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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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DOE OSTI2024

2022 System Optimization Awardee: XFlow Energy Company

Vertical-axis wind turbines offer several cost-saving advantages over traditional horizontal-axis wind turbines, including lower blade manufacturing costs and a simpler mechanical design. Unfortunately, until now, vertical configurations have not been able to match the power production and durability of more conventional horizontal-axis models. XFlow Energy Company (XFlow Energy) plans to improve the design and reduce the production costs of its 25-kilowatt (kW) vertical-axis wind turbine, delivering a levelized cost of energy that is competitive with that of U.S. retail electricity generated by coal-fired power plants. XFlow Energy's efficient turbine also is designed to maximize power production and longevity. Two previous Competitiveness Improvement Project (CIP) awards funded XFlow Energy's development of a 10-meter vertical-axis turbine prototype.

CIP↗
NASA NTRS2004

Numerical, Analytical, Experimental Study of Fluid Dynamic Forces in Seals: Executive Summary and Description of Knowledge-Based System - Volume 1

The objectives of the program were to develop computational fluid dynamics (CFD) codes and simpler industrial codes for analyzing and designing advanced seals for air-breathing and space propulsion engines. The CFD code SCISEAL is capable of producing full three-dimensional flow field information for a variety of cylindrical configurations. An implicit multidomain capability allows the division of complex flow domains to allow optimum use of computational cells. SCISEAL also has the unique capability to produce cross-coupled stiffness and damping coefficients for rotordynamic computations. The industrial codes consist of a series of separate stand-alone modules designed for expeditious parametric analyses and optimization of a wide variety of cylindrical and face seals. Coupled through a Knowledge-Based System (KBS) that provides a user-friendly Graphical User Interface (GUI), the industrial codes are PC based using an OS/2 operating system. These codes were designed to treat film seals where a clearance exists between the rotating and stationary components. Leakage is inhibited by surface roughness, small but stiff clearance films, and viscous pumping devices. The codes have demonstrated to be a valuable resource for seal development of future air-breathing and space propulsion engines.

Liang, Anita D.↗
NASA NTRS2019

High-Performance Computing Optimization for Aladyn – Adaptive Neural Network Molecular Dynamics Mini-Application

This report provides a description and performance evaluation of the optimization techniques for high performance computing (HPC) implementation of the open source Computational Materials mini-application Aladyn (https://github.com/nasa/aladyn). Aladyn is a basic molecular dynamics code written in FORTRAN 2003, which is designed to demonstrate the use of adaptive neural networks (ANNs) in atomistic simulations. The role of ANNs is to efficiently reproduce the very complex energy landscape resulting from the atomic interactions in materials with the accuracy of the more expensive quantum mechanics-based calculations. The ANN is trained on a large set of atomic structures calculated using the density functional theory (DFT) method. While achieving orders of magnitude faster computational performance than DFT, the ANN-based approach was still very computationally demanding compared to the conventional approach of using empirically fitted energy functions. After its initial development, Aladyn was evaluated and optimized by experts at the NASA Advanced Supercomputing (NAS) division to exploit modern supercomputer architectures. The code has been optimized for execution on multicore central processing units (CPUs), including Intel® Skylake microarchitecture, and on graphic accelerators, such as Nvidia® V100 graphic processing units (GPUs), using Open Multi-Processing (OpenMP) and Open Accelerators (OpenACC) programming interfaces. The optimization achieved a speedup of 4.7 times the baseline version on CPU performance and an additional 2.4 times on CPU+GPU performance. Atomistic computer simulations are a fundamental tool in materials research to model material properties form physics-based first principles. Atomic interaction, governed by Quantum Mechanics (QM) require sophisticated and highly computationally demanding mathematical models to calculate [1]. Classical methods use approximate functional forms, empirically fitted through a set of variable parameters to emulate atomic energies as direct functions of atomic coordinates [2]. While empirical potentials are computationally much simpler, allowing simulations of large-scale systems of up to a trillion (1012) atoms [3], they are substantially less accurate compared to quantum calculations and applicable only to very specific atomic configurations or predefined crystallographic phases. A recently suggested approach is to use heuristic machine learning methods [4], such as those based on Adaptive Neural Networks (ANNs) to predict atomic energies, after being trained on a sufficiently large database of QM-calculated structures [5,6]. This approach reduces significantly the computational complexity, allowing for simulations of orders of magnitude larger systems compared to QM-based methods without compromising accuracy. Still, compared to classical methods using empirical energy functions, ANN methods remain two- to three orders of magnitude more computationally demanding. Hence, the computational cost of simulations, together with the need for extensive training of ANNs, still makes the practical implementation of ANN-based methods quite challenging. The purpose of the Aladyn mini-application software [7], available as open source at https://github.com/nasa/aladyn, is to be a testbed for exploring possible optimization strategies to develop highly scalable parallel algorithms for ANN-based atomistic simulations. Aladyn is aimed at utilizing the architecture of the high-end modern highperformance computing (HPC) hardware based on multicore central processing units (CPUs) equipped with graphic processing unit (GPU) accelerators. Specifically, the goal is to optimize the performance on a single HPC compute node, before implementing scaling to multi-node parallelization using message passing interface (MPI). At the same time, the open source code of Aladyn can serve as a training model for students and professors in academia.

