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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 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Monte Carlo Hauser-Feshbach computer code system to model nuclear reactions: YAHFC

A computer program framework, YAHFC, to model low-energy nuclear reactions is presented. The framework allows for reactions with incident particles ranging from protons/neutrons to alphas and is designed to address reactions that ultimately lead to the formation of compound nuclear systems that then decay statistically as outlined in concepts of Hauser and Feshbach. Additionally, instead of a reaction, it is also possible to model the decay of a nuclear system with an initial excitation and population. The code models nuclear decays with a Monte Carlo process that tracks the decay of each state. This allows for an exact representation of the spectra for all emitted particles in each of the final exit channels and the possibility of generating reaction data for simulation purposes. The program is interfaced with the optical model code system FRESCOX to calculate transmission coefficients as well as the effects of coupled channels and other direct excitations via the distorted wave Born approximation (DWBA). Modules are included to account for nuclear processes such as width corrections, pre-equilibrium emission, and fission. The program is controlled by a series of input commands and while a set of input parameters exists for each projectile and target, the input commands allow for complete control over each input parameter. Extensive data files are produced and a program is provided that converts YAHFC data files into nuclear data library entries in the generalized nuclear data structure (GNDS).

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Identification of Distribution of Elastic Constants of SiC/SiC Braided Tubes

There has been considerable interest in chemical vapor infiltration (CVI) manufactured silicon carbide fiber and silicon carbide matrix (SiCf/SiCm) composite tubes due to their superior mechanical properties. There are many parameters in the manufacturing of SiCf/SiCm composite tubes such as elastic constants of constituent SiC fiber and SiC matrix, braiding angle, porosity, and others whose values vary over a range. A sensitivity analysis will be carried out to determine the dominant parameters among the abovementioned ones. We intend to determine the variability in elastic constants of SiCf/SiCm composite tubes considering the variability in dominant parameters. A chosen number of combinations of dominant variables will be sampled through Latin hypercube sampling (LHS) method. The elastic constants of composite tube namely Young's moduli in circumferential and longitudinal direction, Poisson's ratio and shear modulus will be calculated at sampled points through finite element (FE) analysis. A polynomial response surface (PRS) will be built for each of the elastic constants to be used a surrogate to FE analysis. once the predictive accuracy of PRS is verified at the validation set of data points, the PRS will be invoked 105 times in Monte Carlo simulations (MCS). The variability can be calculated based on MCS-generated data which will be presented at the conference.

42 ENGINEERING↗
DOE OSTI2021

Discrete stochastic model of point defect-dislocation interaction for simulating dislocation climb

Dislocation climb is an important high temperature process in metals plasticity, responsible for the phenomena such as creep, swelling, or hardening. Climb is defined by the ability of dislocations to leave their original glide plane by interacting with point defects. As such, dislocation climb is controlled by point defect diffusion/absorption/emission, all of which involve thermal activation. The existing thermodynamically consistent models for climb are generally formulated in a continuum framework, through the definition of effective defect fluxes and climb propensities in response to thermodynamic driving forces. However, the point-wise discrete nature of vacancies (and/or self-interstitials) confers a highly discrete nature to the climb dynamics, which is also strongly affected by elastic forces. The combination of discreteness, thermal activation, and elasticity makes this process too challenging for direct atomistic methods such as molecular dynamics. Here we develop a kinetic Monte Carlo model that captures vacancy generation and transport kinetics acting in conjuction with the evolving elastic fields provided by discrete dislocation dynamics simulations. The two models are coupled via the applied stresses and stress gradients generated by dislocation structures at vacancy locations. Our simulations reveal two surprising results. First that climb is dominated by vacancy emission even when the background vacancy concentration is much higher than the equilibrium one. And, second, that climb velocities might be much faster than otherwise believed when one uses the classical theories of climb. These effects are due to the locality of vacancy-dislocation processes, which are not captured in classical treatments that assume smooth vacancy fluxes and homogeneous concentrations. We apply the method to study elementary climb processes in body-centered cubic iron and furnish climb mobility functions to be used in parametric dislocation dynamics and/or crystal plasticity simulations. Here, we apply the technique to study non-conservative plastic bypass of spherical precipitates by edge dislocations and point out the differences between our discrete approach and existing continuum formulations.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Particle-in-cell simulations of burning inertial confinement fusion capsule implosions

