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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 685 records · Page 38
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

When Photoelectrons Meet Gas Molecules: Determining the Role of Inelastic Scattering in Ambient Pressure X-ray Photoelectron Spectroscopy

Inelastic photoelectron scattering (IPES) by gas molecules, a critical phenomenon observed in ambient pressure X-ray photoelectron spectroscopy (APXPS), complicates spectral interpretation due to kinetic energy loss in the primary spectrum and the appearance of additional features at higher binding energies. In this study, we systematically investigate IPES in various gas environments using APXPS, providing detailed insights into interactions between photoelectrons emitted from solid surfaces and surrounding gas molecules. Corelevel XPS spectra of Au, Ag, Zn, and Cu metals were recorded over a wide kinetic energy range in the presence of CO 2 , N 2 , Ar, and H 2 gases, demonstrating the universal nature of IPES across different systems. Additionally, we analyzed spectra of scattering effects induced by gas-phase interactions without metal solids. In two reported CO 2 -reduction systems (p-GaN/ Au/Cu and p-Si/TaO x /Cu), we elucidated that IPES is independent of the composition, structure, or size of the solid materials. Using metal foil platforms, we further developed an analytical model to extract electron excitation cross sections of gas molecules. These findings enhance our understanding of IPES mechanisms and enable the predictions of IPES structures in other solid–gas systems, providing a valuable reference for future APXPS studies and improving the accuracy of spectral analysis in gas-rich catalytic interfaces.

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

Leveraging Natural Language Processing and Generative Models in Molecular Chemistry: Property Prediction and Novel Compound Generation

The accurate prediction of molecular properties is important for the rational design and the advancement of green chemistry and sustainable materials research. However, the predictive power of traditional computational chemistry methods is limited due to computational restrictions. Here, in this study, we examine an alternative approach to the accurate prediction of properties of organic compounds: natural language processing (NLP)-based molecular embedding. Using viscosity, partition coefficient (log P), and enthalpy of vaporization as test properties through a survey of comprehensive datasets comprising 5695 data points for viscosity, 25 870 data points for log P, and 2296 data points for enthalpy of vaporization. These are important properties for the design of greener, safer, and sustainable chemical processes. Models were trained using NLP methods such as Mol2vec and fine-tuned ChemBERTa, and results were compared with traditional input featurization techniques such as Morgan fingerprints and quantum chemistry derived sigma profiles and DFT features. Among the various machine learning models, Mol2vec demonstrated superior predictive capabilities, achieving the highest correlation coefficient (R 2 = 0.945) and lowest RMSE (0.106 mPa s) for viscosity, as well as high accuracy for log P and enthalpy of vaporization predictions. These findings establish the Mol2vec featurization technique, graph-convolutional neural networks (GCNN), and fine-tuned ChemBERTa model as powerful tools for predictive modeling of organic compounds properties, offering a significant improvement over previously used featurization techniques and opening up strategies for very-high-throughput computational screening. Finally, we integrated ML models with hybrid language-model-based generative adversarial networks (LM-GAN) to generate novel molecular sequences with desirable properties for different research applications. The ability to computationally design solvents with lower viscosity, lower log P, and lower enthalpy of vaporization offers a data-driven route to accelerating the discovery of sustainable alternatives to traditionally toxic solvents.

ChemBERTa↗
DOE OSTI2020

An Observational Comparison of Level of Neutral Buoyancy and Level of Maximum Detrainment in Tropical Deep Convective Clouds