Yamakov, Vesselin I.↗
NASA NTRS2013

A Cross-Track Cloud-Scanning Dual-Frequency Doppler (C2D2) Radar for the Proposed ACE Mission and Beyond

The National Resource Council’s Earth Science Decadal Survey” (NRCDS) has identified the Aerosol/Climate/Ecosystems (ACE) Mission as a priority mission for NASA Earth science. The NRC recommended the inclusion of "a cross-track scanning cloud radar with channels at 94 GHz and possibly 34 GHz for measurement of cloud droplet size, glaciation height, and cloud height". Several radar concepts have been proposed that meet some of the requirements of the proposed ACE mission but none have provided scanning capability at both 34 and 94 GHz due to the challenge of constructing scanning antennas at 94 GHz. In this paper, we will describe a radar design that leverages new developments in microwave monolithic integrated circuits (MMICs) and micro-machining to enable an electronically-scanned radar with both Ka-band (35 GHz) and W-band (94-GHz) channels. This system uses a dual-frequency linear active electronically-steered array (AESA) combined with a parabolic cylindrical reflector. This configuration provides a large aperture (3m x 5m) with electronic-steering but is much simpler than a two-dimension AESA of similar size. Still, the W-band frequency requires element spacing of approximately 2.5 mm, presenting significant challenges for signal routing and incorporation of MMICs. By combining (Gallium Nitride) GaN MMIC technology with micro-machined radiators and interconnects and silicon-germanium (SiGe) beamforming MMICs, we are able to meet all the performance and packaging requirements of the linear array feed and enable simultaneous scanning of Ka-band and W-band radars over swath of up to 100 km.

Aerosol/Climate/Ecosystems (ACE) Mission↗
NASA NTRS2019

Development Concepts for Mars Ascent Vehicle (MAV) Solid and Hybrid Vehicle Systems

The Advanced Concepts Office (ACO) at Marshall Space Flight Center (MSFC) has conducted ongoing studies and trades into options for both hybrid and solid vehicle systems for potential Mars Ascent Vehicle (MAV) concepts for the Jet Propulsion Laboratory (JPL). Two MAV propulsion options are being studied for use in a potential Mars Sample Retrieval (MSR) campaign. The following paper describes the current concepts for hybrid and solid propulsion vehicles for MAV as part of a potential MSR campaign, and provides an overview of the ongoing studies and trades for both hybrid and solid vehicle system concepts. Concepts and options under consideration for vehicle subsystems include reaction control system (RCS), separation, and structures will be described in terms of technology readiness level (TRL), benefit to the vehicle design, and associated risk. A hybrid propulsion system, which uses a solid fuel core and liquid oxidizer, is currently being developed by JPL with support from MSFC. This type of hybrid propulsion vehicle would allow the MAV to be more flexible at the cost of higher complexity, in contrast to the solid propulsion vehicle that is simpler, but allows less flexibility. The solid propulsion vehicle study performed by MSFC in 2018 further refined the solid propulsion system sizing as well as added definition to vehicle subsystem concepts, including the RCS, structures and configuration, interstage and separation, aerodynamics, and power/avionics. The studies were performed using an iterative concept design methodology, engaging subject matter experts from across MSFC’s propulsion and vehicle systems disciplines as well as seeking trajectory feedback from analysts at JPL.

