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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 613 records · Page 34
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Multi-Functional Distributed Fiber Sensors for Pipeline Monitoring and Methane Detections. Final Report

As an abundant and cheap fossil energy source, natural gas has become a significant energy supply to support the United States’ economy. However, the large-scale extraction and utilization of natural gas also impose significant challenges on methane leakage. This problem is exacerbated by aging gas utility delivery systems, including interstate high-pressure pipelines, storage, and transmission facilities. This project aims to develop a cost-effective fiber optical sensing method that can perform multi-parameter real-time measurements of natural gas pipelines across long interrogation distances up to 100 km with 1-meter spatial resolution. This sensing tool can evaluate overall pipeline efficiency and reduce methane emissions for mid-stream methane infrastructures. To accomplish this objective, research and development efforts funded by this project have resulted in the following accomplishments: This project successfully has developed new functional sensory polymer materials using Metal-Organic Frameworks (MOFs) that can be coated on optical fiber through the reel-to-reel coating process. Functional polymer-coated optical fibers can perform sensitive methane detection through evanescence wave interaction and strain-based measurements to achieve 1% detection sensitivities. The new sensors fibers support both distributed measurements and multiplexed fiber sensors array for multi-point measurements. This project developed and optimized a new multi-core optical fiber that supports simultaneous and distributed measurements of strain and temperatures with 1-meter spatial resolutions across up to 100-km interrogation distance. This new fiber, combined with sensory polymercoated fiber, could perform both distributed temperature and methane detections. This project developed a new artificial intelligence big data algorithm approach that can effectively analyze high-resolution data harnessed by distributed fiber sensors to protect natural gas pipelines against external threats and detect internal defects induced by corrosion. Working with our industry partner, this project developed new optical fibers that support fiber sensor fabrications through polymer coating after the fibers are drawn. These new fibers eliminate the need for direct sensor fabrication when the fiber is fabricated on a fiber draw tower, which drastically expands fiber sensors' applicability. This research project has significantly advanced the distributed fiber sensing technology. It will dramatically increase the applicability and adaptability of distributed fiber sensors for a wide array of applications in energy, sustainability, and environmental science, including structural health monitoring of natural gas pipelines, oil infrastructures, hydrogen facilities, and environmental monitoring of carbon storage sites, water supply systems, and others.

03 NATURAL GAS↗
DOE OSTI2021

Analysis of Approximations in Modeling of BWR Bundle Void Distributions

In boiling water reactors, complex heterogeneous bundle designs, control blades adjacent to the corner of bundles, and the presence of boiling can lead to complex internal void distributions. A few approximations exist to model these void distributions. They could be modeled using a 1D axial solver in which each axial node is assumed to be at an average void, or each pin cell could be modeled with its own void concentration. In the latter case, the void could be discretized in pin-centered or coolant-centered channels. The goal of this project was to quantify the effect of using the different approximations for modeling internal void distributions on neutronics calculations. Using 3D void distributions calculated with CTF, Monte Carlo Neutral Particle (MCNP) transport code models were created for GE-9 and GE-14 lattices. For each model, the internal void distribution from CTF at a given axial node was selected, and a lattice calculation was carried out with MCNP. Comparisons between models using a lattice-averaged void, or using a void distribution in coolant-centered channels, showed large differences in reactivity which in some cases were well above 1,000 pcm, and it also showed differences in normalized fission rates greater than 20%. It was also found that using a lattice average void can lead to a significant difference in the estimation of the worth of a control blade. The differences found when comparing results from models using pin-centered and coolant-centered channels were up to 200 pcm in reactivity and up to 1.4% in the normalized fission rates. In addition to these two sets of comparisons, MCNP models were set up so that each subchannel had a saturated liquid component around the fuel pins and a saturated vapor component in the center to approximate annular flow. In comparison to the models using coolant-centered subchannels, up to 1–3% differences in normalized fission rates could be found.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2023

