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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 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Model selection and signal extraction using Gaussian Process regression

We present a novel computational approach for extracting localized signals from smooth background distributions. We focus on datasets that can be naturally presented as binned integer counts, demonstrating our procedure on the CERN open dataset with the Higgs boson signature, from the ATLAS collaboration at the Large Hadron Collider. Our approach is based on Gaussian Process (GP) regression — a powerful and flexible machine learning technique which has allowed us to model the background without specifying its functional form explicitly and separately measure the background and signal contributions in a robust and reproducible manner. Unlike functional fits, our GP-regression-based approach does not need to be constantly updated as more data becomes available. We discuss how to select the GP kernel type, considering trade-offs between kernel complexity and its ability to capture the features of the background distribution. We show that our GP framework can be used to detect the Higgs boson resonance in the data with more statistical significance than a polynomial fit specifically tailored to the dataset. Finally, we use Markov Chain Monte Carlo (MCMC) sampling to confirm the statistical significance of the extracted Higgs signature.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Analytical gradient-based optimization of CALPHAD model parameters

The calibration of CALPHAD (CALculation of PHAse Diagrams) models involves the solution of a very challenging high-dimensional multiobjective optimization problem. Traditional approaches to parameter fitting predominantly rely on gradient-free methods, which while robust, are computationally inefficient and often scale poorly with model complexity. In this work, we introduce and demonstrate a generalizable framework for analytic gradient-based optimization of the parameters of the CALPHAD model enabled by the recently formalized Jansson derivative technique. This method allows for efficient evaluation of gradients of thermodynamic properties at equilibrium with respect to model parameters, even in the presence of arbitrarily complex internal degrees of freedom. Leveraging these semi-analytic gradients, we employ the conjugate gradient (CG) method to optimize thermodynamic model parameters for four binary alloy systems: Cu-Mg, Fe-Ni, Cr-Ni, and Cr-Fe. Across all systems, CG achieves comparable or superior optimality relative to Bayesian ensemble Markov Chain Monte Carlo (MCMC) with improvements in computational efficiency ranging from one to three orders of magnitude. Furthermore, our results establish a new paradigm for CALPHAD assessments in which high fidelity data-rich model calibration becomes tractable using deterministic gradient-informed algorithms.

CALPHAD↗
DOE OSTI2023

Thermodynamic modeling of the Nb-Ni system with uncertainty quantification using PyCalphad and ESPEI

Here, the Nb–Ni system is remodeled with uncertainty quantification (UQ) using software tools of PyCalphad and ESPEI (the Extensible, Self-optimizing Phase Equilibria Infrastructure) with the presently implemented capability of modeling site fraction based on Wyckoff positions. The five- and three-sublattice models are used to model the topologically close pack (TCP) μ-Nb 7 Ni 6 and δ-NbNi 3 phases according to their Wyckoff positions. The inputs for CALPHAD-based thermodynamic modeling include the thermochemical data as a function of temperature predicted by first-principles and phonon calculations based on density functional theory (DFT), ab initio molecular dynamics (AIMD) simulations, together with phase equilibrium and site fraction data in the literature. In addition to phase diagram and thermodynamic properties, the CALPHAD-based predictions of site fractions of Nb in μ-Nb 7 Ni 6 agree well with experimental data. Furthermore, the UQ estimation using the Markov Chain Monte Carlo (MCMC) method as implemented in ESPEI is applied to study the uncertainty of site fraction in μ-Nb 7 Ni 6 and enthalpy of mixing (ΔH mix ) in liquid.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes

Uncertainty propagation in complex engineering systems often poses significant computational challenges related to modeling and quantifying probability distributions of model outputs, as those emerge as the result of various sources of uncertainty that are inherent in the system under investigation. Gaussian Processes regression (GPs) is a robust meta-modeling technique that allows for fast model prediction and exploration of response surfaces. Multi-fidelity variations of GPs further leverage information from cheap and low fidelity model simulations in order to improve their predictive performance on the high fidelity model. In order to cope with the high volume of data required to train GPs in high dimensional design spaces, a common practice is to introduce latent design variables that are typically projections of the original input space to a lower dimensional subspace, and therefore substitute the problem of learning the initial high dimensional mapping, with that of training a GP on a low dimensional space. Here in this paper, we present a Bayesian approach to identify optimal transformations that map the input points to low dimensional latent variables. The \projection" mapping consists of an orthonormal matrix that is considered a priori unknown and needs to be inferred jointly with the GP parameters, conditioned on the available training data. The proposed Bayesian inference scheme relies on a two-step iterative algorithm that samples from the marginal posteriors of the GP parameters and the projection matrix respectively, both using Markov Chain Monte Carlo (MCMC) sampling. In order to take into account the orthogonality constraints imposed on the orthonormal projection matrix, a Geodesic Monte Carlo sampling algorithm is employed, that is suitable for exploiting probability measures on manifolds. We extend the proposed framework to multi-fidelity models using GPs including the scenarios of training multiple outputs together. We validate our framework on three synthetic problems with a known lower-dimensional subspace. The benefits of our proposed framework, are illustrated on the computationally challenging aerodynamic optimization of a last-stage blade for an industrial gas turbine, where we study the effect of an 85-dimensional shape parameterization of a three-dimensional airfoil on two output quantities of interest, specifically on the aerodynamic efficiency and the degree of reaction

42 ENGINEERING↗
DOE OSTI2024

Transient anisotropic kernel for probabilistic learning on manifolds

PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an Itô equation from a stochastic dissipative Hamiltonian dynamical system, acting as the MCMC generator, for which the KDE-estimated probability measure with the training dataset is the invariant measure. PLoM performs a projection on a reduced-order vector basis related to the training dataset, using the diffusion maps (DMAPS) basis constructed with a time-independent isotropic kernel. In this paper, we propose a new ISDE projection vector basis built from a transient anisotropic kernel, providing an alternative to the DMAPS basis to improve statistical surrogates for stochastic manifolds with heterogeneous data. The construction ensures that for times near the initial time, the DMAPS basis coincides with the transient basis. For larger times, the differences between the two bases are characterized by the angle of their spanned vector subspaces. The optimal instant yielding the optimal transient basis is determined using an estimation of mutual information from Information Theory, which is normalized by the entropy estimation to account for the effects of the number of realizations used in the estimations. Consequently, this new vector basis better represents statistical dependencies in the learned probability measure for any dimension. Three applications with varying levels of statistical complexity and data heterogeneity validate the proposed theory, showing that the transient anisotropic kernel improves the learned probability measure.

Diffusion maps↗
DOE OSTI2025

Accelerating Hamiltonian Monte Carlo for Bayesian inference in neural networks and neural operators

Hamiltonian Monte Carlo (HMC) is a powerful and accurate method to sample from the posterior distribution in Bayesian inference. However, HMC techniques are computationally demanding for Bayesian neural networks due to the high dimensionality of the network’s parameter space and the non-convexity of their posterior distributions. Therefore, various approximation techniques, such as variational inference (VI) or stochastic gradient MCMC, are often employed to infer the posterior distribution of the network parameters. Such approximations introduce inaccuracies in the inferred distributions, resulting in unreliable uncertainty estimates. In this work, we propose a hybrid approach that combines inexpensive VI and accurate HMC methods to efficiently and accurately quantify uncertainties in neural networks and neural operators. The proposed approach leverages an initial VI training on the full network. We examine the influence of individual parameters on the prediction uncertainty, which shows that a large proportion of the parameters do not contribute substantially to uncertainty in the network predictions. This information is then used to significantly reduce the dimension of the parameter space, and HMC is performed only for the subset of network parameters that strongly influence prediction uncertainties. This yields a framework for accelerating the full batch HMC for posterior inference in neural networks. We demonstrate the efficiency and accuracy of the proposed framework on deep neural networks and operator networks, showing that inference can be performed for large networks with tens to hundreds of thousands of parameters. Finally, we show that this method can effectively learn surrogates for complex physical systems by modeling the operator that maps from upstream conditions to wall-pressure data on a cone in hypersonic flow.

