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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1978

An improved analytical model of the separated region on nozzle boattails

A viscous flow calculation is formulated for modelling the effects of separated, reverse flow regions on nozzle afterbody pressures and drags at subsonic Mach numbers. A discriminating streamline approach is used. Effects of skin friction, axial pressure gradient, and nozzle jet entrainment on the separated discriminating streamline shape are included. The viscous flow calculation is coupled to a potential flow calculation. The combined analysis is found to accurately predict both the magnitude of the measured afterbody pressures and drag, and also their variation with Reynolds number, Mach number and afterbody shape.

Presz, W. M., Jr.↗
DOE OSTI2022

Continued Verification of MOOSE Structural Mechanics Tools for Modeling Core Bowing Phenomena in Fast Reactors

Under the U.S. Department of Energy Office of Nuclear Energy’s Advanced Modeling and Simulation (NEAMS) Program, an integrated multiphysics approach is being developed to model the core bowing phenomena important to liquid metal-cooled fast reactors. Core bowing is an important passive safety mechanism whereby increased power (which leads to temperature and flux gradients) influences the core to bow into less reactive configurations when the restraint system is properly designed. The phenomenon includes a complex interplay of radiation transport, duct temperature calculations involving fluid flow and heat transfer, and thermo-mechanical responses to the induced temperature and flux gradients. Structural material properties are also important to determining inelastic response to longer term flux gradients which cause irradiation creep and swelling. While core bowing provides a strong negative reactivity feedback when the restraint system is designed properly, it also results in additional forces between assemblies which increase the loads required to extricate them during refueling or control rod movement. Therefore, the restraint system must be designed with these tradeoffs in mind. The first stage of the work, which commenced in FY21 and continues through FY22, assesses thermo-mechanical modeling tools for producing core bowing predictions consistent with conventional tools. The Multiphysics Object Oriented Simulation Environment (MOOSE) Tensor Mechanics and Contact Modules are employed. This status report describes work on additional thermo-mechanical benchmark verification problems with increased complexity from the examples demonstrated in FY21. Several benchmark verification examples were selected from the IAEA verification and validation report. These examples involve clusters of ducts representative of a sector of a hexagonal reactor core which bow into each other and cause contact and load pad elevations, as well as single ducts subjected to irradiation fields undergoing swelling and subsequent bowing. The MOOSE-based results were compared to both IAEA benchmark participants’ results, analytic equations as available, and NUBOW-3D, a beam model code developed by Argonne National Laboratory. In every case, the MOOSE results agreed with other simulations results, providing additional verification basis of the tools for this particular physics application.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

FY23 Status Report on MOOSE-Based Approaches to Modeling Core Bowing in Fast Reactors

Under the U.S. Department of Energy Office of Nuclear Energy’s Advanced Modeling and Simulation (NEAMS) Program, an integrated multiphysics approach is being developed to model the core bowing phenomena important to liquid metal-cooled fast reactors. Core bowing is an important passive safety mechanism whereby increased power (which leads to temperature and flux gradients) influences the core to bow into less reactive configurations when the restraint system is properly designed. The phenomenon includes a complex interplay of radiation transport, duct temperature calculations involving fluid flow and heat transfer, and thermo-mechanical responses to the induced temperature and flux gradients. Structural material properties are also important to determining inelastic response to longer term flux gradients which cause irradiation creep and swelling. While core bowing provides a strong negative reactivity feedback when the restraint system is designed properly, it also results in additional forces between assemblies which increase the loads required to extricate them during refueling or control rod movement. Therefore, the restraint system must be designed with these tradeoffs in mind. The first stage of the work, which commenced in FY21 and continues through FY23, assesses thermo-mechanical modeling tools for producing core bowing predictions consistent with conventional tools. The Multiphysics Object Oriented Simulation Environment (MOOSE) Tensor Mechanics and Contact Modules are employed. This status report describes work on additional thermo-mechanical benchmark verification problems with increased complexity from the examples demonstrated in FY21 and FY22. Several benchmark verification examples were selected from the IAEA verification and validation report with increased number of ducts and more complex contact interaction behavior. These examples involve a full symmetric sector with restraint rings at multiple load pad locations to simulate a limited free-bow restraint system concept, as well as irradiation induced swelling and creep effects in a sector. In addition, improvements to the contact module sideset assignment were assessed and compared with previous MOOSE results to verify the contact behavior. The MOOSE-based results were compared to IAEA benchmark participants’ results. In every case, the MOOSE results agreed with other simulations results for estimating bowing behavior, providing additional verification basis of the tools for this particular physics application. Estimation of contact forces were mostly in agreement, with a few outlier results. A plan was suggested for dealing with the discrepancies with estimating contact force values. In addition, a 1-way coupling demonstration was performed using subchannel analysis code Pronghorn-SC and MOOSE on an ABR-1000-design sodium-cooled fast reactor assembly to evaluate coolant and duct wall temperatures.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2018