Out-of-distribution generalization for learning quantum dynamics

Generalization bounds are a critical tool to assess the training data requirements of Quantum Machine Learning (QML). Recent work has established guarantees for in-distribution generalization of quantum neural networks (QNNs), where training and testing data are drawn from the same data distribution. However, there are currently no results on out-of-distribution generalization in QML, where we require a trained model to perform well even on data drawn from a different distribution to the training distribution. Here, we prove out-of-distribution generalization for the task of learning an unknown unitary. In particular, we show that one can learn the action of a unitary on entangled states having trained only product states. Since product states can be prepared using only single-qubit gates, this advances the prospects of learning quantum dynamics on near term quantum hardware, and further opens up new methods for both the classical and quantum compilation of quantum circuits.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Test and Validate Distributed Coaxial Cable Sensors for in situ Condition Monitoring of Coal-Fired Boiler Tubes

This project aims to test, validate, and advance the technology readiness level (from TRL5 to TRL7) of a novel low-cost distributed stainless-steel/ceramic coaxial cable sensing (SSC-CCS) technology for in situ monitoring of the boiler tube temperature in existing coal-fired power plants. The novel SSC-CCS sensing technology and associated condition-based monitoring (CBM) software to be demonstrated in this project will lead to an improved understanding of the boiler tube failure mechanisms and a prognostic system to improve the overall performance, reliability, and flexibility of the nation’s coal-fired power plant fleet. A boiler tube monitoring system with distributed coaxial cable temperature sensors and a sensor acquisition system was constructed. The high-temperature coaxial cable sensor with a length of 1.3m was made by using a quartz tube (1mm inner diameter (ID) and 6mm outer diameter (OD)) to concentrically separate a 304 stainless-steel (SS) rod (1mm OD) and SS tube (7.94mm OD and 6.16mm ID). The sensor acquisition system includes a vector network analyzer (VNA), a radio frequency (RF) power amplifier, multiple switches and a USB hub. The distributed stainless-steel quartz coaxial cable sensor (SSQ-CCS) had a linear response to temperature with a resolution uncertainty of σ = 0.77℃. To withstand the harsh conditions of 3,300 steam pressures and 800℃ high temperatures, the sensor was shielded by a protective tube made of the same material as the boiler tube. The protection tube had an OD of 1.5 inches and a thickness of 0.25 inches. In the laboratory tests, the sensor showed good sensitivity and fast response. The drift was bounded between +0.33% and -0.67% during a test at 600℃ for 350 hours, indicating good stability of the sensor. A field test was conducted where four sensors were welded on four superheat tubes (SH-Ts) at a coal-fired power station over 400 days. Conventional thermocouples were welded to the superheater tubes alongside the coaxial cable sensors for the purpose of comparison. Two sensors were capable of distributed sensing, with three multiplexed sensing sections. The other two sensors were single section. During the 400-day test period, the power plant experienced startups and shutdowns. At the steady state operations, the temperature of the boiler tube is about 600℃ (1112°F). The sensors recorded the entire coal-firing processes (start-up, steady state, and shut-down) and the glitch event. A GSM modem and a Watchdog were added to the system to ensure reliable data recording. The GSM modem sent daily messages to plant managers and Clemson team to inform the status of the sensor system. If the system was not normally working, the Watchdog would reboot the system automatically. The new coaxial cable based distributed sensing technology has been proven to be successful in both laboratory and field tests. A comprehensive four-stage multi-physics computational framework has been developed to assist the design, optimization, installation, and operation of SSQ-CCS. With the consideration of various operation conditions, we predict the distributions of flue gas temperatures within coal-fired boilers, the temperature correlation between the boiler tube and SSQ-CCS, and the safety of SSQ-CCS. A conditional-based monitoring system is implemented as well. The computational framework developed in this work can guide the future operation of coal-fired plants and other power plants for the safety prediction of boiler operations.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2025