Tropical deep convective clouds are important drivers of large-scale atmospheric circulation representing the main vertical transport pathway through the depth of the troposphere for heat, momentum, water, and chemical species. The strength and depth of this transport are impacted by the convective updraft size and intensity that are driven by buoyancy, dynamical forcing, and mixing of environmental air, i.e., entrainment. In this paper, we identify tropical deep convective systems with well-defined forward anvils using Atmospheric Radiation Measurement (ARM) ground-based profiling radars, at three ARM fixed-sites in the Tropical Western Pacific (TWP; i.e., Manus, Nauru, Darwin) and three ARM Mobile Facility deployments in Niamey, Niger; Gan Island, Maldives; and Manacapuru, Brazil. We use the difference between the level of neutral buoyancy (LNB) and the level of maximum detrainment (LMD) as a proxy for the effective bulk convective entrainment (εproxy). The LNB, the theoretical height that a parcel raised above the level of free convection would reach with no mixing, is calculated based on pre-convection radiosonde measurements using parcel theory. The LMD is the height of the maximum reflectivity observed in forward anvil clouds by profiling radars. Deep convective systems over the TWP show higher LNBs that extend to 16.3 km on average and larger εproxy (median LNB minus LMD up to 6.5 km) compared to their continental counterparts in the Amazon and West Africa. Oceanic conditions show larger convective available potential energy (CAPE) coupled with higher moisture at low levels which favors larger εproxy. In contrast, continental cases initiate and develop under high convective inhibition, steeper environmental lapse rate, and high wind shear conditions, which show smaller offset between LNB and LMD. Deep convective cases that promote significant cold pools at the surface experience less εproxy. Using a Random Forest regression algorithm, CAPE is associated with the highest feature importance score for predicting convective εproxy, followed by low-level relative humidity. For continental cases, the low-level wind shear also indicates higher importance.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Comparison of Conventional and Constrained Variational Methods for Computing Large-Scale Budgets and Forcing Fields

Analyses of atmospheric heat and moisture budgets serve as an effective tool to study convective characteristics over a region and to provide large-scale forcing fields for various modeling applications. This paper examines two popular methods for computing large-scale atmospheric budgets: the conventional budget method (CBM) using objectively gridded analyses based primarily on radiosonde data and the constrained variational analysis (CVA) approach which supplements vertical profiles of atmospheric fields with measurements at the top of the atmosphere and at the surface to conserve mass, water, energy, and momentum. Successful budget computations are dependent on accurate sampling and analyses of the thermodynamic state of the atmosphere and the divergence field associated with convection and the large-scale circulation that influences it. Utilizing analyses generated from data taken during Dynamics of the Madden-Julian Oscillation (DYNAMO) field campaign conducted over the central Indian Ocean from October to December 2011, we evaluate the merits of these budget approaches and examine their limitations. While many of the shortcomings of the CBM, in particular effects of sampling errors in sounding data, are effectively minimized with CVA, accurate large-scale diagnostics in CVA are dependent on reliable background fields and rainfall constraints. Finally, for the DYNAMO analyses examined, the operational model fields used as the CVA background state provided wind fields that accurately resolved the vertical structure of convection in the vicinity of Gan Island. However, biases in the model thermodynamic fields were somewhat amplified in CVA resulting in a convective environment much weaker than observed.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Near-field terahertz nonlinear optics with blue light

Abstract The coupling of terahertz optical techniques to scattering-type scanning near-field microscopy (s-SNOM) has recently emerged as a valuable new paradigm for probing the properties of semiconductors and other materials on the nanoscale. Researchers have demonstrated a family of related techniques, including terahertz nanoscopy (elastic scattering, based on linear optics), time-resolved methods, and nanoscale terahertz emission spectroscopy. However, as with nearly all examples of s-SNOM since the technique’s inception in the mid-1990s, the wavelength of the optical source coupled to the near-field tip is long, usually at energies of 2.5 eV or less. Challenges in coupling of shorter wavelengths (i.e., blue light) to the nanotip has greatly inhibited the study of nanoscale phenomena in wide bandgap materials such as Si and GaN. Here, we describe the first experimental demonstration of s-SNOM using blue light. With femtosecond pulses at 410 nm, we generate terahertz pulses directly from bulk silicon, spatially resolved with nanoscale resolution, and show that these signals provide spectroscopic information that cannot be obtained using near-infrared excitation. We develop a new theoretical framework to account for this nonlinear interaction, which enables accurate extraction of material parameters. This work establishes a new realm of possibilities for the study of technologically relevant wide-bandgap materials using s-SNOM methods.

