accumulation slowed down, or even ceased, during the neoglacial climate cooling after the mid-Holocene, but increased again in the 20th century. The model-estimated peat depths ranged from 1.1 to 2.7 m, similar to the field-based estimate of 2.29 m for the region. We found that the changes in vegetation and their distributions were the main factors in determining the spatial variations of SOC accumulation during different time periods. Warmer summer temperature and stronger radiation seasonality, along with higher precipitation in the HTM and the 20th century, might have resulted in the extensive peatland expansion and carbon accumulation.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2006

An Overview of NASA's IM&S Verification and Validation Process Plan and Specification for Space Exploration

NASA's Exploration Systems Mission Directorate (ESMD) is implementing a management approach for modeling and simulation (M&S) that will provide decision-makers information on the model's fidelity, credibility, and quality. This information will allow the decision-maker to understand the risks involved in using a model's results in the decision-making process. This presentation will discuss NASA's approach for verification and validation (V&V) of its models or simulations supporting space exploration. This presentation will describe NASA's V&V process and the associated M&S verification and validation (V&V) activities required to support the decision-making process. The M&S V&V Plan and V&V Report templates for ESMD will also be illustrated.

Gravitz, Robert M.↗
DOE OSTI2023

Exploring the formation of gold/silver nanoalloys with gas-phase synthesis and machine-learning assisted simulations

While nanoalloys are of paramount scientific and practical interest, the main processes leading to their formation are still poorly understood. Key structural features in the alloy systems, including the crystal phase, chemical ordering, and morphology, are challenging to control at the nanoscale, making it difficult to extend their use to industrial applications. Here, in this contribution, we focus on the gold/silver system that has two of the most prevalent noble metals and combine experiments with simulations to uncover the formation mechanisms at the atomic level. Nanoparticles were produced using a state-of-the-art inert-gas aggregation source and analyzed using transmission electron microscopy and energy-dispersive X-ray spectroscopy. Machine-learning-assisted molecular dynamics simulations were employed to model the crystallization process from liquid droplets to nanocrystals. Our study finds a preponderance of nanoparticles with five-fold symmetric morphology, including icosahedra and decahedra which is consistent with previous results on mono-metallic nanoparticles. However, we observed that gold atoms, rather than silver atoms, segregate at the surface of the obtained nanoparticles for all the considered alloy compositions. These segregation tendencies are in contrast to previous studies and have consequences on the crystallization dynamics and the subsequent crystal ordering. We finally showed that the underpinning of this surprising segregation dynamics is due to charge transfer and electrostatic interactions rather than surface energy considerations.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Learning the solution operator of parametric partial differential equations with physics-informed DeepONets

Partial differential equations (PDEs) play a central role in the mathematical analysis and modeling of complex dynamic processes across all corners of science and engineering. Their solution often requires laborious analytical or computational tools, associated with a cost that is markedly amplified when different scenarios need to be investigated, for example, corresponding to different initial or boundary conditions, different inputs, etc. In this work, we introduce physics-informed DeepONets, a deep learning framework for learning the solution operator of arbitrary PDEs, even in the absence of any paired input-output training data. We illustrate the effectiveness of the proposed framework in rapidly predicting the solution of various types of parametric PDEs up to three orders of magnitude faster compared to conventional PDE solvers, setting a previously unexplored paradigm for modeling and simulation of nonlinear and nonequilibrium processes in science and engineering.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2012

Exploring the Use of Computer Simulations in Unraveling Research and Development Governance Problems

Understanding Research and Development (R&D) enterprise relationships and processes at a governance level is not a simple task, but valuable decision-making insight and evaluation capabilities can be gained from their exploration through computer simulations. This paper discusses current Modeling and Simulation (M&S) methods, addressing their applicability to R&D enterprise governance. Specifically, the authors analyze advantages and disadvantages of the four methodologies used most often by M&S practitioners: System Dynamics (SO), Discrete Event Simulation (DES), Agent Based Modeling (ABM), and formal Analytic Methods (AM) for modeling systems at the governance level. Moreover, the paper describes nesting models using a multi-method approach. Guidance is provided to those seeking to employ modeling techniques in an R&D enterprise for the purposes of understanding enterprise governance. Further, an example is modeled and explored for potential insight. The paper concludes with recommendations regarding opportunities for concentration of future work in modeling and simulating R&D governance relationships and processes.