2 Kelvin helium distribution system for the Electron Ion Collider’s 10 o’clock satellite refrigerator

The Electron-Ion Collider (EIC) at Brookhaven National Laboratory (BNL) will involve superfluid helium cooling of superconducting magnets and Superconducting Radio Frequency (SRF) cavities at several sites around the existing Relativistic Heavy Ion Collider (RHIC) accelerator tunnel. While the majority of the cooling power for these loads is provided by BNL’s central cryogenic plant, Jefferson Lab is designing satellite equipment which augments the central plant and enables 2 Kelvin operation. The 2 K cryogenic distribution system for the collider’s 10 o’clock location (Interaction Region 10 or IR10) includes all necessary interfaces to the IR10 Satellite Refrigerator, to the overall EIC cryogenic distribution system, and to 12 SRF cryomodules for the electron and hadron storage rings. In addition to providing the required cooling capacities in all operating modes, the IR10 2 K cryogenic distribution system also stabilizes the supply temperature and enables safe connection and disconnection of individual IR10 cryomodules. Moreover, the layout of the IR10 2 K cryogenic distribution system copes with challenging spatial constraints and adapts to the process configuration and routing of existing RHIC cryogenic distribution components which will be re-used for EIC. This paper gives a full overview of the IR10 satellite cryogenic distribution system design, and highlights some of the challenges encountered.

Laverdure, Nathaniel [Thomas Jefferson National Ac↗
DOE OSTI2023

Transmission and Distribution Real-Time Analysis Software for Monitoring and Control: Design and Simulation Testing

The US electric grid is facing operational, stability, and security challenges. Transmission system operators need some measure of visibility into distribution system renewable generation. Distribution system generation needs to support transmission system voltage. The grid is experiencing an expansion in measurement systems. How to take full advantage of this expansion and defend against attacks, both cyber and physical, poses additional challenges. This paper introduces software designed to meet these challenges. At the center of the software is an Integrated System Model (ISM) that spans from transmission to secondary distribution. The ISM is employed in real-time abnormality detection, voltage stability forecasting, and multi-mode control. The software architecture along with selected analysis modules is presented. Testing results are presented for: 1—attacks on utility infrastructure; 2—energy savings from optimal control; 3—distribution system control response during a low voltage transmission system event; 4—cyber-attacks on PV inverters, where physical inverters are used in hardware-in-the-simulation-loop studies. Contributions of this work include real-time analysis that spans from three-phase transmission through secondary distribution; an approach for detecting abnormalities that employs measurements from three independent measurement systems; and a multi-mode distribution system control that responds to cyber-attacks, physical attacks, equipment failures, and transmission system needs.

14 SOLAR ENERGY↗
DOE OSTI2023

The Pantheon+ Analysis: Forward Modeling the Dust and Intrinsic Color Distributions of Type Ia Supernovae, and Quantifying Their Impact on Cosmological Inferences

Abstract Recent studies have shown that the observed color distributions of Type Ia supernovae (SNe Ia) can be well described by a combination of a dust distribution and an intrinsic color distribution. Using the Pantheon+ sample of 1701 SN Ia, we apply a new forward-modeling fitting method (Dust2Dust) to measure the parent dust and color distributions, including their dependence on host-galaxy mass. At each fit step, the SN Ia selection efficiency is determined from a large simulated sample that is reweighted to reflect the proposed distributions. We use five separate metrics to describe the goodness of fit: distribution of fitted light-curve color c , cosmological residual trends with c , cosmological residual scatter with c , fitted color–luminosity relationship β SALT2 , and intrinsic scatter σ int . We present the results and the uncertainty in 12-dimensional space. Furthermore, we measure that the uncertainty on this modeling propagates to an upper threshold uncertainty in the equation of state of dark energy w of 0.014(1) for the Pantheon+ cosmology analysis and contributes negligible uncertainty to the Hubble constant H 0 . The Dust2Dust code is made publicly available at https://github.com/djbrout/dustdriver .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

How to Build an Empirical Speed Distribution for Dark Matter in the Solar Neighborhood

The dark matter flux in a direct detection experiment depends on its local speed distribution. This distribution has been inferred from simulations of Milky Way–like galaxies, but such models serve only as proxies, given that no simulation directly captures the detailed evolution of our own Galaxy. This motivates alternative approaches that obtain this distribution directly from observations. In this work, we utilize 98 Milky Way analogues from the TNG50 simulation to develop and validate a procedure for inferring the dark matter speed distribution using the kinematics of nearby stars. We find that the dark matter that originated from old mergers, plus that from recent nonluminous accretions, is well described by a Maxwell–Boltzmann speed distribution centered at the local standard-of-rest velocity. Meanwhile, recently accreted dark matter from massive mergers has speeds that can be traced from the associated stellar debris of these events. The stellar populations systematically underestimate the velocity dispersion of their dark matter counterparts, but a simple kinematic boost brings the two into good alignment. Using the TNG50 host galaxies, we demonstrate that combining these two contributions provides an accurate reconstruction of the local dark matter speeds. As an application of the procedure to our own Galaxy, we utilize stellar kinematic data from Gaia to quantify how the dark matter remnants from the Milky Way’s last major merger impact its speed distribution in the solar neighborhood.

