Search NASA⌕ Search

SEARCH · Search NASA

Results for “Reliability Prediction”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Energetic Stability of Freestanding and Metal-Supported Borophenes: Quantum Monte Carlo and Density-Functional Theory Calculations

Quantum Monte Carlo (QMC) and density-functional theory (DFT) calculations were carried out to study cohesion energetics of two-dimensional (2D) sheets of boron atoms called borophenes. Our QMC calculations confirmed the polymorphism among free-standing borophenes that was reported in previous DFT studies. Although Perdew-Burke-Ernzerhof (PBE) calculations significantly overestimate the cohesive energies of 2D boron sheets, DFT-PBE relative energetics with respect to each other among various free-standing borophenes are found to be in quantitative agreement with the corresponding QMC results. This suggests that one can make reliable predictions for relative stability of different boron sheets through DFT-PBE calculations. Our analysis of PBE formation energies of borophenes on metal surfaces shows that the polymorphic range is extended for borophenes on the Ag(111) and the Au(111) surfaces beyond that of free-standing borophenes, reflecting recent experimental synthesis of beta(12) and chi(3) boron sheets on the Ag(111) surface. We have also found that a hexagonal borophene can be stabilized through charge transfer from a metal surface and is energetically favored on the AI(111) surface over other borophene structures. Finally, it is found that the bilayer formation could be energetically favored over its monolayer form for the borophene-Au system, especially for borophenes with hexagonal hole densities eta lower than 1/9. This leads to our prediction that in addition to its monolayer form, a bilayer eta = 1/12 borophene can be synthesized on the Au(111) surface, opening a new possibility for borophene-based electronic devices.

ab initio↗
DOE OSTI2019

Dye Nanoaggregate Structures in MK-2, N3, and N749 Dye···TiO 2 Interfaces That Represent Dye-Sensitized Solar Cell Working Electrodes

The working electrode of a dye-sensitized solar cell (DSSC) consists of dye molecules adsorbed onto nanoparticles of a semiconductor such as TiO 2 . A reliable prediction of the DSSC photovoltaic performance of a given dye requires in-depth knowledge about the precise structure of the dye···TiO 2 interface. In this work, X-ray reflectometry (XRR) and grazing-incidence small-angle X-ray scattering (GISAXS) are employed to determine the dye···TiO 2 interfacial structure and associated dye aggregation behavior of three high-performance DSSC dyes, i.e., a metal-free organic dye, MK-2, and the two archetypal ruthenium-based organometallic dyes, N3 and N749 (Black Dye). Results show that all three dyes form nanoaggregates in dye···TiO 2 interfaces. We determine the dye nanoaggregate separations, sizes, distribution densities, and the extent of short-range order within each dye self-assembly in the longitudinal and lateral directions. Dye···TiO 2 composites fabricated using dye solutions of varying concentrations are analyzed. We find that nanoaggregates of the three dyes are separated by several hundred nanometers (158-203 nm) in dye···TiO 2 interfaces that have been fabricated using concentrated dye solutions (0.5 mM or 1.0 mM). MK-2 and N749 dyes also display smaller interparticle separations. Dye nanoparticle diameters are of the order of 156-198 nm, sizes that are comparable to the largest interparticle separations. Thus, no extraneous dye particles can be fitted into gaps between particles, so the dye self-assembly is saturated. Self-assemblies of all three dyes exhibit both lateral and longitudinal short-range order; N3 displays a particularly short coherence length along the TiO 2 surface, with extensive structured disorder along the longitudinal direction. The operation of DSSC working electrodes would therefore seem to be dependent on a dye self-assembly that may exhibit several levels of structural granularity and dye aggregation effects.