Stability, Magnetism and Hardness of Iron Carbides from First-Principles Calculations

The stability, electronic structure, magnetism and hardness of iron carbides in eight different space groups were investigated by the first-principles calculations. The electron-ion interactions were modelled using the Vanderbilt ultra-soft pseudopotentials (USPP) and norm-conserving pseudopotentials (NCPP). Generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) formulation was used to describe the exchange-correlation energy. The calculated lattice parameters for USPP were in good agreement with experimental results. The formation enthalpy, cohesive energy and elastic constants revealed that with the exception of hex-Fe{sub 2}C, all Fe-C compounds were both thermodynamically and mechanically stable. Meanwhile, the iron carbides’ bulk and shear moduli were also calculated and compared. Furthermore, magnetic properties were studied and it was found that the average magnetic moment of Fe{sub 23}C{sub 6} (2.123 μ{sub B}) was the largest among these compounds, with a value close to that of pure α-Fe (2.24 μ{sub B}μ. As expected, hex-Fe{sub 2}C is a paramagnetic metal based on the obtained densities of states and partial densities of states. Furthermore, the magnetic behaviour of iron carbides with polymorphic transformation was almost identical. Finally, several semi-empirical methods were used for estimating the hardness of the iron carbides.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

A Stochastic Gradient Descent Approach for Stochastic Optimal Control

In this work, we introduce a stochastic gradient descent approach to solve the stochastic optimal control problem through stochastic maximum principle. The motivation that drives our method is the gradient of the cost functional in the stochastic optimal control problem is under expectation, and numerical calculation of such an expectation requires fully computation of a system of forward backward stochastic differential equations, which is computationally expensive. By evaluating the expectation with single-sample representation as suggested by the stochastic gradient descent type optimisation, we could save computational efforts in solving FBSDEs and only focus on the optimisation task which aims to determine the optimal control process.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1996

Comprehensive Analysis of Two Downburst-Related Aircraft Accidents

Although downbursts have been identified as the major cause of a number of aircraft takeoff and landing accidents, only the 1985 Dallas/Fort Worth (DFW) and the more recent (July 1994) Charlotte, North Carolina, landing accidents provided sufficient onboard recorded data to perform a comprehensive analysis of the downburst phenomenon. The first step in the present analysis was the determination of the downburst wind components. Once the wind components and their gradients were determined, the degrading effect of the wind environment on the airplane's performance was calculated. This wind-shear-induced aircraft performance degradation, sometimes called the F-factor, was broken down into two components F(sub 1) and F(sub 2), representing the effect of the horizontal wind gradient and the vertical wind velocity, respectively. In both the DFW and Charlotte cases, F(sub 1) was found to be the dominant causal factor of the accident. Next, the aircraft in the two cases were mathematically modeled using the longitudinal equations of motion and the appropriate aerodynamic parameters. Based on the aircraft model and the determined winds, the aircraft response to the recorded pilot inputs showed good agreement with the onboard recordings. Finally, various landing abort strategies were studied. It was concluded that the most acceptable landing abort strategy from both an analytical and pilot's standpoint was to hold constant nose-up pitch attitude while operating at maximum engine thrust.