47 OTHER INSTRUMENTATION↗
DOE OSTI2020

Controlling spin current polarization through non-collinear antiferromagnetism

The interconversion of charge and spin currents via spin-Hall effect is essential for spintronics. Energy-efficient and deterministic switching of magnetization can be achieved when spin polarizations of these spin currents are collinear with the magnetization. However, symmetry conditions generally restrict spin polarizations to be orthogonal to both the charge and spin flows. Spin polarizations can deviate from such direction in nonmagnetic materials only when the crystalline symmetry is reduced. Here, we show control of the spin polarization direction by using a non-collinear antiferromagnet Mn 3 GaN, in which the triangular spin structure creates a low magnetic symmetry while maintaining a high crystalline symmetry. We demonstrate that epitaxial Mn 3 GaN/permalloy heterostructures can generate unconventional spin-orbit torques at room temperature corresponding to out-of-plane and Dresselhaus-like spin polarizations which are forbidden in any sample with two-fold rotational symmetry. Our results demonstrate an approach based on spin-structure design for controlling spin-orbit torque, enabling high-efficient antiferromagnetic spintronics.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Efficient and bright white light-emitting diodes based on single-layer heterophase halide perovskites

At present, electric lighting accounts for ~15% of global power consumption and thus the adoption of efficient, low-cost lighting technologies is important. Here, halide perovskites have been shown to be good emitters of pure red, green and blue light, but an efficient source of broadband white electroluminescence suitable for lighting applications is desirable. Here, we report a white light-emitting diode (LED) strategy based on solution-processed heterophase halide perovskites that, unlike GaN white LEDs, feature only one broadband emissive layer and no phosphor. Our LEDs operate with a peak luminance of 12,200 cd m –2 at a bias of 6.6 V and a maximum external quantum efficiency of 6.5% at a current density of 8.3 mA cm –2 . Systematic in situ and ex situ characterizations reveal that the mechanism of efficient electroluminescence is charge injection into the α phase of CsPbI 3 , α to δ charge transfer and α–δ balanced radiative recombination. Future advances in fabrication technology and mechanistic understanding should lead to further improvements in device efficiency and luminance.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Black carbon aerosol number and mass concentration measurements by picosecond short-range elastic backscatter lidar

Abstract Black carbon aerosol emissions are recognized as contributors to global warming and air pollution. There remains, however, a lack of techniques to remotely measure black carbon aerosol particles with high range and time resolution. This article presents a direct and contact-free remote technique to estimate the black carbon aerosol number and mass concentration at a few meters from the emission source. This is done using the Colibri instrument based on a novel technique, referred to here as Picosecond Short-Range Elastic Backscatter Lidar (PSR-EBL). To address the complexity of retrieving lidar products at short measurement ranges, we apply a forward inversion method featuring radiometric lidar calibration. Our method is based on an extension of a well-established light-scattering model, the Rayleigh–Debye–Gans for Fractal-Aggregates (RDG-FA) theory, which computes an analytical expression of lidar parameters. These parameters are the backscattering cross-sections and the lidar ratio for black carbon fractal aggregates. Using a small-scale Jet A-1 kerosene pool fire, we demonstrate the ability of the technique to quantify the aerosol number and mass concentration with centimetre range-resolution and millisecond time-resolution.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2024

FORSE+: Simulating non-Gaussian CMB foregrounds at 3 arcmin in a stochastic way based on a generative adversarial network

We present FORSE+, a Python package that produces non-Gaussian diffuse Galactic thermal dust emission maps at arcminute angular scales and that has the capacity to generate random realizations of small scales. This represents an extension of the FORSE (Foreground Scale Extender) package, which was recently proposed to simulate non-Gaussian small scales of thermal dust emission using generative adversarial networks (GANs). With the input of the large-scale polarization maps from observations, FORSE+ has been trained to produce realistic polarized small scales at 3′ following the statistical properties, mainly the non-Gaussianity, of observed intensity small scales, which are evaluated through Minkowski functionals. Furthermore, by adding different realizations of random components to the large-scale foregrounds, we show that FORSE+ is able to generate small scales in a stochastic way. In both cases, the output small scales have a similar level of non-Gaussianity compared with real observations and correct amplitude scaling as a power law. These realistic new maps will be useful, in the future, to understand the impact of non-Gaussian foregrounds on the measurements of the cosmic microwave background (CMB) signal, particularly on the lensing reconstruction, de-lensing, and the detection of cosmological gravitational waves in CMB polarizationB-modes.