Shpigel, Tal [Princeton Univ., NJ (United States)]↗
DOE OSTI2021

Novel estimation of aerosol processes with particle size distribution measurements: a case study with the TOMAS algorithm v1.0.0

Atmospheric aerosol microphysical processes are a significant source of uncertainty in predicting climate change. Specifically, aerosol nucleation, emissions, and growth rates, which are simulated in chemical transport models to predict the particle size distribution, are not understood well. However, long-term size distribution measurements made at several ground-based sites across Europe implicitly contain information about the processes that created those size distributions. This work aims to extract that information by developing and applying an inverse technique to constrain aerosol emissions as well as nucleation and growth rates based on hourly size distribution measurements. We developed an inverse method based upon process control theory into an online estimation technique to scale aerosol nucleation, emissions, and growth so that the model–measurement bias in three measured aerosol properties exponentially decays. The properties, which are calculated from the measured and predicted size distributions, used to constrain aerosol nucleation, emission, and growth rates are the number of particles with a diameter between 3 and 6 nm, the number with a diameter greater than 10 nm, and the total dry volume of aerosol ( N 3–6 , N 10 , V dry ), respectively. In this paper, we focus on developing and applying the estimation methodology in a zero-dimensional “box” model as a proof of concept before applying it to a three-dimensional simulation in subsequent work. The methodology is first tested on a dataset of synthetic and perfect measurements that span diverse environments in which the true particle emissions, growth, and nucleation rates are known. The inverse technique accurately estimates the aerosol microphysical process rates with an average and maximum error of 2 % and 13 %, respectively. Next, we investigate the effect that measurement noise has on the estimated rates. The method is robust to typical instrument noise in the aerosol properties as there is a negligible increase in the bias of the estimated process rates. Finally, the methodology is applied to long-term datasets of in situ size distribution measurements in western Europe from May 2006 through June 2007. At Melpitz, Germany, and Hyytiälä, Finland, the average diurnal profiles of estimated 3 nm particle formation rates are reasonable, having peaks near noon local time with average peak values of 1 and 0.15 cm –3 s –1 , respectively. The normalized absolute error in estimated N 3–6 , N 10 , and V dry at three European measurement sites is less than 15 %, showing that the estimation framework developed here has potential to decrease model–measurement bias while constraining uncertain aerosol microphysical processes.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Evaluating precipitation distributions at regional scales: a benchmarking framework and application to CMIP5 and 6 models

Abstract. As the resolution of global Earth system models increases, regional-scale evaluations are becoming ever more important. This study presents a framework for quantifying precipitation distributions at regional scales and applies it to evaluate Coupled Model Intercomparison Project (CMIP) 5 and 6 models. We employ the Intergovernmental Panel on Climate Change (IPCC) sixth assessment report (AR6) climate reference regions over land and propose refinements to the oceanic regions based on the homogeneity of precipitation distribution characteristics. The homogeneous regions are identified as heavy-, moderate-, and light-precipitating areas by K-means clustering of Integrated Multi-satellitE Retrievals for Global Precipitation Measurement (GPM) version 6 final run product (IMERG) precipitation frequency and amount distributions. With the global domain partitioned into 62 regions, including 46 land and 16 ocean regions, we apply 10 established precipitation distribution metrics. The collection includes metrics focused on the maximum peak, lower 10th percentile, and upper 90th percentile in precipitation amount and frequency distributions; the similarity between observed and modeled frequency distributions; an unevenness measure based on cumulative amount; average total intensity on all days with precipitation; and number of precipitating days each year. We apply our framework to 25 CMIP5 and 41 CMIP6 models, as well as six observation-based products of daily precipitation. Our results indicate that many CMIP5 and 6 models substantially overestimate the observed light-precipitation amount and frequency, as well as the number of precipitating days, especially over midlatitude regions outside of some land regions in the Americas and Eurasia. Improvement from CMIP5 to 6 is shown in some regions, especially in midlatitude regions, but it is not evident globally, and over the tropics most metrics point toward degradation.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Comparing Measured Driver Behavior Distributions to Results from Car-Following Models using SUMO and Real-World Vehicle Trajectories from Radar: SUMO Default vs. Radar-Measured CF model Parameters