Bayesian inference↗
DOE OSTI2025

MOOSE ProbML: Parallelized probabilistic machine learning and uncertainty quantification for computational energy applications

Here, this paper presents the development and demonstration of massively parallel probabilistic machine learning (ML) and uncertainty quantification (UQ) capabilities within the Multiphysics Object-Oriented Simulation Environment (MOOSE), an open-source computational platform for parallel finite element and finite volume analyses. In addressing the computational expense and uncertainties inherent in complex multiphysics simulations, this paper integrates Gaussian process (GP) variants, active learning, Bayesian inverse UQ, adaptive forward UQ, Bayesian optimization, evolutionary optimization, and Markov chain Monte Carlo (MCMC) within MOOSE. It also elaborates on the interaction among key MOOSE systems — Sampler, MultiApp, Reporter, and Surrogate — in enabling these capabilities. The modularity offered by these systems enables development of a multitude of probabilistic ML and UQ algorithms in MOOSE. Example code demonstrations include parallel active learning and parallel Bayesian inference via active learning. The impact of these developments is illustrated through five applications relevant to computational energy applications: UQ of nuclear fuel fission product release, using parallel active learning Bayesian inference; very rare events analysis in nuclear microreactors using active learning; advanced manufacturing process modeling using multi-output GPs (MOGPs) and dimensionality reduction; fluid flow using deep GPs (DGPs); and tritium transport model parameter optimization for fusion energy, using batch Bayesian optimization. These capabilities are part of the MOOSE framework.

97 - MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Improvement of two-phase closure models in CTF using Bayesian inference

Under the Consortium for Advanced Simulation of Light Water Reactors (CASL) program, extensive capabilities have been developed in CTF to analyze light-water reactors (LWRs) for normal operating conditions, departure from nucleate boiling (DNB), and system transients. However, further improvements are required in the modeling and simulation of boiling water reactors (BWRs), which is a focus of the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. In this work, CTF validation results were used to optimize selected modeling coefficients by calibrating to experimental data using a Bayesian inference approach. Here, calibration studies were conducted to improve (vapor) void fraction prediction without worsening the two-phase pressure drop prediction, as well as to improve the two-phase pressure drop prediction. Calibration was performed for interfacial drag and wall shear models. Surrogates were developed to alleviate the computational expense required for sampling the parameter space using Markov chain Monte Carlo (MCMC). An assessment performed with calibrated models demonstrated an improvement of CTF in its prediction of key parameters such as void fraction and two-phase pressure drop.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

Discrepancy quantification between experimental and simulated data of CO 2 adsorption isotherm using hierarchical Bayesian estimation

Here, to quantitatively analyze the inconsistencies commonly observed between experimental and simulated adsorption isotherms, parameter estimation of adsorption isotherm models was conducted by hierarchical Bayesian estimation with parameter uncertainties being quantified as probability distributions. The estimation method was implemented using Markov Chain Monte Carlo (MCMC) to analyze multiple data sets obtained from different sources, including a publicly available database. To describe the discrepancies of experimental and simulated adsorption data, the simulation data was set as the reference to which experimental measurements were compared. We applied the proposed approach to analyze CO 2 adsorption isotherms that are measured and simulated on zeolite 13X and MIL-101(Cr). In these case studies, the discrepancy of CO 2 adsorption isotherm was successfully quantified between experimental measurements and predictions given by molecular simulations using Grand Canonical Monte Carlo (GCMC), where uncertainties were quantified as probability distributions. Furthermore, experimental data sets that agree well with the GCMC simulation have been identified, providing insights into experimental and measurement methods as well as choosing the right assumptions in the molecular simulation.