McCollum, Lisa Tunstill↗
NASA NTRS1990

Using input command pre-shaping to suppress multiple mode vibration

Spacecraft, space-borne robotic systems, and manufacturing equipment often utilize lightweight materials and configurations that give rise to vibration problems. Prior research has led to the development of input command pre-shapers that can significantly reduce residual vibration. These shapers exhibit marked insensitivity to errors in natural frequency estimates and can be combined to minimize vibration at more than one frequency. This paper presents a method for the development of multiple mode input shapers which are simpler to implement than previous designs and produce smaller system response delays. The new technique involves the solution of a group of simultaneous non-linear impulse constraint equations. The resulting shapers were tested on a model of MACE, an MIT/NASA experimental flexible structure.

Hyde, James M.↗
NASA NTRS1987

Aerodynamic analysis for aircraft with nacelles, pylons, and winglets at transonic speeds

A computational method has been developed to provide an analysis for complex realistic aircraft configurations at transonic speeds. Wing-fuselage configurations with various combinations of pods, pylons, nacelles, and winglets can be analyzed along with simpler shapes such as airfoils, isolated wings, and isolated bodies. The flexibility required for the treatment of such diverse geometries is obtained by using a multiple nested grid approach in the finite-difference relaxation scheme. Aircraft components (and their grid systems) can be added or removed as required. As a result, the computational method can be used in the same manner as a wind tunnel to study high-speed aerodynamic interference effects. The multiple grid approach also provides high boundary point density/cost ratio. High resolution pressure distributions can be obtained. Computed results are correlated with wind tunnel and flight data using four different transport configurations. Experimental/computational component interference effects are included for cases where data are available. The computer code used for these comparisons is described in the appendices.

Boppe, Charles W.↗
NASA NTRS2000

Effective Potential Energies and Transport Cross Sections for Interactions of Hydrogen and Nitrogen

The interaction energies for N2-He and N2-H2 are calculated by accurate ab initio methods. The virial coefficient and differential scattering cross section for N2-H2 are calculated; the theoretical results are compared with experimental data. The transport collision integrals for N2-H2 and N2-N2 interactions are calculated and tabulated; the results yield transport coefficients that compare well with measured data. Transport coefficients are found to be determined accurately from the interaction energies for a specific configuration of the molecule formed from the interaction partners. Comparisons with results of measurement and accurate calculations demonstrate that the transport properties of complex molecular interactions can be determined rapidly and fairly accurately from the interaction energies of simpler system using combination rules for the short-range parameters of effective interaction energies and the coefficients for the long-range forces. The coefficients for a two-parameter temperature expansion of diffusion and viscosity are tabulated for a realistic universal potential energy that is based primarily on the results of very accurate calculations of the He-He interaction energy.

Stallcop, James R.↗
DOE OSTI2024

DESS (Differential Equation System Solver) [SWR-24-48]

The Differential Equation System Solver (DESS) is a Rust crate implementing fixed-step and adaptive-step solvers and designed especially for modeling physical systems. Seven explicit ordinary differential equation (ODE) solver methods have been added so far: Euler’s, Heun’s, Midpoint, Ralston’s, Classic Runge-Kutta, Bogacki-Shampine, and Cash-Karp. These comprise five fixed-step methods and two adaptive-step methods. Few solver packages are implemented in the Rust ecosystem and none are intended specifically for physical system modeling, so the goal of DESS is to create a Rust ODE solver crate designed to easily specify and model physical systems with modular, configurable solver options. In addition to allowing users to directly input equations to solve, DESS allows users to optionally specify and define relationships between nodes in their system, which the package then translates into a system of equations via the Rust macro system, leading to simpler and more intuitive code.

