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DOE OSTI2024

Triplet‐Singlet Emission of d‐Block Metal Complexes Characterized by Spin‐Orbit Natural Transition Orbitals

Abstract Spin‐orbit natural transition orbital (SO‐NTO) methodology, recently developed in our group for complete and restricted active space (CAS/RAS) wavefunction calculations, is applied to analyze triplet‐to‐singlet emission in transition metal complexes. The lowest‐energy (longest‐wavelength) spin‐forbidden transition is studied for for [Ir(pbt)2(acac)] and [Re(CO)4(pbt)] and the complexes [W(CO)4(bpy)] and [Mo(CO)4(bpy)]. For the latter complexes, spin‐forbidden transitions from higher spin‐triplet levels are additionally analyzed. SO‐NTOs are compared with spin‐free NTOs for the transitions under consideration. The major assignment of a spin‐forbidden transition is obtained from the spin‐free NTO analysis, while the source of intensity of the electronic transition is revealed by the SO‐NTOs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

The role of interannual ENSO events in decadal timescale transitions of the Interdecadal Pacific Oscillation

The build-up of decadal timescale upper ocean heat content in the off-equatorial western tropical Pacific can provide necessary conditions for interannual El Niño/Southern Oscillation (ENSO) events to contribute to decadal timescale transitions of tropical Pacific SSTs to the opposite phase of the Interdecadal Pacific Oscillation (IPO). This can be viewed as a corollary to subseasonal westerly wind burst events contributing to El Niño interannual timescale transitions. In this work, a long pre-industrial control run with CESM1 is analyzed to show that there is a greater chance of ENSO activity to contribute to an IPO transition when off-equatorial western Pacific Ocean heat content reaches either a maximum (for El Niño to contribute to a transition to positive IPO) or minimum (for La Niña to contribute to a transition to negative IPO) as seen in observations. If above a necessary ocean heat content threshold, the convergence associated with westerly anomaly near-equatorial surface winds associated with El Niño activity can draw that heat content equatorward to sustain anomalously warm western and central Pacific SSTs. These are associated with positive precipitation and convective heating anomalies, a Gill-type response and wind stress curl anomalies that continue to feed warm water into the near-equatorial western Pacific. These conditions then sustain the decadal-timescale transition to positive IPO (with the opposite sign for transition to negative IPO). Associated central equatorial Pacific convective heating anomalies produce SLP and wind stress anomalies in the North and South Pacific that can excite westward-propagating off-equatorial oceanic Rossby waves to contribute to the western Pacific thermocline depth and consequent heat content anomalies.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Determining the effects of U/Pu ratio on subsolidus phase transitions in U-Pu-Zr metallic fuel alloys

Here, ternary alloys consisting primarily of uranium, plutonium, and zirconium (U-Pu-Zr) are among the leading candidate fuel systems considered for fast spectrum nuclear reactors. Despite historical operation data from the testing of U-Pu-Zr rods in the Experimental Breeder Reactor-II, considerable uncertainty about the evolution of phases and microstructure across the ternary composition space exists. Due to sluggish kinetics and other difficulties in handling metal actinide specimens, quantitative measurements of phase-transitions in U-Pu-Zr alloys remain sparse in scientific literature, with most investigators reporting either phase-transition temperatures or phase identification data, but not both from the same specimens. The purpose of this paper is to critically compare experimental and calculated phase transition data and correlate with the microstructure and phase characterization data of as-cast and annealed U-Pu-Zr alloys. Phase transition peaks were measured using differential scanning calorimetry in the subsolidus regions (723-948 K) of three ternary U-Pu-Zr alloys with the same zirconium concentration but various U/Pu ratios. Overlapping peaks were deconvoluted using a Frazier-Suzuki peak fitting algorithm, and the critical peak temperatures and enthalpies were calculated. In general, increasing concentrations of Pu were associated with enhanced thermal stability of the body-centered cubic γ phase upon both heating and cooling. Experimental phase transition temperatures in this study tended to agree well with the predictions of the established ternary phase diagrams and other reported phase transition temperatures in literature. Additionally, the TAF-ID thermodynamic database was used to calculate a U-Pu-40 at.% Zr pseudobinary diagram as well as ternary diagrams from 773 to 973 K. The equilibrium phase transition temperatures tended to be considerably lower than measured peak temperatures upon both heating and cooling. Recommendations for improving the quality of data in future U-Pu-Zr characterization studies are also discussed.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2022

Hydrogen trapping and storage in the group IVB-VIB transition metal carbides

The transition metal carbides have been known to act as traps for hydrogen in steels as well as potential materials for hydrogen storage. Despite numerous experimental and a few theoretical studies, what impacts hydrogen trapping and storage in the transition metal carbides is not well understood. In this work, we use density functional theory to systematically investigate the bulk trapping and storage capabilities of the transition metal carbides. We specifically examine how trapping and storage changes with the transition metal, from the group IVB to group VIB, carbon concentration, and structure. Our results demonstrate a strong correlation between the trap energy and the number of valence electrons in the transition metal, suggesting that the group IVB transition metal carbides are the best carbides for trapping and storage. Hydrogen preferentially sits at octahedral interstices, e.g., the carbon vacancies, in the structure except in certain cases near the Me 2 C concentration when tetrahedral interstices, devoid of nearest neighbor carbon, can become more favorable. Our results further demonstrate that the lower the carbon concentration, the more hydrogen the transition metal carbide can store. These results demonstrate which carbides will act as the best traps for hydrogen.