Anomalies observed in the neutron spectral shift of high-yield shots at the National Ignition Facility (NIF) suggest the presence of suprathermal ions [E. P. Hartouni et al., Nat. Phys. 19, 72 (2023)], implying that kinetic effects play a significant role in burning inertial confinement fusion (ICF) plasmas. Furthermore, recent measurements of reaction-in-flight (RIF) neutrons offer a direct probe of the stopping power in the burning fuel region of high-energy alpha particles and up-scattered fuel ions. We have developed the particle-in-cell code PICNIC, an exactly energy-conserving particle-in-cell Monte-Carlo collision (PIC-MCC) code to simulate the burn stage in ICF. We present results from 1D spherical simulations of NIF shot N210808, which was the first to exceed the Lawson criterion for laser fusion. We find that the suprathermal ions generated by large-angle Rutherford and nuclear elastic scattering (NES) with fusion alphas produce an alpha knock-on neutron (AKN) signal consistent with the extent and relative yield of the AKN spectrum identified in ignition experiments at the NIF. Furthermore, we also find that the inclusion of large-angle scattering physics does not explain the anomalously large spectral shift observed in the experiment.

High-energy-density plasmas↗
DOE OSTI2020

Direct observation of the momentum distribution and renormalization factor in lithium

We have measured the momentum distribution and renormalization factor $Z_{k_F}$ in liquid and solid lithium by high-resolution Compton scattering. High-resolution data over a wide momentum range exhibit a clear feature of the renormalization and a sharp drop of momentum densities at the Fermi momentum $k_F$. In this work, these results are compared with those computed by quantum Monte Carlo simulation performed both on a disordered crystal and a liquid exhibiting very good agreement. Asymptotic behavior of the experimental and theoretical momentum distributions are examined to estimate $Z_{k_F}$. The experimentally obtained $Z_{k_F} = 0.43^{+0.11}_{–0.01}$ for liquid Li and $0.54^{+0.11}_{–0.02}$ for solid Li are in good agreement with theoretical results of 0.54 ± 0.01 and 0.64 ± 0.01, respectively.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2026

Quantitative 14 N NMR with Monte Carlo Uncertainty Analysis of Nitrate/Nitrite in Alkaline Nuclear Waste

While monitoring of nitrate and nitrite concentrations is important for managing corrosion in nuclear waste systems, existing analytical methods are hindered by turbidity, spectral interference, and delays from sample handling. Here, we demonstrate quantitative 14 N nuclear magnetic resonance (qNMR) spectroscopy as a direct, matrix-tolerant approach for nitrate and nitrite detection at natural abundance. Monte Carlo resampling was integrated into the workflow to quantify random error, establish precision–time tradeoffs, and separate noise-limited uncertainty from systematic bias arising from shimming, transmitter offset, or excitation pulse conditions. Quantification of nitrate and nitrite were validated in controlled alkaline matrix challenges and in 18-component Hanford-type simulants. These results establish 14 N qNMR as a practical, uncertainty-bounded tool for monitoring redox-active nitrogen species in chemically complex environments and provide a generalizable framework for quantitative analysis of quadrupolar nuclei.

Graham, Trent R. [Pacific Northwest National Labor↗
DOE OSTI2022

Mach Conference, 2022.