Astronomy & Astrophysics↗
DOE OSTI2020

Recovery from plasma etching-induced nitrogen vacancies in p-type gallium nitride using UV/O 3 treatments

Plasma etching of p-type GaN creates n-type nitrogen vacancy (V N ) defects at the etched surface, which can be detrimental to device performance. In mesa isolated diodes, etch damage on the sidewalls degrades the ideality factor and leakage current. Here a treatment was developed to recover both the ideality factor and leakage current, which uses UV/O 3 treatment to oxidize the damaged layers followed by HF etching to remove them. The temperature dependent I–V measurement shows that the reverse leakage transport mechanism is dominated by Poole–Frenkel emission at room temperature through the etch-induced V N defect. Depth resolved cathodoluminescence confirms that the damage is limited to first several nanometers and is consistent with the V N defect.

42 ENGINEERING↗
DOE OSTI2020

Electron and hole mobility of rutile GeO 2 from first principles: An ultrawide-bandgap semiconductor for power electronics

Rutile germanium dioxide (r-GeO 2 ) is a recently predicted ultrawide-bandgap semiconductor with potential applications in high-power electronic devices, for which the carrier mobility is an important material parameter that controls the device efficiency. We apply first-principles calculations based on density functional and density functional perturbation theory to investigate carrier-phonon coupling in r-GeO 2 and predict its phonon-limited electron and hole mobilities as a function of temperature and crystallographic orientation. The calculated carrier mobilities at 300 K are μ elec , ⊥ c → = 244 cm 2 V –1 s –1 , μ elec , ∥ c → = 377 cm 2 V –1 s –1 , μ hole , ⊥ c → = 27 cm 2 V –1 s –1 , and μ hole , ∥ c → = 29 cm 2 V –1 s –1 . At room temperature, carrier scattering is dominated by the low-frequency polar-optical phonon modes. The predicted Baliga figure of merit of n -type r-GeO 2 surpasses several incumbent semiconductors such as Si, SiC, GaN, and β -Ga 2 O 3 , demonstrating its superior performance in high-power electronic devices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Neutron radiation hardness of aluminum gallium nitride UV LEDs at various wavelengths

Radiation hard diagnostics are critical to the success of nuclear fusion at National Ignition Facility, Z, ITER, and prolonged space explorations. We have first demonstrated the exceptional proton radiation hardness of initial GaN devices and qualified their space flight and deployment for missions such as the Laser Interferometer Space Antenna and International Space Station. We have further conducted neutron radiation hardness experiments at Los Alamos Neutron Science Center by opening a new high fluence beam station. During 2014–2016, we irradiated multiple Aluminum Gallium Nitride (AlGaN) LEDs with a maximum fluence of 2.4 × 1013 neutrons/cm2 in 3 years and generated 54 161 current–voltage (I–V) scan traces. Our data processing program analyzes each and all I–V traces. In addition, we retrieved local temperature records to analyze and remove temperature effects in the outdoor environment. The I–V curve families of AlGaN UV LEDs with emitting wavelengths of 265, 275, and 310 nm were compared. The I–V curves of 265 nm AlGaN UV LEDs have the smallest deviations from the average value, while the I–V curves for 310 nm AlGaN LEDs showed the largest deviations from the average value. We have reached another important recommendation for the optimal use of multiple AlGaN optoelectronic devices or imaging arrays for inertially confined fusion diagnostics: Shorter wavelength devices at 265 nm exhibit more consistent radiation hardness performance than the 310 nm devices. Higher aluminum content LEDs or AlxGa1–xN devices with higher mole fraction x for generating shorter wavelengths have better radiation hardness for fusion diagnostics.