08 HYDROGEN↗

DOE OSTI2020

Spin Transitions and Compressibility of ε-Fe 7 N 3 and γ'-Fe 4 N: Implications for Iron Alloys in Terrestrial Planet Cores

Iron nitrides are possible constituents of the cores of Earth and other terrestrial planets. Pressure-induced magnetic changes in iron nitrides and effects on compressibility remain poorly understood. In this work, we report synchrotron X-ray emission spectroscopy (XES) and X-ray diffraction (XRD) results for ε-Fe 7 N 3 and γ'-Fe 4 N up to 60 GPa at 300 K. The XES spectra reveal completion of high- to low-spin transition in ε-Fe 7 N 3 and γ'-Fe 4 N at 43 and 34 GPa, respectively. The completion of the spin transition induces stiffening in bulk modulus of ε-Fe 7 N 3 by 22% at ~40 GPa, but has no resolvable effect on the compression behavior of γ'-Fe 4 N. Fitting pressure-volume data to the Birch-Murnaghan equation of state yields V 0 = 83.29 ± 0.03 (Å 3 ), K 0 = 232 ± 9 GPa, K 0 ' = 4.1 ± 0.5 for nonmagnetic ε-Fe 7 N 3 above the spin transition completion pressure, and V 0 = 54.82 ± 0.02 (Å 3 ), K 0 = 152 ± 2 GPa, K 0 ' = 4.0 ± 0.1 for γ'-Fe 4 N over the studied pressure range. By reexamining evidence for spin transition and effects on compressibility of other candidate components of terrestrial planet cores, Fe 3 S, Fe 3 P, Fe 7 C 3 , and Fe 3 C based on previous XES and XRD measurements, we located the completion of high- to low-spin transition at ~67, 38, 50, and 30 GPa at 300 K, respectively. The completion of spin transitions of Fe 3 S, Fe 3 P, and Fe 3 C induces elastic stiffening, whereas that of Fe 7 C 3 induces elastic softening. Changes in compressibility at completion of spin transitions in iron-light element alloys may influence the properties of Earth's and planetary cores.

58 GEOSCIENCES↗

DOE OSTI2026

Intrinsic topological Weyl phase transition induced by a magnetostructural transformation in a kagome magnet

Topological phase transitions provide a unique window into the interplay between structure, magnetism, and Weyl physics in magnetic Weyl semimetals. However, realizing an intrinsic Weyl phase transition between two distinct Weyl states near room temperature remains challenging. Here, we demonstrate that a magnetostructural transition effectively induces such a transition in the kagome magnet Mn3Ga. High-resolution neutron diffraction, magnetization characterizations and first-principles calculations reveal that Mn3Ga undergoes a chiral antiferromagnetic transition below 485 K, followed by a magnetostructural transition to a monoclinic structure with highly canted antiferromagnetic order near room temperature. These cooperative changes in lattice and magnetic symmetries reorganize Weyl nodes, driving a transition from a primary type-II Weyl state to a distinct Weyl state, accompanied by dramatic variations in the anomalous Hall effect and appearance of topological Hall effect. Our findings open a new pathway for discovering novel topological Weyl states and advancing potential spintronic applications.

Yang, Tsung-Han [ORNL] (ORCID:000000030186302X)↗

DOE OSTI2022

Machine learning for impurity charge-state transition levels in semiconductors from elemental properties using multi-fidelity datasets

Quantifying charge-state transition energy levels of impurities in semiconductors is critical to understanding and engineering their optoelectronic properties for applications ranging from solar photovoltaics to infrared lasers. While these transition levels can be measured and calculated accurately, such efforts are time-consuming and more rapid prediction methods would be beneficial. Here, we significantly reduce the time typically required to predict impurity transition levels using multi-fidelity datasets and a machine learning approach employing features based on elemental properties and impurity positions. We use transition levels obtained from low-fidelity (i.e., local-density approximation or generalized gradient approximation) density functional theory (DFT) calculations, corrected using a recently proposed modified band alignment scheme, which well-approximates transition levels from high-fidelity DFT (i.e., hybrid HSE06). Further, the model fit to the large multi-fidelity database shows improved accuracy compared to the models trained on the more limited high-fidelity values. Crucially, in our approach, when using the multi-fidelity data, high-fidelity values are not required for model training, significantly reducing the computational cost required for training the model. Our machine learning model of transition levels has a root mean squared (mean absolute) error of 0.36 (0.27) eV vs high-fidelity hybrid functional values when averaged over 14 semiconductor systems from the II–VI and III–V families. As a guide for use on other systems, we assessed the model on simulated data to show the expected accuracy level as a function of bandgap for new materials of interest. Finally, we use the model to predict a complete space of impurity charge-state transition levels in all zinc blende III–V and II–VI systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Complete definition of 𝑁 →Δ transition generalized parton distributions