14 SOLAR ENERGY↗
DOE OSTI2026

Ripening of Rh Nanoparticle Catalysts in Reverse Water–Gas Shift via a Data-Driven Model Combining Physics, Theory, and Experiment

Degradation via sintering is an ongoing challenge that impedes the broad commercial success of supported metallic nanoparticle catalysts. To mitigate degradation via informed catalyst design and process operations, here we aim to disambiguate the underlying mechanisms of sintering by combining theory and experiment in a quantitative framework. While mechanistic sintering models exist, they only model a single sintering pathway, even though multiple sintering mechanisms can occur simultaneously or dominate at different stages of the process. Data-driven machine learning models have emerged as a means to represent complex processes through data regression. However, machine learning models have very large data needs and lack mechanistic insights due to their black-box encoding. To develop an interpretive model of catalyst degradation via sintering, we constructed a hybrid model combining mechanistic “physics-based” models and data-driven methods to obtain both reliable predictions and mechanistic insights regarding experimentally observed sintering phenomena. Focusing on nanoparticle sintering in the Rh–TiO 2 catalyst for the reverse water–gas shift (RWGS) reaction, the hybrid model couples a mechanistic term for Ostwald ripening with energy values calculated via density functional theory (DFT) with a parametric, data-driven discrepancy function term for unmodeled mechanisms. The hybrid model is trained using Bayesian inference with data collected from small-angle X-ray scattering (SAXS) in situ experiments wherein average nanoparticle diameter versus time was measured at three relevant operating temperatures. The calibrated hybrid model results show that an Ostwald ripening-only model parameterized with fixed DFT energies does not fully capture the time and temperature dependence of the SAXS-observed sintering kinetics, and that an additional functional contribution, or DFT energy calibration, is required to reconcile simulation and experiment. Analysis of the hybrid-model error confirms that the hybrid model outperforms both the purely mechanistic and purely data-driven alternatives in terms of expected predictive accuracy for time-evolving average particle sizes. Furthermore, the results support the hypothesis that the Ostwald ripening mechanism is less important for explaining the sintering phenomena as operating temperature increases under an assumed fixed DFT parameterization. This could be explained in one of two ways: either latent, unmodeled sintering mechanisms dominate at higher temperatures, or the DFT uncertainty increases with temperature. The proposed modeling approach directly links theory to experiments and simulations via a statistical hybrid modeling framework and can be extended to other catalytic systems to improve predictive models and mechanistic understanding.

Bayesian hybrid modeling↗
DOE OSTI2022

Multilevel Computational Studies Reveal the Importance of Axial Ligand for Oxygen Reduction Reaction on Fe–N–C Materials

The systematic improvement of Fe–N–C materials for fuel cell applications has proven challenging, due in part to an incomplete atomistic understanding of the oxygen reduction reaction (ORR) under electrochemical conditions. Herein, a multilevel computational approach, which combines ab initio molecular dynamics simulations and constant potential density functional theory calculations, is used to assess proton-coupled electron transfer (PCET) processes and adsorption thermodynamics of key ORR intermediates. These calculations indicate that the potential-limiting step for ORR on Fe–N–C materials is the formation of the Fe III –OOH intermediate. They also show that an active site model with a water molecule axially ligated to the iron center throughout the catalytic cycle produces results that are consistent with the experimental measurements. In particular, reliable prediction of the ORR onset potential and the Fe(III/II) redox potential associated with the conversion of Fe III –OH to Fe II and desorbed H 2 O requires an axial H 2 O co-adsorbed to the iron center. The observation of a five-coordinate rather than four-coordinate active site has significant implications for the thermodynamics and mechanism of ORR. Importantly, these findings highlight the importance of solvent–substrate interactions and surface charge effects for understanding the PCET reaction mechanisms and transition-metal redox couples under realistic electrochemical conditions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Spin-phonon relaxation from a universal ab initio density-matrix approach

Designing new quantum materials with long-lived electron spin states urgently requires a general theoretical formalism and computational technique to reliably predict intrinsic spin relaxation times. We present a new, accurate and universal first-principles methodology based on Lindbladian dynamics of density matrices to calculate spin-phonon relaxation time of solids with arbitrary spin mixing and crystal symmetry. This method describes contributions of Elliott-Yafet and D’yakonov-Perel’ mechanisms to spin relaxation for systems with and without inversion symmetry on an equal footing. We show that intrinsic spin and momentum relaxation times both decrease with increasing temperature; however, for the D’yakonov-Perel’ mechanism, spin relaxation time varies inversely with extrinsic scattering time. We predict large anisotropy of spin lifetime in transition metal dichalcogenides. The excellent agreement with experiments for a broad range of materials underscores the predictive capability of our method for properties critical to quantum information science.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Advancing simulations of coupled electron and phonon nonequilibrium dynamics using adaptive and multirate time integration