Shen, J.↗
DOE OSTI2019

The effect of self-interaction error on electrostatic dipoles calculated using density functional theory

Spurious electron self-interaction in density functional approximations (DFAs) can lead to inaccurate predictions of charge transfer in heteronuclear molecules that manifest as errors in calculated electrostatic dipoles. Here we show the magnitude of these errors on dipoles computed for a diverse set of 47 molecules taken from the recent benchmark study of Hait and Head-Gordon. We compare results of Perdew-Wang local spin density approximation (PW92), Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and strongly constrained and appropriatedly normed (SCAN) meta-GGA dipole calculations, along with those of their respective self-interaction-corrected (SIC) counterparts, to reference values from accurate wave function-based methods. The SIC calculations were carried out using the Fermi-Löwdin orbital (FLO-SIC) approach. We find that correcting for self-interaction generally increases the degree of charge transfer, thereby increasing the size of calculated dipole moments. The FLO-SIC-PW92 and FLO-SIC-PBE dipoles are in better agreement with reference values than their uncorrected DFA counterparts, particularly for strongly ionic molecules where significant improvement is seen. Applying FLO-SIC to SCAN does not improve dipole values overall. In addition, we show that removing self-interaction improves the description of the dipole for stretched-bond geometries and recovers the physically correct separated atom limit of zero dipole. Finally, we find that the best agreement between the FLO-SIC-DFA and reference dipoles occurs when the molecular geometries are optimized using the FLO-SIC-DFA.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2018

Electronic and Magnetic Properties of Ru{sub 2}FeSi{sub 1−x}Ge{sub x} Alloys in B2 Structure: a First-Principle Study

In this letter, structural, electronic structure, and magnetic properties of Ru{sub 2}FeSi{sub 1−x}Ge{sub x} were analyzed for the first time in the stable B2 phase by using the first-principle calculation. The exchange and correlation potentials are described by using generalized gradient approximations (GGA) The B{sub 2} structure is found to be energetically more favorable than the L2{sub 1} structure. It was found that the lattice parameters increase linearly with increasing x. For all the range studied, the Ru{sub 2}FeSi{sub 1−x}Ge{sub x} alloys exhibit a metallic character. The origin of magnetic moment is due to the exchange splitting of Fe-d states, and the atomic magnetic moment of the present material originates mainly from Fe atom.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2018

Influence of Hydrogen Sulfide Exposure on the Transport and Structural Properties of the Metal–Organic Framework ZIF-8

In this paper, the interaction between hydrogen sulfide and ZIF-8 was studied via structural characterizations and guest molecule diffusion measurements. It was found that hydrogen sulfide reacts with the ZIF-8 external particle surface to form a surface barrier that excludes the uptake of larger molecules (ethanol) and slows down the uptake of smaller molecules (carbon dioxide). Nonetheless, bulk transport properties were unaltered, as supported by pulsed field gradient nuclear magnetic resonance studies. Dispersion-corrected density functional theory calculations revealed that H 2 S is consumed by reactions occurring at the ZIF external surface. These reactions result in water and defect formation, both of which were found to be exothermic and independent of both crystallographic facets ({001} and {110}) and surface termination. Finally, we concluded that these surface reactions lead to structural and chemical changes to the ZIF-8 external surface that generate surface barriers to molecular transport.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Atmospheric Turbulent Intermittency Over the Arctic Sea‐Ice Surface During the MOSAiC Expedition

Turbulent motions in the Arctic stable boundary layer are characterized by intermittency, but they are rarely investigated due to limited observations, in particular over the sea‐ice surface. In the present study, we explore the characteristics of turbulent intermittency over the Arctic sea‐ice surface using data collected during the Multidisciplinary drifting Observation for the Study of Arctic Climate expedition from October 2019 to September 2020. We first develop a new algorithm, which performs well in identifying the spectral gap over the Arctic sea‐ice surface. Then the characteristics of intermittency are investigated. It is found that the strength of intermittency increases under the conditions of light surface wind speed, small surface wind speed gradient, and strong surface air temperature gradient. The momentum flux, sensible heat flux, and latent heat flux calculated by raw eddy‐covariance fluctuations are overestimated by 3%, 10%, and 24%, respectively, because submesoscale motions are included. Furthermore, the characteristics of the atmospheric boundary layer structure under various intermittency conditions reveal that strong low‐level jets are favorable to surface turbulent motions that result in weak intermittency, while strong temperature inversions above the surface layer suppress surface turbulent motions and lead to strong intermittency.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