In this study, the physical principles governing car-following (CF) behavior and their impact on traffic flow at signalized intersections are investigated. High temporal-resolution radar data is used to provide valuable insights into actual CF behavior, including acceleration, deceleration, and time headway distribution. Demand-calibrated SUMO simulations are run using empirical CF parameter distributions, and three CF models are evaluated: IDM, EIDM, and Krauss. By emulating radar data in SUMO and processing simulated vehicle traces, discrepancies between empirical and simulated parameter distributions are identified. Further analysis includes comparisons with default SUMO CF model parameters. The findings reveal that measured accelerations differ from CF model parameter accelerations and using the empirical value ($\mu = 0.89m/s^2$) leads to unrealistic simulations that fail volume-based calibration. Default parameters for all three models reasonably approximate the mean and median of measured parameters, but fail to capture the true distribution shape, partly due to homogeneity when using default parameters. The results show that it is more effective to simulate with the default parameters provided by SUMO rather than using measurements of real-world distributions without additional calibration. Future work will investigate closing the loop between the measured real-world and SUMO distributions using traditional calibration tactics, as well as assess the impact of calibrated vs. default CF parameters on simulation outputs like fuel consumption.

Schrader, Max (ORCID:0000000336737672)↗
DOE OSTI2023

Improved Particle Heat Transfer by way of Bimodal Particle Distributions for High Temperature Solar Thermal Energy

High temperature solar thermal facilities are looking to increase operating temperatures through novel heat transfer media, one such being solid particles. These particles operating at high temperatures will require transferring their thermal energy into another working fluid like supercritical carbon dioxide which can be used in advanced power cycles. Achieving high heat transfer between the particles and supercritical carbon dioxide is essential to high efficiency and low-cost operation. Therefore, optimizing the thermal conductivity of these particles is one potential way to ensure high performance. Traditionally, unimodal particle distributions have been employed in high temperature particle solar power plants. However, ambient temperature testing of bimodal particle distributions has revealed a superior thermal conductivity when compared to its unimodal counterpart at the same temperature. This data was obtained by certified, off-the-shelf instruments that can effectively simulate the conditions a particle would be exposed to in a high temperature solar thermal system. Data obtained in this way suggests that the increased thermal conductively imputed by a bimodal particle distribution is significant at working temperatures in solar facilities. Furthermore, the thermal conductivity of these bimodal particle distributions peaks when the best combination of large and small particles is applied. At high temperatures, binary particle distributions are compared to monodispersed distributions of larger particles where heat transfer is more prolific due to the increased surface radiation. Various thermal conductivity, porosity and heat exchanger models are explored in conjunction with data acquired up to 700 C.

Stout, Dallin (ORCID:0009000294586091)↗
NASA NTRS1972

Entropy and chemical change. 1: Characterization of product (and reactant) energy distributions in reactive molecular collisions: Information and enthropy deficiency

Optimal means of characterizing the distribution of product energy states resulting from reactive collisions of molecules with restricted distributions of initial states are considered, along with those for characterizing the particular reactant state distribution which yields a given set of product states at a specified total energy. It is suggested to represent the energy-dependence of global-type results in the form of square-faced bar plots, and of data for specific-type experiments as triangular-faced prismatic plots. The essential parameters defining the internal state distribution are isolated, and the information content of such a distribution is put on a quantitative basis. The relationship between the information content, the surprisal, and the entropy of the continuous distribution is established. The concept of an entropy deficiency, which characterizes the specificity of product state formation, is suggested as a useful measure of the deviance from statistical behavior. The degradation of information by experimental averaging is considered, leading to bounds on the entropy deficiency.