Electronic structure calculations in the time domain provide a deeper understanding of nonequilibrium dynamics in materials. The real-time Boltzmann equation (rt-BTE), used in conjunction with accurate interactions computed from first principles, has enabled reliable predictions of coupled electron and lattice dynamics. However, the timescales and system sizes accessible with this approach are still limited, with two main challenges being the different timescales of electron and phonon interactions and the cost of computing collision integrals. As a result, only a few examples of these calculations exist, mainly for two-dimensional (2D) materials. Here we leverage adaptive and multirate time integration methods to achieve a major step forward in solving the coupled rt-BTEs for electrons and phonons. Relative to conventional (non-adaptive) time-stepping, our approach achieves a 10x speedup for a target accuracy, or greater accuracy by 3–6 orders of magnitude for the same computational cost, enabling efficient calculations in both 2D and bulk materials. This efficiency is showcased by computing the coupled electron and lattice dynamics in graphene up to ~100 ps, as well as modeling ultrafast lattice dynamics and thermal diffuse scattering maps in bulk materials (silicon and gallium arsenide). In addition to improved efficiency, our adaptive method can resolve the characteristic rates of different physical processes, thus naturally bridging different timescales. This enables simulations of longer timescales and provides a framework for modeling multiscale dynamics of coupled degrees of freedom in matter. Our work opens new opportunities for quantitative studies of nonequilibrium physics in materials, including driven lattice dynamics with phonons coupled to electrons, spin, and other degrees of freedom.

Yao, Jia [California Institute of Technology (CalT↗
DOE OSTI2026

Characterization of volcanic tuff pores pre- and post-underground nuclear detonation using ultra-small and small angle neutron scattering

The ability to accurately model the subsurface transport of radionuclides is fundamental to the remote detection and characterization of underground nuclear explosion (UNE) events. Developing more sophisticated transport models presents a significant opportunity to enhance monitoring capabilities, particularly in the reliable prediction of signature migration. Experimentally determined characterization of geologic materials associated with transport properties is the pertinent base information for such robust model development and calibration. Here, we report results from an unprecedented study demonstrating changes to the pore and fracture network structures in geological materials in response to UNEs over nanometer to micrometer length scales. Volcanic tuffs of five different lithological formations from pre- and post-UNE environments were collected from the Nevada National Security Site. Combined ultra-small and small-angle neutron scattering techniques were used to characterize the tuff pore structure. The results demonstrate measurable differences in the specific surface area and porosity of samples pre- and post-shot from texturally similar lithological formations, indicating that pore properties can serve as a direct physical signature of a UNE. The results also provide experimentally determined transport parameters in support of advanced model development through the integration of gas migration, hydrodynamic simulations, and geologic framework models.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Machine-learning-accelerated multimodal characterization and multiobjective design optimization of natural porous materials

Natural porous materials such as nanoporous clays are used as green and low-cost adsorbents and catalysts. The key factors determining their performance in these applications are the pore morphology and surface activity, which are typically represented by properties such as specific surface area, pore volume, micropore content and pH. The latter may be modified and tuned to specific applications through material processing and/or chemical treatment. Characterization of the material, raw or processed, is typically performed experimentally, which can become costly especially in the context of tuning of the properties towards specific application requirements and needing numerous experiments. In this work, we present an application of tree-based machine learning methods trained on experimental datasets to accelerate the characterization of natural porous materials. The resulting models allow reliable prediction of the outcomes of experimental characterization of processed materials (R2 from 0.78 to 0.99) as well as identification of key factors contributing to those properties through feature importance analysis. Furthermore, the high throughput of the models enables exploration of processing parameter–property correlations and multiobjective optimization of prototype materials towards specific applications. We have applied these methodologies to pinpoint and rationalize optimal processing conditions for clays exploitable in acid catalysis. One of such identified materials was synthesized and tested revealing appreciable acid character improvement with respect to the pristine material. Specifically, it achieved 79% removal of chlorophyll-a in acid catalyzed degradation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Ion irradiation and modification: The role of coupled electronic and nuclear energy dissipation and subsequent nonequilibrium processes in materials