Balaban, Mariusz A.↗
DOE OSTI2016

TOUGH3: A new efficient version of the TOUGH suite of multiphase flow and transport simulators

The TOUGH suite of nonisothermal multiphase flow and transport simulators has been updated by various developers over many years to address a vast range of challenging subsurface problems. The increasing complexity of the simulated processes as well as the growing size of model domains that need to be handled call for an improvement in the simulator's computational robustness and efficiency. Moreover, modifications have been frequently introduced independently, resulting in multiple versions of TOUGH that (1) led to inconsistencies in feature implementation and usage, (2) made code maintenance and development inefficient, and (3) caused confusion to users and developers. TOUGH3—a new base version of TOUGH—addresses these issues. It consolidates both the serial (TOUGH2 V2.1) and parallel (TOUGH2-MP V2.0) implementations, enabling simulations to be performed on desktop computers and supercomputers using a single code. New PETSc parallel linear solvers are added to the existing serial solvers of TOUGH2 and the Aztec solver used in TOUGH2-MP. The PETSc solvers generally perform better than the Aztec solvers in parallel and the internal TOUGH3 linear solver in serial. TOUGH3 also incorporates many new features, addresses bugs, and improves the flexibility of data handling. So due to the improved capabilities and usability, TOUGH3 is more robust and efficient for solving tough and computationally demanding problems in diverse scientific and practical applications related to subsurface flow modeling.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

An HPC-Based Hydrothermal Finite Element Simulator for Modeling Underground Response to Community-Scale Geothermal Energy Production

Geothermal heat, as renewable energy, shows great advantage with respect to its environmental impact due to its significantly lower CO2 emissions than conventional fossil fuel. Open and closed-loop geothermal heat pumps, which utilize shallow geothermal systems, are an efficient technology for cooling and heating buildings, especially in urban areas. Integrated use of geothermal energy technologies for district heating, cooling, and thermal energy storage can be applied to optimize the subsurface for communities to provide them with multiple sustainable energy and community resilience benefits. The utilization of the subsurface resources may lead to a variation in the underground environment, which might further impact local environmental conditions. However, very few simulators can handle such a highly complex set of coupled computations on a regional or city scale. We have developed high-performance computing (HPC) based hydrothermal finite element (FE) simulator that can simulate the subsurface and its hydrothermal conditions at a scale of tens of km. The HPC simulator enables us to investigate the subsurface thermal and hydrologic response to the built underground environment (such as basements and subways) at the community scale. In this study, a coupled hydrothermal simulator is developed based on the open-source finite element library deal.II. The HPC simulator was validated by comparing the results of a benchmark case study against COMSOL Multiphysics, in which Aquifer Thermal Energy Storage (ATES) is modeled and a process of heat injection into ATES is simulated. The use of an energy pile system at the Treasure Island redevelopment site (San Francisco, CA, USA) was selected as a case study to demonstrate the HPC capability of the developed simulator. The simulator is capable of modeling multiple city-scale geothermal scenarios in a reasonable amount of time.

Sun, Xiang↗
DOE OSTI2018

One- and Two-Equation Models to Simulate Ion Transport in Charged Porous Electrodes

Energy storage in porous capacitor materials, capacitive deionization (CDI) for water desalination, capacitive energy generation, geophysical applications, and removal of heavy ions from wastewater streams are some examples of processes where understanding of ionic transport processes in charged porous media is very important. In this work, one- and two-equation models are derived to simulate ionic transport processes in heterogeneous porous media comprising two different pore sizes. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without Faradaic reactions or specific adsorption of ions. A two-step volume averaging technique is used to derive the averaged transport equations for multi-ionic systems without any further assumptions, such as thin electrical double layers or Donnan equilibrium. A comparison between both models is presented. The effective transport parameters for isotropic porous media are calculated by solving the corresponding closure problems. An approximate analytical procedure is proposed to solve the closure problems. Numerical and theoretical calculations show that the approximate analytical procedure yields adequate solutions. Lastly, a theoretical analysis shows that the value of interphase pseudo-transport coefficients determines which model to use.