Bernstein, R. B.↗
NASA NTRS1973

Energy distribution functions of kilovolt ions parallel and perpendicular to the magnetic field of a modified Penning discharge

The distribution function of ion energy parallel to the magnetic field of a modified Penning discharge has been measured with a retarding potential energy analyzer. These ions escaped through one of the throats of the magnetic mirror geometry. Simultaneous measurements of the ion energy distribution function perpendicular to the magnetic field have been made with a charge exchange neutral detector. The ion energy distribution functions are approximately Maxwellian, and the parallel and perpendicular kinetic temperatures are equal within experimental error. These results suggest that turbulent processes previously observed in this discharge Maxwellianize the velocity distribution along a radius in velocity space and cause an isotropic energy distribution. When the distributions depart from Maxwellian, they are enhanced above the Maxwellian tail.

Roth, R. J.↗
NASA NTRS1973

Path-length distributions of photons diffusely reflected from a semi-infinite atmosphere.

Knowledge of the probability distribution of photon path lengths for diffuse reflection in the continuum from a planetary atmosphere greatly simplifies computation of the shape and strength of absorption lines formed under conditions of multiple scattering. Graphs of this probability distribution are presented for diffuse reflection from a semiinfinite conservative atmosphere for both isotropic and forward-directed scattering, and for several angles of incidence and reflection. Graphs showing distributions for the case of absorption in the continuum are also discussed. The distributions are obtained from an analytic expression for the path-length distribution for nth-order scattering and a tabulation of nth-order intensities. It is found that te shape of the probability distribution depends markedly on the degree of forward scattering due to the phase function. The results are compared with those obtained by a Monte Carlo computation of the sort which might be used if tabulated nth-order intensities were not available.

Appleby, J. F.↗
NASA NTRS1973

The asymptotic distribution of maxima in bivariate samples

The joint distribution (as n tends to infinity) of the maxima of a sample of n independent observations of a bivariate random variable (X,Y) is studied. A method is developed for deriving the asymptotic distribution of the maxima, assuming that X and Y possess asymptotic extreme-value distributions and that the probability element dF(x,y) can be expanded in a canonical series. Applied both to the bivariate normal distribution and to the bivariate gamma and compound correlated bivariate Poisson distributions, the method shows that maxima from all these distributions are asymptotically uncorrelated.

Campbell, J. W.↗
NASA NTRS1975

Pressure distributions on a cambered wing body configuration at subsonic Mach numbers

An investigation was conducted in the Langley high-speed 7- by 10-foot tunnel at Mach numbers of 0.20 and 0.40 and angles of attack up to about 22 deg to measure the pressure distributions on two cambered-wing configurations. The wings had the same planform (aspect ratio of 2.5 and a leading-edge-sweep angle of 44 deg) but differed in amounts of camber and twist (wing design lift coefficient of 0.35 and 0.70). The effects of wing strake on the wing pressure distributions were also studied. The results indicate that the experimental chordwise pressure distribution agrees reasonably well with the design distribution over the forward 60 percent of nearly all the airfoil sections for the lower cambered wing. The measured lifting pressures are slightly less than the design pressures over the aft part of the airfoil. For the highly cambered wing, there is a significant difference between the experimental and the design pressure level. The experimental distribution, however, is still very similar to the prescribed distribution. At angles of attack above 12 deg, the addition of a wing-fuselage strake results in a significant increase in lifting pressure coefficient at all wing stations outboard of the strake-wing intersection.

Henderson, W. P.↗
NASA NTRS1976

Calculations of the Electron Energy Distribution Function in a Uranium Plasma by Analytic and Monte Carlo Techniques

Electron energy distribution functions were calculated in a U235 plasma at 1 atmosphere for various plasma temperatures and neutron fluxes. The distributions are assumed to be a summation of a high energy tail and a Maxwellian distribution. The sources of energetic electrons considered are the fission-fragment induced ionization of uranium and the electron induced ionization of uranium. The calculation of the high energy tail is reduced to an electron slowing down calculation, from the most energetic source to the energy where the electron is assumed to be incorporated into the Maxwellian distribution. The pertinent collisional processes are electron-electron scattering and electron induced ionization and excitation of uranium. Two distinct methods were employed in the calculation of the distributions. One method is based upon the assumption of continuous slowing and yields a distribution inversely proportional to the stopping power. An iteration scheme is utilized to include the secondary electron avalanche. In the other method, a governing equation is derived without assuming continuous electron slowing. This equation is solved by a Monte Carlo technique.

Bathke, C. G.↗