There has been considerable interest in chemical vapor infiltration (CVI) manufactured silicon carbide fiber and silicon carbide matrix (SiC/SiC) composite tubes due to their superior mechanical properties. The SiC/SiC tubes are known to remain stable even after prolonged exposure to radiation. However, there are many variables in the manufacturing of SiC/SiC composite tubes such as elastic constants of constituent SiC fiber and SiC matrix, braiding angle, porosity, and others whose values vary over a range. The intention is to identify the distribution of elastic constants of SiC/SiC tubes which would help model the performance of the novel cladding material. To do so, a sensitivity analysis will be carried out to determine the dominant variables that influence the elastic constants of SiC/SiC tubes. A chosen number of combinations of dominant variables will be sampled through the Latin hypercube sampling (LHS) method. The elastic constants of composite tube namely Young's moduli in the circumferential and longitudinal direction, Poisson's ratio, and shear modulus will be calculated at sampled points through finite element (FE) analysis. A polynomial response surface (PRS) will be built for each of the elastic constants to be used as a surrogate to FE analysis. Once the predictive accuracy of PRS is verified at the validation set of data points, the PRS will be invoked 105 times in Monte Carlo simulations (MCS). The uncertainty can be quantified by calculating the coefficient of variation (CV) based on MCS-generated data which will be presented at the conference.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2025

Machine learning force field model for kinetic Monte Carlo simulations of itinerant Ising magnets

Here, we present a scalable machine learning (ML) framework for large-scale kinetic Monte Carlo (kMC) simulations of itinerant electron Ising systems. As the effective interactions between Ising spins in such itinerant magnets are mediated by conducting electrons, the calculation of energy change due to a local spin update requires solving an electronic structure problem. Such repeated electronic structure calculations could be overwhelmingly prohibitive for large systems. Assuming the locality principle, a convolutional neural network (CNN) model is developed to directly predict the effective local field and the corresponding energy change associated with a given spin update based on Ising configuration in a finite neighborhood. As the kernel size of the CNN is fixed at a constant, the model can be directly scalable to kMC simulations of large lattices. Our approach is reminiscent of the ML force field models widely used in first-principles molecular dynamics simulations. Applying our ML framework to a square-lattice double-exchange Ising model, we uncover unusual coarsening of ferromagnetic domains at low temperatures. Our work highlights the potential of ML methods for large-scale modeling of similar itinerant systems with discrete dynamical variables.

machine learning↗
DOE OSTI2022

On-the-fly response function generation method for composite coarse mesh

The hybrid stochastic deterministic transport code COMET, based on the incident response expansion theory, is used to model reactor cores with high fidelity and formidable computational speed. COMET models a reactor core using a library of incident flux response expansion coefficients that are pre computed for all the unique lattice cells (e.g., fuel assemblies, reflector blocks, etc.) in the core. In order to further improve its computational efficiency in pre-calculating the response library a new response function generation method is developed to compute the response functions for the composite coarse meshes made of a smaller set of unique lattices on the fly within the COMET's deterministic transport core sweep. The efficiency is achieved by eliminating a number of unique lattices that can be made up from the reduced set of unique meshes on the fly. The numerical process consists of the following steps. First, the boundary condition on composite coarse mesh boundaries is projected onto the expansion basis to compute the incident flux moments on external surfaces of all the basic (reduced set of unique) coarse meshes. Secondly, the deterministic sweeping solver in COMET is used to converge on the outgoing/incoming flux expansion moments crossing interfaces between the basic coarse meshes. Thirdly, the response functions for the composite coarse meshes are constructed as a superposition on the fly. The new response function generation method was tested on 88 composite coarse meshes consisting of CANDU fuel bundles and moderator blocks. It was found that response functions generated by the new method agree very well with those generated by direct Monte Carlo calculations. The average and maximum relative differences in the surface-to-surface response coefficients computed by the two methods are 0.10% and 0.20%, respectively. Similarly, the average and maximum relative differences in the response fission densities are 0.13% and 0.43%, respectively. These discrepancies are within one standard deviation of the stochastic uncertainties. The new method is five times faster than the original direct Monte Carlo method. The size of the response function library for the new method is five times smaller than that for the original method, leading to significantly less requirement for the computer hard drive space and memory. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2025