25 ENERGY STORAGE↗
DOE OSTI2025

2D reactive transport model of shale chemical weathering and biogeochemical fluxes along a mountainous hillslope, East River Watershed, Colorado: Input files and simulation results

This data package contains input files and simulation results for a two-dimensional (2D) reactive transport model used to quantitatively analyze the coupled hydrological and biogeochemical processes governing shale weathering and associated biogeochemical fluxes under realistic environmental conditions in the high-elevation East River Watershed. These data support the conclusions presented in Stolze et al. (Water Resources Research, under review), "Model-based interpretation of solute exports and carbon partitioning during shale weathering in a mountainous hillslope". The model simulates atmospheric-subsurface gas exchange, subsurface water flow, and shale weathering processes under dynamic, year-scale conditions along a shale-underlain hillslope located in the East River watershed. The simulations were performed using the PFLOTRAN flow and reactive transport code and executed on the Perlmutter supercomputer to leverage its large-scale parallel computing capabilities. The data package contains two zipped folders, "model_input_files" and "simulation_results", and one readme.txt file. "model_input_files" contains the necessary input files to run the calibrated base-base model presented in Stolze et al. (Water Resources Research, under review). "simulation_results" contains a single hdf5 file ("Output_2D_hillslope_model.h5") which includes the results of simulation performed using the base-case model. This file can be opened with HDFView 3.1.4, Python, or MATLAB. "readme.txt" contains relevant information about the base-case model and provides guidelines on how to run the associated input files provided in the folder "model_input_files". Furthermore, readme.txt provides information regarding the model results provided in "Output_2D_hillslope_model.h5" such as matrix dimensionality and output units. Field datasets used to evaluate model performance were collected at three monitoring wells located along a hillslope transect (PLM1, PLM2, and PLM3). Dissolved ion concentration data were collected from November 2016 to October 2021 for Ca, Mg, DIC, Na, K, SO4 (Dong et al., 2025 - dic_npoc_data_2014_2024.zip - DOI:10.15485/1660459; Williams et al., 2025 - anion_data_2014_2024.zip - DOI:10.15485/1668054; Dong et al., 2025 - cation_data_2014_2024.zip - DOI:10.15485/1668055). Note that we used the files named er_PLM1_xx_yy, er_PLM2_xx_yy, and er_PLM3_xx_yy where xx stands for the name of the aqueous species and yy stands for the depth where the measurements were performed. Soil water content ([0 - 1] m) and water table depth were collected from November 2016 to October 2021 (Wan et al., 2024 - Dynamic_water_table__depthsFig2b.csv and Soil_water_content_Fig4e.csv - DOI:10.15485/2322567). Gaseous CO2 concentration were collected from October 2020 to December 2021(Wan et al., 2024 - Soil_CO2_concentrations_Fig4h.csv - DOI:10.15485/2322567) Gaseous CO2 flux from the subsurface to the atmosphere were collected in the vicinity of PLM2 from October 2019 to May 2022 (Wu et al., 2025). Soil microbial biomass concentration was measured from August 2016 to June 2017 (Sorensen et al., 2019 - 2017_East_River_Pumphouse_Microbial_Biomass__1_.csv - DOI:10.15485/1577267) All field data are published as CSV files compatible with Microsoft Excel, MATLAB, and Python, or as text files. The coordinates of the monitoring wells and the CO2(g) flux sensor in the coordinate system WGS84 are: -PLM1: [38.9197710 ; -106.9492750] -PLM2: [38.9201580 ; -106.9487170] -PLM3: [38.9207843 ; -106.9483668] -PLM4: 38.9210060 ; -106.9479528] -CO2(g) flux sensor: [38.9199180 ; -106.9489906] ------------------------------------------------------------------------------------------- This work was supported by the Watershed Function Science Focus Area at Lawrence Berkeley National Laboratory funded by the US Department of Energy, Office of Science, Biological and Environmental Research under Contract No. DE-AC02-05CH11231. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a Department of Energy User Facility using NERSC award BER-ERCAP 23980, BER-ERCAP 28550, and BER-ERCAP 33789.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2002

Modeling the Stability of Volatile Deposits in Lunar Cold Traps

There are several mechanisms acting at the cold traps that can alter the inventory of volatiles there. Primarily, the lunar surface is bombarded by meteoroids which impact, melt, process, and redistribute the regolith. Further, solar wind and magnetospheric ion fluxes are allowed limited access onto the regions in permanent shadow. Also, although cold traps are in the permanent shadow of the Sun, there is a small flux of radiation incident on the regions from interstellar sources. We investigate the effects of these space weathering processes on a deposit of volatiles in a lunar cold trap through simulations. We simulate the development of a column of material near the surface of the Moon resulting from space weathering. This simulation treats a column of material at a lunar cold trap and focuses on the hydrogen content of the column. We model space weathering processes on several time and spatial scales to simulate the constant rain of micrometeoroids as well as sporadic larger impactors occurring near the cold traps to determine the retention efficiency of the cold traps. We perform the Monte Carlo simulation over many columns of material to determine the expectation value for hydrogen content of the top few meters of soil for comparison with Lunar Prospector neutron data.