42 ENGINEERING↗
DOE OSTI2025

Pervaporative Dehydration of 2,3-Butanediol by Dense Poly(vinylidene fluoride) Hollow Fiber Membranes: Parameter Estimation, Process Design, and Technoeconomic Evaluation under Uncertainty

Pervaporation, combined with other separation processes, can effectively remove water from fermentation product streams, making it highly suitable for purifying alcohols like 2,3-butanediol (BDO). In this study, a dense poly(vinylidene fluoride) (PVDF) hollow fiber membrane module prototype was fabricated for BDO dehydration, achieving >0.2 LMH total flux and >95% BDO rejection. With a Markov chain Monte Carlo (MCMC) approach, Bayesian inference was used to quantify the uncertainty of the permeance parameters. A membrane cascade model was developed to scale up a process that purifies a preconcentrated BDO feed (70 wt %) to high purity (90 wt %). Through propagation of the uncertainty of the parameters and sensitivity analyses of the process variables, a cascade design was recommended. Despite data and model limitations, the framework enabled a reliable system analysis and economic evaluation, validated through tight confidence intervals in key process metrics, establishing the foundation for future applications of Bayesian methods in membrane-based processes.

Animal feed↗
DOE OSTI2025

Using Machine Learning to Generate a GISS ModelE Calibrated Physics Ensemble (CPE)

A neural network (NN) surrogate of the NASA GISS ModelE atmosphere (version E3) is trained on a perturbed parameter ensemble (PPE) spanning 45 physics parameters and 36 outputs. The NN is leveraged in a Markov Chain Monte Carlo (MCMC) Bayesian parameter inference framework to generate a second posterior constrained ensemble coined a “calibrated physics ensemble,” or CPE. The CPE members are characterized by diverse parameter combinations and are, by definition, close to top-of-atmosphere radiative balance, and must broadly agree with numerous hydrologic, energy cycle and radiative forcing metrics simultaneously. Global observations of numerous cloud, environment, and radiation properties (provided by global satellite products) are crucial for CPE generation. The inference framework explicitly accounts for discrepancies (or biases) in satellite products during CPE generation. We demonstrate that product discrepancies strongly impact calibration of important model parameter settings (e.g., convective plume entrainment rates; fall speed for cloud ice). Structural improvements new to E3 are retained across CPE members (e.g., stratocumulus simulation). Notably, the framework improved the simulation of shallow cumulus and Amazon rainfall while not degrading radiation fields, an upgrade that neither default parameters nor Latin Hypercube parameter searching achieved. Analyses of the initial PPE suggested several parameters were unimportant for output variation. However, many “unimportant” parameters were needed for CPE generation, a result that brings to the forefront how parameter importance should be determined in PPEs. From the CPE, two diverse 45-dimensional parameter configurations are retained to generate radiatively-balanced, auto-tuned atmospheres that were used in two E3 submissions to CMIP6.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

A warm super-Neptune around the G-dwarf star TOI-1710 revealed with TESS, SOPHIE, and HARPS-N

We report the discovery and characterization of the transiting extrasolar planet TOI-1710 b. It was first identified as a promising candidate by the Transiting Exoplanet Survey Satellite. Its planetary nature was then established with SOPHIE and HARPS-N spectroscopic observations via the radial-velocity method. The stellar parameters for the host star are derived from the spectra and a joint Markov chain Monte-Carlo adjustment of the spectral energy distribution and evolutionary tracks of TOI-1710. A joint MCMC analysis of the TESS light curve and the radial-velocity evolution allows us to determine the planetary system properties. From our analysis, TOI 1710 b is found to be a massive warm super-Neptune (M p = 28.3 ± 4.7 M ⊕ and R p = 5.34 ± 0.11 R ⊕ ) orbiting a G5V dwarf star (T eff = 5665 ± 55 K) on a nearly circular 24.3-day orbit (e = 0.16 ± 0.08). The orbital period of this planet is close to the estimated rotation period of its host star P rot = 22.5 ± 2.0 days and it has a low Keplerian semi-amplitude K = 6.4 ± 1.0 m s -1 ; we thus performed additional analyses to show the robustness of the retrieved planetary parameters. With a low bulk density of 1.03 ± 0.23 g cm -3 and orbiting a bright host star (J = 8.3, V = 9.6), TOI-1710 b is one of the best targets in this mass-radius range (near the Neptunian desert) for atmospheric characterization via transmission spectroscopy, a key measurement in constraining planet formation and evolutionary models of sub-Jovian planets.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Constraining multi-field inflation using the SPHEREx all-sky survey power spectra