Understanding material responses to energy deposition from energetic charged particles is important for defect engineering, ion-beam processing, ion-beam analysis and modification, geologic aging, space exploration, and nuclear applications. As an incident ion penetrates a solid, its energy is transferred to electrons and to atomic nuclei of the solid. Much of this electronic energy deposition is subsequently transferred to the atomic structure via electron–phonon (e–ph) coupling, leading to local inelastic thermal spikes in which energy dissipation is influenced by the local environment. In addition, intense ionization can lead to high densities of localized electronic excitations in wide-bandgap materials and ceramics that can affect defect dynamics and atomic mobility. Specifically, energy exchange between electrons and atomic nuclei, along with localized electronic excitations, can lead to substantial competitive (ionization-induced annealing), additive (both electronic and nuclear energy depositions contributing to damage production), and synergistic (more damage than the sums of separate damage processes) effects. Although nonmonotonic effects of the e–ph coupling strength and athermal processes are demonstrated for pre-existing defects and residual damage during ion–solid interactions, there is limited understanding of when such electronic effects must be considered in atomic-scale models of damage production and evolution in a broad variety of materials. Complex ceramics and chemically disordered solid solution alloys with different constituent elements allow a systematic evaluation of defect dynamics and irradiation performance with increasing complexity. Current knowledge regarding tuning of bonding characteristics and chemical disorder to control atomic-level dynamics is reviewed. Although a lack of fundamental understanding obstructs the advancement of reliable predictions for ion beam material modification, it highlights challenges and opens research opportunities. Insights into the complex electronic and atomic correlations with extreme energy deposition will strengthen our ability to design materials and predict ion-irradiation-induced damage in a radiation environment, and they may pave the way to better control fundamental processes and design new material functionalities for advanced technologies.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Too big, too small, or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems

Multipole moments are the first-order responses of the energy to spatial derivatives of the electric field strength. The quality of density functional theory prediction of molecular multipole moments thus characterizes errors in modeling the electron density itself, as well as the performance in describing molecules interacting with external electric fields. However, only the lowest non-zero moment is translationally invariant, making the higher-order moments origin-dependent. Therefore, instead of using the 3 × 3 quadrupole moment matrix, we utilize the translationally invariant 3 × 3 matrix of second cumulants (or spatial variances) of the electron density as the quantity of interest (denoted by K). The principal components of K are the square of the spatial extent of the electron density along each axis. A benchmark dataset of the principal components of K for 100 small molecules at the coupled cluster singles and doubles with perturbative triples at the complete basis set limit is developed, resulting in 213 independent K components. The performance of 47 popular and recent density functionals is assessed against this Var213 dataset. Several functionals, especially double hybrids, and also SCAN and SCAN0 predict reliable second cumulants, although some modern, empirically parameterized functionals yield more disappointing performance. The H, Li, and Be atoms, in particular, are challenging for nearly all methods, indicating that future functional development could benefit from the inclusion of their density information in training or testing protocols.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Challenges and transformative opportunities in superconductor vortex physics