Crider, D. H.↗
DOE OSTI2025

CFD modeling of near-wall combustion and unburned methane prediction in natural gas spark ignition engines

Natural gas-powered engines play a critical role in gas drilling, compression, and transmission sectors, but methane (CH 4 ) from engine combustion slip can be significant over their lifespan, contributing to atmospheric pollution and signaling reduced engine efficiency. Here, to address this challenge, computational fluid dynamics (CFD) simulations offer valuable insights into the in-cylinder combustion process, enabling the optimization of combustion strategies and engine designs to minimize unburned CH 4 slip. This study aims to evaluate and improve combustion models for simulating the combustion process and predicting unburned CH 4 concentrations in natural gas spark-ignition (SI) engines, including engines that are part of combined reformer-engine systems. Specifically, the performance of two flamelet-based combustion models—the Extended Coherent Flame Model (ECFM) and the G-equation model—was assessed using experimental engine data collected under varying excess-air ratio (λ) conditions and fuel compositions, including natural gas and syngas blends. In addition, to enhance the predictive capabilities of the G-equation model, a flame-wall interaction (FWI) sub-model was integrated into its framework. The effects of its model parameters, such as quenching and influence distance, on combustion behavior and unburned methane predictions were analyzed in detail. The ECFM tended to predict delayed combustion phasing under diluted mixture conditions, resulting in overprediction of unburned CH 4 concentrations. In contrast, the G-equation model provided reasonable predictions of combustion pressure, while representing higher the CH 4 reduction rate across the operating condition compared to experimental data. Incorporating the FWI sub-model—with the quenching distance calculated based on a pressure-dependent relation (P -0.48 ) and a fixed influence distance of 1.5 mm—further improved the G-equation model’s accuracy in predicting CH 4 reduction rates without compromising its ability to simulate the combustion process.

Combustion model↗
DOE OSTI2022

Dimensional stability of low-cost thermoplastic composite molds

This report addresses a key question for using additively manufactured tooling from carbon fiber reinforced thermoplastic composites. How does the distortion of the molded part change with the print orientation used in the manufacture of tooling? The objective of this research was to discuss the distortion of parts molded with such tooling, in relation to the deflections in the tooling during the compression molding process. This was achieved through both model simulations and experimental molding runs. A vertical orientation was used in printing the mold with a distinctly non-planar surface: this orientation allowed for higher performance along the press closing direction -- higher thermal conductivity, higher stiffness, and lower coefficient of thermal expansion. Epoxy-carbon fiber twill weave fabric prepreg was then compression molded in this mold. The thermomechanical properties of the tooling material and the molded part were measured and used in simulation of mold deformation as well as part distortion. The thermomechanical anisotropy of the mold is quite different from that of the molded part because in the mold, the stiffest direction is the z-axis, while in the molded thermoset part, the z-axis is the weakest direction. The connection between mold deformation during the compression molding process and the final part distortion can be seen from the simulation results. The simulation results for the case where the mold was four times as stiff as the part along the press closing direction compared well with experiment. When the mold stiffness was lowered in relation to the part stiffness, the mold deformation during the compression molding increased; but this led to a smaller extent of part distortion.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Molecular dynamics simulation of gold cluster growth during sputter deposition

We present a molecular dynamics simulation scheme that we apply to study the time evolution of the self-organized growth process of metal cluster assemblies formed by sputter-deposited gold atoms on a planar surface. The simulation model incorporates the characteristics of the plasma-assisted deposition process and allows for an investigation over a wide range of deposition parameters. It is used to obtain data for the cluster properties which can directly be compared with recently published experimental data for gold on polystyrene [M. Schwartzkopf et al., ACS Appl. Mater. Interfaces 7, 13547 (2015)]. While good agreement is found between the two, the simulations additionally provide valuable time-dependent real-space data of the surface morphology, some of whose details are hidden in the reciprocal-space scattering images that were used for the experimental analysis.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