Steuteville, Robin↗
DOE OSTI2017

Chapter 11: Concentrating Solar Power

This chapter summarizes the applications of the supercritical CO2 (sCO2) Brayton cycle in concentrating solar power (CSP) plants. The design and operation of CSP plants are reviewed to highlight the requirements for the power cycle and attributes that are advantageous for the solar-thermal application. The sCO2 Brayton cycle offers the potential of higher cycle efficiency versus superheated or supercritical steam cycles at temperatures relevant for CSP applications. In addition, Brayton cycle systems using sCO2 are anticipated to have smaller weight and volume, lower thermal mass, and less complex power blocks compared with Rankine cycles due to the higher density of the fluid and simpler cycle design. The simpler machinery and compact size of the sCO2 process may also reduce the installation, maintenance, and operation cost of the system. Power cycle capacities in the range of 10-150 MWe are anticipated for the CSP application. In this chapter, we explore sCO2 Brayton cycle configurations that have attributes that are desirable from the perspective of a CSP application, such as the ability to accommodate dry cooling and daily cycling, as well as integration with thermal energy storage.

14 SOLAR ENERGY↗
DOE OSTI2021

HVAC System Performance for Energy Codes (Technical Brief)

The prescriptive path is likely the most widely used approach for commercial code compliance in the United States. Though easy to implement, the prescriptive approach does not discriminate between high-performing and poorly performing heating, ventilation, air conditioning (HVAC) system configurations that are both minimally compliant. To meet aggressive energy and carbon reduction goals, energy codes will need to transition from prescriptive to performance-based approaches, a transition that is riddled with several challenges. HVAC System Performance is a discipline performance path and provides a simpler solution to HVAC system evaluation compared to whole building performance, while keeping tradeoffs limited to specific building systems. The Total System Performance Ratio (TSPR) is a metric for evaluation of overall system efficiency instead of individual component efficiency, a solution that could also eventually facilitate the transition to a 100% performance-based code structure. TSPR is a ratio that compares the annual heating and cooling load of a building to the annual energy consumed by the building’s HVAC system. A web-based calculation tool has been developed for determining a building’s TSPR. Already incorporated into the 2018 Washington State Energy Code, this approach has also been evaluated by the ASHRAE Standard 90.1 Project Committee and has the potential to provide a comprehensive performance-based approach for HVAC system evaluation and analysis

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2021

Total System Performance Ratio—A Systems Based Approach for Evaluating HVAC System Efficiency

The prescriptive path is the most widely used approach for commercial code compliance in the United States. Though easy to implement, prescriptive approaches do not typically discriminate between minimally compliant, high-performing and poorly performing HVAC system configurations. Hence, to meet aggressive energy and carbon reduction goals, it is clear that energy codes will need to transition from prescriptive to performance-based approaches, a transition that is riddled with several challenges. This paper discusses a new HVAC system-based performance approach (HVAC System Performance) which provides a simpler solution to HVAV system evaluation compared to whole building performance, while keeping tradeoffs limited to specific building systems. The Total System Performance Ratio (TSPR) is a metric for evaluation of overall system efficiency instead of individual component efficiency, a solution which could also eventually facilitate the transition to a 100% performance-based code structure. TSPR is a ratio that compares the annual heating and cooling load of a building to the annual energy consumed by the building’s HVAC system. A calculation software tool has been developed for determining a building’s TSPR. Already incorporated into the 2018 Washington State Energy Code, this approach is also being evaluated by ASHRAE Standard 90.l Project Committee and has the potential to provide a comprehensive performance-based approach for HVAC system evaluation and analysis.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
NASA NTRS1983

Application of Calspan pitch rate control system to the Space Shuttle for approach and landing

A pitch rate control system designed for use in the shuttle during approach and landing was analyzed and compared with a revised control system developed by NASA and the existing OFT control system. The design concept control system uses filtered pitch rate feedback with proportional plus integral paths in the forward loop. Control system parameters were designed as a function of flight configuration. Analysis included time and frequency domain techniques. Results indicate that both the Calspan and NASA systems significantly improve the flying qualities of the shuttle over the OFT. Better attitude and flight path control and less time delay are the primary reasons. The Calspan system is preferred because of reduced time delay and simpler mechanization. Further testing of the improved flight control systems in an in-flight simulator is recommended.