In superconductors, the motion of vortices introduces unwanted dissipation that is disruptive to applications. Fortunately, material defects can immobilize vortices, acting as vortex pinning centers, which engenders dramatic improvements in superconductor material properties and device operation. This has motivated decades of research into developing methods of tailoring the disorder landscape in superconductors to increase the strength of vortex pinning. Yet, efficacious materials engineering still eludes us. The electromagnetic properties of real (disordered) superconducting materials cannot yet be reliably predicted, such that designing superconductors for applications remains a largely inefficient process of trial and error. This is ultimately due to large gaps in our knowledge of vortex dynamics: the field is challenged by the extremely complex interplay between vortex elasticity, vortex-vortex interactions, and material disorder. Here, we review obstacles and recent successes in understanding and controlling vortex dynamics in superconducting materials and devices. We further identify major open questions and discuss opportunities for transformative research in the field. This includes improving our understanding of vortex creep, determining and reaching the ceiling for the critical current, advanced microscopy to garner accurate structure-property relationships, frontiers in predictive simulations and the benefits of artificial intelligence, as well as controlling and exploiting vortices in quantum information applications.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

The critical role of synthesis conditions on small polaron carrier concentrations in hematite—A first-principles study

Achieving highly efficient energy conversion with transition metal oxides necessitates overcoming conductivity limitations due to the formation of small polarons. Detailed understanding of the interplay among intrinsic defects, dopants, and electron polarons can help devise strategies for achieving higher carrier concentrations, therefore improving carrier conductivity. This work employs first-principles calculations to reliably predict electron polaron concentrations in a prominent polaronic oxide, hematite (Fe 2 O 3 ), by resolving interactions between charged defects and electron polarons and keeping charge neutrality condition among all charged species. This work addresses that both V O and Fe i can be primary donors in undoped hematite depending on the synthesis conditions, such as synthesis temperature and oxygen partial pressure, despite the fact that V O owns an extremely high ionization energy compared to k B T. Furthermore, from calculations of a plethora of n-type dopants (group IV and V elements), we find that Ti, Ge, Sb, and Nb are able to raise electron polaron concentrations in hematite significantly without considering dopant clustering. However, the magnitude of electron polaron concentration increase would be smaller if the dopant has a high tendency of clustering, such as Ti. Here we reveal the critical role of synthesis conditions on tuning electron polaron concentrations of both undoped and doped hematite. Our theoretical analysis provides important insights and general design principles for engineering more conductive polaronic oxides.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Controlling superconductivity of CeIrIn 5 microstructures by substrate selection

Superconductor/metal interfaces are usually fabricated in heterostructures that join these dissimilar materials. A conceptually different approach has recently exploited the strain sensitivity of heavy-fermion superconductors, selectively transforming regions of the crystal into the metallic state by strain gradients. The strain is generated by differential thermal contraction between the sample and the substrate. Here, we present an improved finite-element model that reliably predicts the superconducting transition temperature in CeIrIn 5 even in complex structures. Different substrates are employed to tailor the strain field into the desired shapes. Using this approach, both highly complex and strained as well as strain-free microstructures are fabricated to validate the model. This enables a high degree of control over the microscopic strain fields and forms the basis for more advanced structuring of superconductors as in Josephson junctions yet also finds natural use cases in any material class in which a modulation of the physical properties on a chip is desirable.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Role of morphology in defect formation and photo-induced carrier instabilities in amorphous indium oxide

Ab initio molecular dynamics liquid-quench simulations and hybrid density functional calculations are performed to model the effects of room-temperature atomic fluctuations and photo-illumination on the structural and electronic properties of amorphous sub-stoichiometric In 2 O 2.96 . A large configurational ensemble is employed to reliably predict the distribution of localized defects as well as their response to the thermal and light activation. The results reveal that the illumination effects on the carrier concentration are greater in amorphous configurations with shorter In–O bond length and reduced polyhedral sharing as compared to the structures with a more uniform morphology. Further, the obtained correlation between the photo-induced carrier density and the reduction in the number of fully coordinated In-atoms implies that metal oxides with a significant fraction of crystalline/amorphous interfaces would show a more pronounced response to illumination. Photo-excitation also produces In–O 2 –In defects that have not been previously found in sub-stoichiometric amorphous oxides; these defects are responsible for carrier instabilities due to overdoping.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2026

Chaos as the cause of randomness in tearing mode onset times in DIII-D ITER baseline scenario plasmas

The first evidence of widespread chaotic plasma dynamics relevant to the onset of disruptive tearing modes is found in DIII-D tokamak discharges. This evidence is obtained by calculating positive maximal Lyapunov exponents from Mirnov signals using the Rosenstein algorithm, implying the chaotic growth and/or rotation of magnetohydrodynamic modes in DIII-D [Luxon, Nucl. Fusion 42, 614 (2002)] plasmas. Such chaotic dynamics offer an explanation for the random tearing mode onset times previously observed in ITER baseline scenario discharges. Further, the Lyapunov timescales associated with this chaos could inhibit reliable prediction of tearing mode onset beyond the angular momentum confinement timescale; this constraint should inform the design of plasma control systems on ITER.