The elphbolt ab initio solver for the coupled electron-phonon Boltzmann transport equations

elphbolt is a modern Fortran (2018 standard) code for efficiently solving the coupled electron–phonon Boltzmann transport equations from first principles. Using results from density functional and density functional perturbation theory as inputs, it can calculate the effect of the non-equilibrium phonons on the electronic transport (phonon drag) and non-equilibrium electrons on the phononic transport (electron drag) in a fully self-consistent manner and obeying the constraints mandated by thermodynamics. It can calculate the lattice, charge, and thermoelectric transport coefficients for the temperature gradient and electric fields, and the effect of the mutual electron–phonon drag on these transport properties. The code fully exploits the symmetries of the crystal and the transport-active window to allow the sampling of extremely fine electron and phonon wave vector meshes required for accurately capturing the drag phenomena. The corray feature of modern Fortran, which offers native and convenient support for parallelization, is utilized. The code is compact, readable, well-documented, and extensible by design.

electron drag↗
DOE OSTI2013

Phase stabilities at a glance: Stability diagrams of nickel dipnictides

In the course of the recent advances in chemical structure prediction, a straightforward type of diagram to evaluate phase stabilities is presented based on an expedient example. Crystal structures and energetic stabilities of dipnictides NiPn{sub 2} (Pn = N, P, As, Sb, Bi) are systematically investigated by first principles calculations within the framework of density functional theory using the generalized gradient approximation to treat exchange and correlation. These dipnictides show remarkable polymorphism that is not yet understood systematically and offers room for the discovery of new phases. Relationships between the concerned structures including the marcasite, the pyrite, the arsenopyrite/CoSb{sub 2}, and the NiAs{sub 2} types are highlighted by means of common structural fragments. Electronic stabilities of experimentally known and related AB{sub 2} structure types are presented graphically in so-called stability diagrams. Additionally, competing binary phases are taken into consideration in the diagrams to evaluate the stabilities of the title compounds with respect to decomposition. The main purpose of the stability diagrams is the introduction of an image that enables the estimation of phase stabilities at a single glance. Beyond that, some of the energetically favored structure types can be identified as potential new phases.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Optical properties of armchair (7, 7) single walled carbon nanotubes

Full potential linearized augmented plane waves method with the generalized gradient approximation for the exchange-correlation potential was applied to calculate the optical properties of (7, 7) single walled carbon nanotubes. The both x and z directions of the incident photons were applied to estimate optical gaps, dielectric function, electron energy loss spectroscopies, optical conductivity, optical extinction, optical refractive index and optical absorption coefficient. The results predict that dielectric function, ε (ω), is anisotropic since it has higher peaks along z-direction than x-direction. The static optical refractive constant were calculated about 1.4 (z-direction) and 1.1 (x- direction). Moreover, the electron energy loss spectroscopy showed a sharp π electron plasmon peaks at about 6 eV and 5 eV for z and x-directions respectively. The calculated reflection spectra show that directions perpendicular to the tube axis have further optical reflection. Moreover, z-direction indicates higher peaks at absorption spectra in low range energies. Totally, increasing the diameter of armchair carbon nanotubes cause the optical band gap, static optical refractive constant and optical reflectivity to decrease. On the other hand, increasing the diameter cause the optical absorption and the optical conductivity to increase. Moreover, the sharp peaks being illustrated at optical spectrum are related to the 1D structure of CNTs which confirm the accuracy of the calculations.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Vertical detachment energies of ammonia cluster anions using self-interaction-corrected methods