Data-Driven Closures and Assimilation for Stiff Multiscale Random Dynamics

Here, we introduce a data-driven and physics-informed framework for propagating uncertainty in stiff, multiscale random ordinary differential equations (RODEs) driven by correlated (colored) noise. Unlike systems subjected to Gaussian white noise, a deterministic equation for the joint probability density function (PDF) of RODE state variables does not exist in closed form. Moreover, such an equation would require as many phase-space variables as there are states in the RODE system. To alleviate this curse of dimensionality, we instead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations for low-dimensional observables/quantities of interest. The unclosed terms take the form of state-dependent conditional expectations, which are directly estimated from data at sparse observation times. However, for systems exhibiting stiff, multiscale dynamics, data sparsity introduces regression discrepancies that compound during RoPDF evolution. This is overcome by introducing a kinetic-like defect term to the RoPDF equation, which is learned by assimilating in sparse, low-fidelity RoPDF estimates. Two assimilation methods are considered, namely nudging and deep neural networks, which are successfully tested against Monte Carlo simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Verification of Upcoming MCNP Features For Estimating Nuclear Data Sensitivities in Fixed Source Simulations [Abstract]

Predictive simulation codes, like the Monte Carlo N-Particle (MCNP) transport code, are used throughout the nuclear community. These simulations are based on nuclear data. Maximizing the accuracy and precision of nuclear data maximizes the accuracy and precision of the overall simulation. This is imperative to applications that rely on simulations. For example, improving nuclear data for special nuclear material improves simulation accuracy in stockpile stewardship applications, which results in larger safety margins and decreased operational costs. The improvement and validation of nuclear data is completed through integral benchmark experiments. Past benchmarks have primarily been limited to focus on the effective multiplication factor ($\kappa$ eff ); broadening the purview of benchmarks beyond $\kappa$ eff -dependent nuclear data addresses nuclear data deficiencies. Different response types depend on different areas of nuclear data. This dependence is quantified as nuclear data sensitivity: the change in response due to perturbation of a contributing parameter. The larger the nuclear data sensitivity of a response, the more the experiment is influenced by the uncertainties of the nuclear data. The optimization of nuclear data sensitivities in future benchmarks would result in more detailed validation of lesser studied areas of nuclear data. Currently, direct sensitivity capabilities are not easily found for all experiment types and parameters. An MCNP tool to directly estimate the cross section sensitivities of tallied values is under development. Additionally, updates have been made to the perturbation feature of MCNP, which can be used in a less direct approach to estimating sensitivities. This work verifies these features to estimate nuclear data sensitivities in fixed source simulations of a 4.5-kg sphere of alpha- phase weapons-grade plutonium surrounded by differing amounts of copper and polyethylene. Integrated estimates made using MCNP’s tools were found to statistically agree with integrated estimates made from manual perturbation of nuclear data proving the validity of the MCNP tools.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Benchmarking Monte Carlo codes for the modelling of low-energy neutron production target reactions

The increasing adoption of accelerator-based neutron sources (ABNS) for applications including neutron capture therapy (NCT) research has highlighted the need for accurate simulation tools. Precise modelling of the neutron production target is crucial to ensure that simulated predictions of neutron beam characteristics used for subsequent beam shaping assembly design are reliable. This work presents a comprehensive benchmarking of four widely-used Monte Carlo codes - Geant4, PHITS, FLUKA (CERN), and MCNP - for modelling low-energy neutron production target reactions. Using their recommended physics models and cross-section libraries, we evaluate each code’s performance in simulating four beam-target reactions: 7 Li(p,n) 7 Be, 9 Be(p,n) 9 B, 9 Be(d,n) 10 B, and C(d,n)N. Predictions of neutron yield, angular distributions, and energy spectra are compared against available thick target experimental data. Results show varying levels of agreement between the codes depending on the reaction type, energy range, and beam characteristics. Geant4, MCNP and PHITS are the overall best performing codes for the simulation of total neutron yield and yield in the forward direction across most reactions. Across energies where experimental benchmarks exist, inter-code discrepancies in total and forward-directed yield are typically 10 to 30%, with larger deviations at near-threshold incident ion energies. PHITS provides the best overall reproduction of experimental spectra, particularly for the 9 Be(p,n) 9 B reaction. Additionally, PHITS demonstrates superior computational performance for most reactions. These findings provide valuable guidance for ABNS design, highlighting the strengths and limitations of each code for the simulation of low-energy neutron production reactions.