Chaotic dynamics↗
DOE OSTI2021

Landsat 8 monitoring of multi-depth suspended sediment concentrations in Lake Erie’s Maumee River using machine learning

Satellite remote sensing has been widely used to map suspended sediment concentration (SSC) in waterbodies. However, due to the complexity of sediment-water interactions, it has been difficult to derive linear and non-linear regression equations to reliably predict SSC, especially when trying to estimate depth of integrated sediment. Herein, this study uses Landsat 8 OLI (Operational Land Imager) sensor to map SSC within the Maumee River in Ohio, USA, at multiple depth intervals (15, 61, 91, and 182 cm). Simple linear least squares regression (LLSR), and three common machine learning models: random forest (RF), support vector regression (SVR), and model averaged neural network (MANN) were used to estimate SSC at the depth intervals. All machine learning models significantly outperformed LLSR while RF performed the best. In both RF and MANN, R2 (coefficient of determination) increases with depth with a maximum R2 of 0.89 and 0.83, respectively, at a depth of 0–182 cm. The results show that machine learning models can implement nonlinear relationships that produce better predictions than traditional linear regression methods in estimating depth integrated SSC, especially when samples are limited.

47 OTHER INSTRUMENTATION↗
DOE OSTI2025

Theoretical tools for neutrino scattering: interplay between lattice QCD, EFTs, nuclear physics, phenomenology, and neutrino event generators

Maximizing the discovery potential of increasingly precise neutrino experiments will require an improved theoretical understanding of neutrino-nucleus cross sections over a wide range of energies. Low-energy interactions are needed to reconstruct the energies of astrophysical neutrinos from supernovae bursts and search for new physics using increasingly precise measurement of coherent elastic neutrino scattering. Higher-energy interactions involve a variety of reaction mechanisms including quasi-elastic scattering, resonance production, and deep inelastic scattering that must all be included to reliably predict cross sections for energies relevant to DUNE and other accelerator neutrino experiments. Refined nuclear interaction models in these energy regimes will also be valuable for other applications, such as measurements of reactor, solar, and atmospheric neutrinos. This manuscript discusses the theoretical status, challenges, required resources, and path forward for achieving precise predictions of neutrino-nucleus scattering and emphasizes the need for a coordinated theoretical effort involved lattice QCD, nuclear effective theories, phenomenological models of the transition region, and event generators.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Constraining the local pulsar population with the cosmic-ray positron fraction

Observations of the TeV halos associated with nearby pulsars indicate that these objects inject significant fluxes of very high-energy electron-positrons pairs into the interstellar medium (ISM), thereby likely providing the dominant contribution to the cosmic-ray positron flux. In this paper, we use the cosmic-ray positron fraction as measured by the AMS-02 Collaboration to constrain the characteristics of the local pulsar population. For reasonable model parameters, we find that we can obtain good agreement with the measured positron fraction up to energies of E e ~ 300 GeV. At higher energies, the positron fraction is dominated by a small number of pulsars, making it difficult to reliably predict the shape of the expected positron fraction. The low-energy positron spectrum supports the conclusion that pulsars typically transfer approximately η ~ 5–20% of their total spindown power into the production of very high-energy electron-positron pairs, producing a spectrum of such particles with a hard spectral index, α ~ 1.5–1.7. Such pulsars typically spindown on a timescale on the order of τ ~ 10 4 years. Our best fits were obtained for models in which the radio and gamma-ray beams from pulsars are detectable to 28% and 62% of surrounding observers, respectively.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