Sun, K. -X. (ORCID:0000000224659235)↗
DOE OSTI2021

Spatially resolved x-ray detection with photonic crystal scintillators

We study the self-collimation phenomenon in photonic crystals (PhC) of wide bandgap materials for ultra-fast and high spatial resolution x-ray detection. We work on various heavy inorganic scintillators: BaF 2 , GaN, ZnO, CsI:Tl, NaI:Tl, LYSO, WO 4 compounds, and plastic scintillators. Conventional scintillator detectors do not rely on a direct detection mechanism; hence, they require intricate design and fabrication processes. We offer a PhC design to observe self-collimation phenomena and overcome the ongoing spatial resolution challenges with these types of materials. We investigate the photonic band diagrams and iso-frequency contours. Fourier transforms based on finite-difference time-domain and frequency domain simulations are done for verifying and analyzing the self-collimation with the selected material. Light extraction efficiency at the PhC–air interface, depending on the truncation distance from the excitation point, is measured. Beam divergence values are calculated at 1 mm propagation distance. The vertical field profiles are obtained to observe the confinement. For the spatial resolution analysis, cross-sectional beam profiles have been examined. Furthermore, Gaussian envelopes are fitted to beam profiles for a consistent data analysis, and full-width-at-half-maximum values are considered. As a result, we theoretically prove and demonstrate the spatially resolved x-ray detection at the sub-micrometer level for a wide range of scintillator materials.

36 MATERIALS SCIENCE↗
DOE OSTI2021

On the design and performance of InGaN/Si double-junction photocathodes

Via a combined theoretical and experimental study, we have investigated the synthesis and performance characteristics of InGaN/Si double-junction photoelectrochemical (PEC) water splitting devices, which promise a theoretical solar-to-hydrogen conversion efficiency ~30% under AM 1.5G one-sun illumination. The double-junction photocathodes consist of a p + -InGaN top light absorber and a Si bottom p-n junction, which are connected through a nanowire tunnel junction. The effect of indium composition of the top light absorber as well as the impact of p-type Mg dopant incorporation on the PEC performance was studied. Experimentally, the sample with 32% indium composition showed a maximum photocurrent density of ~9 mA/cm 2 at 0.4 mV vs reverse hydrogen electrode (RHE) with applied bias photon-to-current efficiency (ABPE) of ~9.5%. An optimum p-type doping level ~1 × 10 17 cm -3 was also identified, which results in the best device performance as a result of optimum surface band bending as well as vertical charge carrier (hole) transport. These results also show a good agreement with our theoretical analysis. This work provides significant insights in advancing the design and development of high efficiency PEC devices for artificial photosynthesis using industry ready materials, e.g., Si and GaN, to achieve large-scale, low-cost onsite hydrogen fuel production.

42 ENGINEERING↗
DOE OSTI2021

Computational Fermi Level Engineering and Doping-Type Conversion of Mg:Ga 2 O 3 via Three-Step Synthesis Process

Gallium oxide (Ga 2 O 3 ) is being actively explored for electronics that can operate at high power, temperature, and frequency as well as for deep-ultraviolet optoelectronics and other applications due to its ultra-wide bandgap (UWBG) and low projected fabrication cost of large-size and high-quality crystals. Efficient n-type doping of monoclinic beta-phase of Ga 2 O 3 has been achieved, but p-type doping faces fundamental obstacles due to compensation, deep acceptor levels, and the polaron transport mechanism of free holes. However, aside from the challenges of achieving p-type conductivity, plenty of opportunity exists to engineer the position of the Fermi level for improved design of Ga 2 O 3 -based devices. We use first-principles defect theory and defect equilibrium calculations to simulate a three-step growth-annealing-quench synthesis protocol for hydrogen-assisted Mg doping in ß-Ga 2 O 3 . The simulations take into account the gas phase equilibrium between H 2 , O 2 , and H 2 O, which determines the H chemical potential. We predict Ga 2 O 3 doping-type conversion to a net p-type regime after growth under reducing conditions in the presence of H2 followed by O-rich annealing, which is a similar process to Mg acceptor activation by H removal in GaN. For equilibrium annealing with re-equilibration of compensating O vacancies, there is an optimal temperature that maximizes the Ga 2 O 3 net acceptor density for a given Mg doping level; the acceptor density is further increased in the non-equilibrium annealing scenario without re-equilibration. After quenching to operating temperature, the Ga 2 O 3 Fermi level drops below mid-gap down to about 1.5 eV above the valence band maximum, creating a significant number of uncompensated neutral Mg Ga 0 acceptors. The resulting free hole concentration in Ga 2 O 3 is very low even at elevated operating temperature (~10 8 cm -3 at 400°C) due to the deep energy level of these Mg acceptors, and hole conductivity is further impeded by the polaron hopping mechanism. However, the Fermi-level reduction and suppression of free electron density in this doping-type converted (NA>ND) Ga 2 O 3 material are important for improved designs of Ga 2 O 3 electronic devices. These results illustrate the power of computational predictions not only for new materials but also for their synthesis science.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Strong effect of scandium source purity on chemical and electronic properties of epitaxial Sc x Al 1–x N/GaN heterostructures