43 PARTICLE ACCELERATORS↗
DOE OSTI2022

Hydrogen Burning of 29 Si and Its Impact on Presolar Stardust Grains from Classical Novae

Abstract Presolar stardust grains found in primitive meteorites are believed to retain the isotopic composition of stellar outflows at the time of grain condensation. Therefore, laboratory measurements of their isotopic ratios represent sensitive probes for investigating open questions related to stellar evolution, stellar explosions, nucleosynthesis, mixing mechanisms, dust formation, and galactic chemical evolution. For a few selected presolar grains, classical novae have been discussed as a potential source. For SiC, silicate, and graphite presolar grains, the association is based on the observation of small N ( 12 C)/ N ( 13 C) and N ( 14 N)/ N ( 15 N) number abundance ratios compared to solar values, and abundance excesses in 30 Si relative to 29 Si, as previously predicted by models of classical novae. We report on a direct measurement of the 29 Si(p, γ ) 30 P reaction, which strongly impacts simulated δ 29 Si values from classical novae. Our new experimental 29 Si(p, γ ) 30 P thermonuclear reaction rate differs from previous results by up to 50% in the classical nova temperature range ( T = 100–400 MK), while the rate uncertainty is reduced by up to a factor of 3. Using our new reaction rate in Monte Carlo reaction network and hydrodynamic simulations of classical novae, we estimate δ 29 Si values with much reduced uncertainties. Our results establish δ 29 Si values measured in presolar grains as a sensitive probe for assessing their classical nova paternity. We also demonstrate that δ 30 Si values from nova simulations are currently not a useful diagnostic tool unless the large uncertainty of the 30 P(p, γ ) 31 S reaction rate can be significantly reduced.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Integrated Neutronics Modeling for Inertial Fusion Energy Systems: Development and Application to LD-FIRST

Lawrence Livermore National Laboratory (LLNL) is proposing a new Laser Driven Fusion Integration Research and Science Test Facility (LD-FIRST) with the goal of providing an experimental testbed for future Inertial Fusion Energy (IFE) systems. However, IFE systems require detailed and accurate multiphysics modeling to quantify material damage, thermal loading, and tritium breeding within complex chamber environments. This article presents the first step in an integrated multiphysics framework that couples meshed CAD-based geometry within Monte Carlo neutronic simulations to enable high-fidelity analysis of IFE chamber concepts, with future coupling to external codes. The neutronics workflow utilizes OpenMC and its third-party capability to use CAD-based geometries through DAGMC and tally on unstructured meshes with Libmesh to evaluate neutron transport behavior, geometric fidelity, and material performance under reactor-relevant conditions. The use of tailored tallies on unstructured meshes in this framework allows direct transfer without interpolating to CFD simulation tools. Two IFE chambers were evaluated, both conceived by LLNL: HYLIFE-II and Laser IFE (LIFE). This work produced high-fidelity conformal surface and volumetric meshes of the HYLIFE-II and LIFE chambers with mapped spatial insight into material damage, thermal loading, and tritium breeding. The HYLIFE-II model was built utilizing available resources and used as a test case to verify that the neutronics framework can handle complex geometries. The LIFE chamber CAD was provided by LLNL and was the main focus of this work. This work analyzes multiple ternary alloy breeding materials for the LIFE chamber, across different 6 Li enrichments to produce data relevant to the LD-FIRST project. This work also investigates the level of model fidelity for the LIFE chamber, and results show that inclusion of detailed first wall and coolant structures increased the predicted tritium breeding ratio (TBR) by ~30%, highlighting the sensitivity of tritium breeding and the need for a high-fidelity simulation framework for IFE chambers. These developments provide a scalable toolset for the design and optimization of next-generation IFE chambers, forming a solid foundation for future coupled multiphysics analysis.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Combinatorial sputter deposition of ultrathick Au-Bi alloy films