Abstract We investigate how well the SPHEREx all-sky survey can constrain local primordial non-Gaussianity beyond the parameterf NL using galaxy power spectra. We forecast joint constraints on the parametersf NL ,g NL andτ NL obtained assuming a simple two-field curvaton model of inflation. The parametersf NL andg NL characterise the squeezed limits of the primordial bispectrum and trispectrum respectively, and lead to a characteristic scale-dependence of the galaxy bias that increases out to arbitrarily large scales. Values of the parameterτ NL > (6/5f NL ) 2 cause the galaxy power spectrum to have a stochastic component which also increases out to arbitrarily large scales. Our MCMC forecasts indicate that SPHEREx can provide joint constraints on any two of the three parametersf NL ,g NL andτ NL . Due to strong degeneracies among these parameters, measurements of the galaxy power spectra alone may not be sufficient to jointly constrain all three. Constraints onf NL ,g NL andτ NL obtained from galaxy power spectrum observations depend on the modelling of underlying nuisance parameters. We study the robustness of our forecast constraints to modelling choices and note that even with relatively conservative modelling assumptions, SPHEREx galaxy power spectra can provide strong evidence of local non-Gaussianity, even if the particular values off NL andg NL cannot be measured precisely.

Astronomy & Astrophysics↗
DOE OSTI2021

SDSS-IV MaStar: theoretical atmospheric parameters for the MaNGA stellar library

ABSTRACT We calculate the fundamental stellar parameters effective temperature, surface gravity, and iron abundance – Teff, log g, [Fe/H] – for the final release of the Mapping Nearby Galaxies at APO (MaNGA) Stellar Library (MaStar), containing 59 266 per-visit-spectra for 24 290 unique stars at intermediate resolution (R ∼ 1800) and high S/N (median = 96). We fit theoretical spectra from model atmospheres by both MARCS and BOSZ-ATLAS9 to the observed MaStar spectra, using the full spectral fitting code pPXF. We further employ a Bayesian approach, using a Markov Chain Monte Carlo (MCMC) technique to map the parameter space and obtain uncertainties. Originally in this paper, we cross match MaStar observations with Gaia photometry, which enable us to set reliable priors and identify outliers according to stellar evolution. In parallel to the parameter determination, we calculate corresponding stellar population models to test the reliability of the parameters for each stellar evolutionary phase. We further assess our procedure by determining parameters for standard stars such as the Sun and Vega and by comparing our parameters with those determined in the literature from high-resolution spectroscopy (APOGEE and SEGUE) and from lower resolution matching template (LAMOST). The comparisons, considering the different methodologies and S/N of the literature surveys, are favourable in all cases. Our final parameter catalogue for MaStar cover the following ranges: 2592 ≤ Teff ≤ 32 983 K; −0.7 ≤ log g ≤ 5.4 dex; −2.9 ≤ [Fe/H] ≤ 1.0 dex and will be available with the last SDSS-IV Data Release, in 2021 December.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Abundance matching analysis of the emission-line galaxy sample in the extended Baryon Oscillation Spectroscopic Survey

We present the measurements of the small-scale clustering for the emission-line galaxy (ELG) sample from the extended Baryon Oscillation Spectroscopic Survey (eBOSS) in the Sloan Digital Sky Survey IV (SDSS-IV). We use conditional abundance matching method to interpret the clustering measurements from 0.34 to 70 h -1 Mpc. In order to account for the correlation between properties of ELGs and their environment, we add a secondary connection between star formation rate of ELGs and halo accretion rate. Three parameters are introduced to model the ELG [O II ] luminosity and to mimic the target selection of eBOSS ELGs. The parameters in our models are optimized using Markov Chain Monte Carlo (MCMC) method. We find that by conditionally matching star formation rate of galaxies and the halo accretion rate, we are able to reproduce the eBOSS ELG small-scale clustering within 1σ error level. Our best-fitting model shows that the eBOSS ELG sample only consists of ~ 12 per cent of all star-forming galaxies, and the satellite fraction of eBOSS ELG sample is 19.3 per cent. We show that the effect of assembly bias is ~ 20 per cent on the two-point correlation function and ~ 5 per cent on the void probability function at scale of r ~ 20 h -1 Mpc.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Accelerating cosmological inference with Gaussian processes and neural networks – an application to LSST Y1 weak lensing and galaxy clustering