Weingarten, N. C.↗
DOE OSTI2025

Unified Universal Control and Coordination of Inverter-Based Resources, and Validation for a PV + Battery Hybrid Plant

As renewable energy deployment grows, hybrid power plants (HPPs) combining photovoltaic (PV) and battery systems must evolve to offer both energy and grid stability services. These systems typically include a mix of grid-following (GFL) and grid-forming (GFM) inverters, presenting unique coordination and control challenges. This Department of Energy–funded project developed and validated a Unified Universal Control and Coordination (UUCC) framework for such PV + battery hybrid plants, enabling seamless and stable operation, including ultrafast black start, autonomous synchronization, and robust frequency and voltage regulation, under different grid conditions. The project significantly advanced the understanding of inverter-based resource (IBR) control by developing and validating three complementary system-level approaches for hybrid GFL/GFM operation: 1. A combined Virtual Resistance (VR)-based GFL and Virtual Oscillator Control (VOC)-based GFM method, where each inverter type is governed by a specialized control strategy. Together, these achieve stable, fast-response coordination, eliminating inrush current and enabling smooth black start and grid synchronization across a wide range of grid strengths. 2. A Deadbeat-based UUCC strategy, which uses discrete-time, switching-cycle-level control for both GFL and GFM inverters. This approach replaces traditional PI/PLL control with a control parameter-free, high-bandwidth framework that supports stable LVRT and instantaneous synchronization under all conditions. 3. A benchmark comparison with Siemens’ commercial GFM microgrid controller, which provided a fast baseline platform. The commercial approach decoupled v & f control was implemented on a commercial microgrid controller.The baseline commercial benchmark helped highlight superior transient response and black start performance offered by the deadbeat and VOC approaches. These technical contributions offer substantial improvements over conventional inverter control schemes, which often rely on slow phase-locked loop (PLL)-based synchronization, require careful control parameters tuning, and prone to unstable in weak grids with GFL inverters and in stiff grid with GFM inverters therefore challenging for hybrid GFL+GFM under all grid conditions. The deadbeat-based UUCC framework enables simpler, faster, and more robust operation of hybrid IBR systems using wide-bandgap (WBG) devices such as SiC power semiconductors. The rapid expansion of hybrid distributed energy resources (DERs), including residential and commercial PV-BESS installations such as Tesla Powerwall, PV with vehicle-to-grid (V2G) capability, and other integrated configurations, presents complex operational challenges for medium-voltage radial distribution feeders. These networks are subject to frequent disturbances such as faults, switching operations, rapid reclosing sequences, and feeder reconfigurations, all of which introduce dynamic stress on IBRs. In addition, planned feeder segmentation and deliberate islanding for resilience will require DERs that can autonomously perform blackstart, establish voltage and frequency references, and resynchronize with the main grid. The advanced deadbeat-based UUCC control and blackstart functionalities developed in this project directly address these requirements, enabling decentralized and autonomous operation of inverter-dominated DERs in distribution systems under a wide range of fault and reconfiguration scenarios. From a public benefit perspective, these innovations enable more reliable and cost-effective integration of renewable energy into distribution networks. The ability to autonomously black start and stabilize grids under varying grid conditions support accelerates recovery from outages and support decentralized resilient energy systems. By reducing system complexity and improving performance, this project lays critical groundwork for future inverter-dominated power grids that are clean, reliable, and accessible to all.

14 SOLAR ENERGY↗
DOE OSTI2020

Derivation of K-matrix reaction theory in a discrete basis formalism

The usual derivations of the S and K matrices for two-particle reactions proceed through the Lippmann–Schwinger equation with formal definitions of the incoming and outgoing scattering states. Here we present a simpler alternative derivation that is carried out completely in the Hamiltonian representation, using a discrete basis of configurations for the scattering channels as well as the quasi-bound configurations of the combined fragments. We use matrix algebra to derive an explicit expression for the K matrix in terms of the Hamiltonian of the internal states of the compound system and the coupling between the channels and the internal states. The formula for the K matrix includes explicitly a real dispersive shift matrix to the internal Hamiltonian that is easily computed in the formalism. That expression is applied to derive the usual form of the S matrix as a sum over poles in the complex energy plane. Some extensions and limitations of the discrete-basis Hamiltonian formalism are discussed in the concluding remarks and in the Appendix.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS2010

Dynamic Optimization of Multi-Spacecraft Relative Navigation Configurations in the Earth-Moon System

In this paper, the notion of relative navigation introduced by Hill, Lo and Born is analyzed for a large class of periodic orbits in the Earth-Moon three-body problem, due to its potential in supporting Moon exploration efforts. In particular, a navigation metric is introduced and used as a cost function to optimize over a class of periodic orbits. While the problem could be solve locally as an optimal control problem, a dynamical based approach that allows for a global/systematic view of the problem is proposed. First, the simpler problem of multiple spacecraft placement on a given periodic orbit is solved before the notion of continuation and bifurcation analysis is used to expand the range of solutions thus obtained.