Epitaxial multilayer heterostructures of Sc x Al 1–x N/GaN with Sc contents x = 0.11–0.45 are found to exhibit significant differences in structural quality, chemical impurity levels, and electronic properties depending on the starting Sc source impurity levels. A higher purity source leads to a 2–3 orders of magnitude reduction in the carbon, oxygen, and fluorine unintentional doping densities in MBE-grown Sc x Al 1–x N/GaN multilayers. Electrical measurements of Sc x Al 1–x N/n + GaN single heterostructure barriers show a 5–7 orders of magnitude reduction in the electrical leakage for films grown with a higher purity Sc source at most Sc contents. The measured chemical and electrical properties of epitaxial Sc x Al 1–x N highlight the importance of the starting Sc source material purity for epitaxial device applications that need these highly piezoelectric and/or ferroelectric transition-metal nitride alloys.

36 MATERIALS SCIENCE↗
DOE OSTI2021

A perspective on the electro-thermal co-design of ultra-wide bandgap lateral devices

Fundamental research and development of ultra-wide bandgap (UWBG) semiconductor devices are under way to realize next-generation power conversion and wireless communication systems. Devices based on aluminum gallium nitride (AlxGa1-xN, x is the Al composition), ..beta..-phase gallium oxide (..beta..-Ga2O3), and diamond give promise to the development of power switching devices and radio frequency power amplifiers with higher performance and efficiency than commercial wide bandgap semiconductor devices based on gallium nitride (GaN) and silicon carbide (SiC). However, one of the most critical challenges for the successful deployment of UWBG device technologies is to overcome adverse thermal effects that impact the device performance and reliability. Overheating of UWBG devices originates from the projected high power density operation and poor intrinsic thermal properties of AlxGa1-xN and ..beta..-Ga2O3. This Perspective delineates the need and process for the "electro-thermal co-design" of laterally configured UWBG electronic devices and provides a comprehensive review of current state-of-the-art thermal characterization methods, device thermal modeling practices, and both device- and package-level thermal management solutions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

A unified thermionic and thermionic-field emission (TE–TFE) model for ideal Schottky reverse-bias leakage current

We present a unified thermionic emission (TE) and thermionic-field emission (TFE) model for the ideal reverse-bias leakage current in Schottky junctions. The unified TE–TFE analytical model advances upon previous analytical TFE models by Murphy–Good and Padovani–Stratton, which are the two most widely adopted models by the community, in two major aspects: (i) the applicability of the TFE expression therein is extended to near-zero surface electric fields by an error-function correction, allowing for the calculation of the total current by a nontrivial sum of TE (over-the-barrier current) and TFE (below-the-barrier current) contributions; therefore, an accurate description of the TE-to-TFE transition region is captured analytically for the first time; (ii) image-force lowering is considered with much-simpler correction terms. Comparisons with the reference numerical model show that the unified TE–TFE model has excellent accuracy, as well as a 10 000× reduction in computation time. The unified model is further tested against experimental data from Schottky barrier diodes based on Si, 4H-SiC, GaN, and β-Ga 2 O 3 , revealing accurate extractions of barrier heights and correct descriptions of the ideal reverse leakage characteristics. With the extended applicable range, improved accuracy, and computational efficiency, the unified TE–TFE model is highly valuable for the design and analysis of devices based on Schottky junctions, as well as for potential integration in technology computer-aided design (TCAD) tools.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