We report gold-bismuth alloys are of interest as catalysts and catalytic sensing systems, electrochemical sensors, superconductors, and hohlraums for magnetically assisted inertial confinement fusion implosions. Radiation-hydrodynamics simulations with the Lasnex code of laser-driven hohlraums predict higher x-ray drive from Au-Bi alloys compared with cases of Au-Ta or pure Au and Bi hohlraums. Here, we use direct current magnetron sputtering in Ar gas, with co-sputtering from two elemental targets, to deposit Au-Bi alloys with Bi content of 9–77 at.% and thicknesses up to ~20 µm. Films are characterized by a combination of x-ray diffraction, Rutherford backscattering, scanning electron microscopy, substrate-curvature-based residual stress, and electronic transport measurements. Experiments are complemented by Monte Carlo simulations of ballistic sputtering and gas phase transport of depositing species and Ar gas atoms. Results show that all films are polycrystalline, with three distinct compositional regimes dominated by Au, Au 2 Bi, and Bi crystallographic phases. A metallic behavior of the temperature dependence of electrical resistivity is observed for all the films. Films with Bi content above ~30 at.% exhibit porosity, which is tolerable to hohlraum x-ray drive based on Lasnex simulations.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Model-free Rayleigh weight from x-ray Thomson scattering measurements

X-ray Thomson scattering (XRTS) has emerged as a powerful tool for the diagnostics of matter under extreme conditions. In principle, it gives one access to important system parameters such as the temperature, density, and ionization state, but the interpretation of the measured XRTS intensity usually relies on theoretical models and approximations. In this context, a key property is given by the Rayleigh weight that describes the electronic localization around the ions. Here, we show that it is possible to extract the Rayleigh weight directly from the experimental data without the need for any model calculations or simulations. As a practical application, we consider an experimental measurement of strongly compressed Be at the National Ignition Facility [Döppner et al., Nature 618, 270–275 (2023)]. We demonstrate that experimental results for the Rayleigh weight open up new avenues for the interpretation of XRTS experiments by matching the measurement with ab initio simulations such as density functional theory or path integral Monte Carlo. Interestingly, this new procedure leads to significantly lower density compared to previously used chemical models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Energy Gap Closure of Crystalline Molecular Hydrogen with Pressure

Here, we study the gap closure with pressure of crystalline molecular hydrogen. The gaps are obtained from grand-canonical quantum Monte Carlo methods properly extended to quantum and thermal crystals, simulated by coupled electron ion Monte Carlo methods. Nuclear zero point effects cause a large reduction in the gap (~2 eV). Depending on the structure, the fundamental indirect gap closes between 380 and 530 GPa for ideal crystals and 330–380 GPa for quantum crystals. Beyond this pressure the system enters into a bad metal phase where the density of states at the Fermi level increases with pressure up to ~450–500 GPa when the direct gap closes. Our work partially supports the interpretation of recent experiments in high pressure hydrogen.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

Proof that Combining the Forced-collision and DXTRAN Monte Carlo Variance-reduction Techniques is Fair

This report provides mathematical proof that the MCNP DXTRAN (also known in other codes as forced-flight) and forced-collision variance-reduction techniques do not bias the expected value of Monte Carlo simulation estimates when combined. Proof is also provided that the techniques are unbiased when used independently. To prove that the techniques are unbiased, this report derives the first History Score Moment Equation (HSME) for non-multiplying media employing only forced collisions, only DXTRAN, and then both the variance-reduction techniques combined, as defined next. The HSMEs are found by forming the History Score Probability Density Functions (HSPDFs) and then taking the first score moment. Following its derivation, each HSME with a variance-reduction technique employed is reduced to the HSME for an analog simulation. Because the HSME represents the expected contribution to estimators in the simulation, reducing the HSME with variance reduction to the analog HSME shows that the simulation is unbiased despite the variance-reduction technique considered in the HSME. Analysis regarding higher-score moments of each technique is the subject of prior work and is not addressed herein. Throughout this work, the phase space p is defined to be the particle position x, direction-of-flight unit vector $\hat{Ω}$, energy E, and statistical weight w, $$p ≡ (x; \hat{Ω}; E; w).$$ A reduced phase-space excluding the statistical weight of the particle, $$r ≡ (x; \hat{Ω}; E),$$ is also used.

97 MATHEMATICS AND COMPUTING↗