Systems with weakly bound extra electrons impose great challenges to semilocal density functional approximations (DFAs), which suffer from self-interaction errors. Small ammonia clusters are one such example of weakly bound anions where the extra electron is weakly bound. We applied two self-interaction correction (SIC) schemes, viz., the well-known Perdew–Zunger and the recently developed locally scaled SIC (LSIC) with the local spin density approximation (LSDA), Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA), and the SCAN meta-GGA functionals to calculate the vertical detachment energies (VDEs) of small ammonia cluster anions (NH3)n–. Our results show that the LSIC significantly reduces the errors in calculations of VDE with LSDA and PBE-GGA functionals leading to better agreement with the reference values calculated with coupled cluster singles and doubles with perturbative triples [CCSD(T)]. Accurate prediction of VDE as an absolute of the highest occupied molecular orbital (HOMO) is challenging for DFAs. Our results show that VDEs estimated from the negative of HOMO eigenvalues with the LSIC-LSDA and Perdew–Zunger SIC-PBE are within 11 meV of the reference CCSD(T) results. Here, the LSIC method performs consistently well for the VDE estimates, from both the total energy differences and the absolute HOMO eigenvalues.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Active control of Alfvén eigenmodes by external magnetic perturbations with different spatial spectra

Alfvén eigenmodes have been suppressed and excited in tokamak plasmas by (just) modifying the poloidal spectra of externally applied static magnetic perturbations. This effect is observed experimentally when toroidal spectra of n = 2, n = 4 as well as a mixed spectrum of n = 2 and n = 4 is applied. Under the n = 2 magnetic perturbations, the modes are excited or suppressed by modifying the coil phasing between the upper and the lower set of coils. Regardless of the absolute rotation, an even parity for the n = 4 perturbation is observed to reduce the amplitude of the Alfvénic instabilities, while an odd parity amplifies it. To combine the stabilizing (and destabilizing) effect of n = 2 and n = 4, a mixed spectrum is applied, finding similar reduction (and amplification) trends. However, the impact on the mode amplitude is more subtle, due to the reduced coil current required for a mixed spectrum. The signal level on the fast-ion loss detector is sensitive to the applied poloidal spectrum, which is consistent with Hamiltonian full-orbit modelling of an edge resonant transport layer activated by the 3D perturbative fields. An internal redistribution of the fast-ion population is induced, modifying the phase-space gradients driving the Alfvénic instabilities, and ultimately determining their existence. The calculated edge resonant layers for both n = 2 and n = 4 toroidal spectra are consistent with the observed suppressed and excited phases. Moreover, hybrid kinetic-magnetohydrodynamic (MHD) simulations reveal that this edge resonant transport layer overlaps in phase-space with the population responsible for the fast-ion drive. The results presented here may help to control fast-ion driven Alfvénic instabilities in future burning plasmas with a significant fusion born alpha particle population.

Alfvén waves↗
DOE OSTI2026

Density limit in peeling-limited pedestals at and above the Greenwald value in DIII-D high poloidal beta plasmas

High pressure, peeling limited pedestals with pedestal normalized beta β N,ped >2 and pedestal top density n ped at or above the Greenwald density nG have been achieved in DIII-D high poloidal beta plasmas, with high global normalized beta β N >3 and energy confinement H 98 ~1.2-1.7. Higher β N allows higher pedestal density above the Greenwald value and higher pedestal pressure, even with a low injected torque. MHD modeling confirms that the experimental profiles lie near the peeling-mode unstable boundary with high normalized pressure gradient βMHD and high edge current density. Experimental analysis and stability calculations indicate that the high poloidal beta with strong Shafranov shift, high βMHD and weak/negative magnetic shear improves the pedestal stability by decoupling the peeling and ballooning modes and stabilizing the ballooning modes, thus facilitating access to the second stable region of peeling-ballooning mode. The access to the second stable peeling-ballooning stability region opens Super-H-like channels without extremely strong shaping or strong torque injection. The high-pressure peeling pedestal allows the pedestal density to go beyond the Greenwald limit with strong ExB shear maintained: pedestal pressure increases with pedestal density even when n ped >n G , until reaching the ideal MHD instability boundary, where giant ELMs occur. The giant ELMs are dominated by a strong n=1 component and cause a large reduction of the edge pressure, but a negligible change of the core pressure, consistent with kink/peeling-mode induced instability. The pedestal recovers from the collapse and typically sustains a high baseline density, around the Greenwald limit, during the whole discharge duration. Experiments also found that internal transport barriers and n ped ~0.9nG, peeling limited pedestals could be simultaneously achieved in high β N plasmas, while an internal feedback between ITB strength and pedestal performance is found.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