ABSTRACT Studying the impact of systematic effects, optimizing survey strategies, assessing tensions between different probes and exploring synergies of different data sets require a large number of simulated likelihood analyses, each of which cost thousands of CPU hours. In this paper, we present a method to accelerate cosmological inference using emulators based on Gaussian process regression and neural networks. We iteratively acquire training samples in regions of high posterior probability which enables accurate emulation of data vectors even in high dimensional parameter spaces. We showcase the performance of our emulator with a simulated 3×2 point analysis of LSST-Y1 with realistic theoretical and systematics modelling. We show that our emulator leads to high-fidelity posterior contours, with an order of magnitude speed-up. Most importantly, the trained emulator can be re-used for extremely fast impact and optimization studies. We demonstrate this feature by studying baryonic physics effects in LSST-Y1 3×2 point analyses where each one of our MCMC runs takes approximately 5 min. This technique enables future cosmological analyses to map out the science return as a function of analysis choices and survey strategy.

Astronomy & Astrophysics↗
DOE OSTI2023

The evolution of the Milky Way’s thin disc radial metallicity gradient with K2 asteroseismic ages

ABSTRACT The radial metallicity distribution of the Milky Way’s disc is an important observational constraint for models of the formation and evolution of our Galaxy. It informs our understanding of the chemical enrichment of the Galactic disc and the dynamical processes therein, particularly radial migration. We investigate how the metallicity changes with guiding radius in the thin disc using a sample of red giant stars with robust astrometric, spectroscopic, and asteroseismic parameters. Our sample contains 668 stars with guiding radii 4 < Rg < 11 kpc and asteroseismic ages covering the whole history of the thin disc with precision ${\approx} 25 {{\, \rm per\ cent}}$. We use MCMC analysis to measure the gradient and its intrinsic spread in bins of age and construct a hierarchical Bayesian model to investigate the evolution of these parameters independently of the bins. We find a smooth evolution of the gradient from ≈−0.07 dex kpc−1 in the youngest stars to ≈−0.04 dex kpc−1 in stars older than 10 Gyr, with no break at intermediate ages. Our results are consistent with those based on asteroseismic ages from CoRoT, with that found in Cepheid variables for stars younger than 1 Gyr, and with open clusters for stars younger than 6 Gyr. For older stars we find a significantly lower metallicity in our sample than in the clusters, suggesting a survival bias favouring more metal-rich clusters. We also find that the chemical evolution model of Chiappini '09 is too metal poor in the early stages of disc formation. Our results provide strong new constraints for the growth and enrichment of the thin disc and radial migration, which will facilitate new tests of model conditions and physics.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Indirect Measurement of the 23 Na(p, γ) 24 Mg Direct Capture Reaction Rate via ( 3 He,d) Spectroscopy

The cross section of the $^{23}\text{Na}(p,γ)^{24}\text{Mg}$ reaction is dominated by direct capture at low energies relevant for stellar burning. Such cross sections can be constrained using spectroscopic factors($C^2S$) or asymptotic normalization coefficients(ANCs) from transfer reactions. In this work, the $^{23}\text{Na}(^3\text{He},d)^{24}\text{Mg}$ reaction was measured at $E_{lab}=21$ MeV to extract spectroscopic factors for $^{24}\text{Mg}$ states with excitation energies in $E_x=7 - 12$ MeV using the Enge split-pole spectrograph at the Triangle Universities Nuclear Laboratory. A new non-resonant astrophysical S factor and the direct capture reaction rate for the $^{23}\text{Na}(p,γ)$ reaction are calculated and presented based on this measurement. The new rate at $T<0.04$ GK is 43$\%$ smaller than in previous studies. Rigorous treatments of uncertainties are presented using a Bayesian Markov Chain Monte Carlo (MCMC) method. Finally, sources of uncertainties for computing the direct capture cross section are also discussed in detail.

electromagnetic moments↗