Three Body Problem↗
DOE OSTI2024

Implementing Ordinary Differential Equation Solvers in Rust Programming Language for Modeling Vehicle Powertrain Systems: Preprint

Efficient and accurate ordinary differential equation (ODE) solvers are necessary for powertrain and vehicle dynamics modeling. However, current commercial ODE solvers can be financially prohibitive, leading to a need for accessible, effective, open-source ODE solvers designed for powertrain modeling. Rust is a compiled programming language that has the potential to be used for fast and easy-to-use powertrain models, given its exceptional computational performance, robust package ecosystem, and short time required for modelers to become proficient. However, of the three commonly used (>3,000 downloads) packages in Rust with ODE solver capabilities, only one has more than four numerical methods implemented, and none are designed specifically for modeling physical systems. Therefore, the goal of the Differential Equation System Solver (DESS) was to implement accurate ODE solvers in Rust designed for the component-based problems often seen in powertrain modeling. DESS is a text-based software package that provides a flexible framework for building and solving systems of ODEs. This allows DESS to be included as a dependency for automotive powertrain models that require a variety of solvers and solver configurations. Seven explicit ODE solver methods have been implemented in DESS: Euler’s, Heun’s, midpoint, Ralston’s, classic Runge-Kutta, Bogacki-Shampine, and Cash-Karp. These represent five fixed-step methods and two adaptive-step methods. This paper shows that the solver implementations increase accuracy and computational efficiency compared to Euler's method when modeling a system of three thermal masses in Rust. DESS also includes features designed for modeling component-based physical systems. Users can define relationships between nodes in their system, which the package then translates into a system of equations, leading to simpler and more intuitive code. In the case of a three-thermal-mass system, the user can specify node thermal properties (e.g., thermal capacitance), how nodes are interconnected, and thermal conductance between nodes rather than providing a system of equations. The core contribution from this work is an open-source, text-based Rust package with ODE solvers for automotive powertrain modeling to support cost-free, fast, and accurate simulation.

ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2018

Emergence of the mass discrepancy-acceleration relation from dark matter-baryon interactions

The observed tightness of the mass discrepancy-acceleration relation (MDAR) poses a fine-tuning challenge to current models of galaxy formation. We propose that this relation could arise from collisional interactions between baryons and dark matter (DM) particles, without the need for modification of gravity or ad hoc feedback processes. We assume that these interactions satisfy the following three conditions: (i) the relaxation time of DM particles is comparable to the dynamical time in disk galaxies; (ii) DM exchanges energy with baryons due to elastic collisions; (iii) the product between the baryon-DM cross section and the typical energy exchanged in a collision is inversely proportional to the DM number density. As a proof of principle, we present an example of a particle physics model that gives a DM-baryon cross section with the desired density and velocity dependence. For consistency with direct detection constraints, our DM particles must be either very light (m $\ll$ m b ) or very heavy (m$\gg$ m b ), corresponding respectively to heating and cooling of DM by baryons. In both cases, our mechanism applies and an equilibrium configuration can in principle be reached. In this exploratory paper, we focus on the heavy DM/cooling case because it is technically simpler, since the average energy exchanged turns out to be approximately constant throughout galaxies. Under these assumptions, we find that rotationally-supported disk galaxies could naturally settle to equilibrium configurations satisfying a MDAR at all radii without invoking finely tuned feedback processes. We also discuss issues related to the small scale clumpiness of baryons, as well as predictions for pressure-supported systems. Here we argue in particular that galaxy clusters do not follow the MDAR despite being DM-dominated because they have not reached their equilibrium configuration. Finally, we revisit existing phenomenological, astrophysical and cosmological constraints on baryon-DM interactions in light of the unusual density dependence of the cross section of DM particles.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