PRISM3D: a 3-D reference seismic model for Iberia and adjacent areas

SUMMARY We present PRISM3D, a 3-D reference seismic model of P- and S-wave velocities for Iberia and adjacent areas. PRISM3D results from the combination of the most up-to-date earth models available for the region. It extends horizontally from 15°W to 5°E in longitude, 34°N to 46°N in latitude and vertically from 3.5 km above to 200 km below sea level, and is modelled on a regular grid with 10 and 0.5 km of grid node spacing in the horizontal and vertical directions, respectively. It was designed using models inferred from local and teleseismic body-wave tomography, earthquake and ambient noise surface wave tomography, receiver function analysis and active source experiments. It includes two interfaces, namely the topography/bathymetry and the Mohorovičić (Moho) discontinuity. The Moho was modelled from previously published receiver function analysis and deep seismic sounding results. To that end we used a probabilistic surface reconstruction algorithm that allowed to extract the mean of the Moho depth surface along with its associated standard deviation, which provides a depth uncertainty estimate. The Moho depth model is in good agreement with previously published models, although it presents slightly sharper gardients in orogenic areas such as the Pyrenees or the Betic-Rif system. Crustal and mantle P- and S-wave wave speed grids were built separately on each side of the Moho depth surface by weighted average of existing models, thus allowing to realistically render the speed gradients across that interface. The associated weighted standard deviation was also calculated, which provides an uncertainty estimation on the average wave speed values at any point of the grid. At shallow depths (<10 km), low P and S wave speeds and high VP/VS are observed in offshore basins, while the Iberian Massif, which covers a large part of western Iberia, appears characterized by a rather flat Moho, higher than average VP and VS and low VP/VS. Conversely, the Betic-Rif system seems to be associated with low VP and VS, combined with high VP/VS in comparison to the rest of the study area. The most prominent feature of the mantle is the well known high wave speed anomaly related to the Alboran slab imaged in various mantle tomography studies. The consistency of PRISM3D with previous work is verified by comparing it with two recent studies, with which it shows a good general agreement.The impact of the new 3-D model is illustrated through a simple synthetic experiment, which shows that the lateral variations of the wave speed can produce traveltime differences ranging from –1.5 and 1.5 s for P waves and from –2.5 and 2.5 s for S waves at local to regional distances. Such values are far larger than phase picking uncertainties and would likely affect earthquake hypocentral parameter estimations. The new 3-D model thus provides a basis for regional studies including earthquake source studies, Earth structure investigations and geodynamic modelling of Iberia and its surroundings.

Arroucau, P.↗
DOE OSTI2023

Core-excited states of SF 6 probed with soft-x-ray femtosecond transient absorption of vibrational wave packets

A vibrational wavepacket in SF 6 is created by impulsive stimulated Raman scattering with a few-cycle infrared pulse and mapped simultaneously onto five sulfur core-excited states using table-top soft x-ray transient absorption spectroscopy between 170 to 200 eV. The femtosecond vibrations induce real-time energy shifts of the x-ray absorption, whose amplitude depend strongly on the nature of the core-excited state. The pump laser intensity is used to control the number of vibrational states in the superposition, thereby accessing core-excited levels for various extensions of the S-F stretching motion. This enables the determination of the relative core-level potential energy gradients for the symmetric stretching mode, in good agreement with TDDFT calculations. This experiment demonstrates a new means of characterizing core-excited potential energy curves.

74 ATOMIC AND MOLECULAR PHYSICS↗