We revisit the definition of the leading-twist chiral-even generalized parton distributions (GPDs) for 𝑁 →Δ baryon transitions. We identify and address deficiencies in previous definitions of the transition GPDs inspired by the transition form factors of the vector and axial vector currents. Through systematic analysis of all possible covariant structures, respecting discrete symmetries and the baryon spinor equations of motion, we derive complete sets of independent structures for the transition matrix elements of the vector and axial vector partonic operators. They contain additional structures proportional to the light-cone vector, corresponding to transition GPDs of vanishing first moment, which were not included in previous parametrizations. Their presence is confirmed independently by the light-front multipole expansion and the cross-channel SO(3) partial-wave analysis of the transition matrix elements. Our analysis provides a complete definition of the 𝑁 →Δ transition GPDs for use in theoretical and phenomenological studies.

baryons↗

DOE OSTI2024

First-order structural phase transition at low temperature in GaPt 5 P and its rapid enhancement with pressure

Single crystals of X Pt 5 ⁢P (X = Al, Ga, and In), belonging to the 1-5-1 family of compounds, were grown from a Pt-P solution at high temperatures, and measurements of the ambient pressure, temperature-dependent magnetization, resistivity, and x-ray diffraction were made. Additionally, the ambient-pressure Hall resistivity and temperature-dependent resistance under pressure were measured on GaPt 5 ⁢P. All three compounds have a tetragonal P4/mmm crystal structure at room temperature with metallic transport and weak diamagnetism over the 2–300 K temperature range. Surprisingly, at ambient pressure, both the transport and magnetization measurements on GaPt 5 ⁢P show a steplike feature in the 70–90 K region, suggesting a possible structural phase transition. Neither AlPt 5 ⁢P nor InPt 5 ⁢P have any signatures of a phase transition in their temperature-dependent electrical resistance and magnetization data. Both the hysteretic nature and sharpness of the features in the GaPt 5 ⁢P data suggest that the transition is first-order. Further, single-crystal x-ray diffraction measurements provided further details of the structural transition with a possibility of a crystal symmetry different from P⁢4/mmm below the transition temperature. The transition is characterized by anisotropic changes in the lattice parameters and a volume collapse with respect to the high-temperature tetragonal crystal structure. Furthermore, satellite peaks are observed at two distinct and nonequivalent wave vectors (0, 0, 0.5) and (0.5, 0.5, 0.5), and density functional theory calculations present phonon softening, especially at (0.5, 0.5, 0.5), as a possible driving mechanism. Additionally, we find that the structural transition temperature increases rapidly with increasing pressure, reaching room temperature by ~2.2 GPa, highlighting the high degree of pressure sensitivity of GaPt 5 ⁢P and fragile nature of its room-temperature structure. Even though the volume collapse and extreme pressure sensitivity suggest chemical pressure should drive a similar structural change in AlPt 5⁢ P, where both unit-cell dimensions and volume are smaller, its structure is found to be the same as that of the room-temperature GaPt 5 ⁢P. Overall, GaPt 5 ⁢P stands out as a sole member of the 1-5-1 family of compounds for which a temperature-driven structural change has been observed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2020

Relativistic hybrid stars with sequential first-order phase transitions and heavy-baryon envelopes

We compute the mass, radius and tidal deformability of stars containing phase transitions from hadronic to quark phase(s). These quantities are computed for three types of hadronic envelopes: purely nuclear, hyperonic, and Δ-resonance--hyperon admixed matter. We consider either a single first-order phase transition to a quark phase with a maximally stiff equation of state (EOS) or two sequential first-order phase transitions mimicking a transition from hadronic to a quark matter phase followed by a second phase transition to another quark phase. We explore the parameter space which produces low-mass twin and triplet configurations where equal-mass stars have substantially different radii and tidal deformabilities. We demonstrate that while for purely hadronic stiff EOS the obtained maximum mass is inconsistent with the upper limit on this quantity placed by GW170817, the inclusion of the hyperonic and Δ-resonance degrees of freedom, as well as the deconfinement phase transition at sufficiently low density, produces a configuration of stars consistent with this limit. The obtained hybrid star configurations are in the mass range relevant for the interpretation of the GW170817 event. We compare our results for the tidal deformability with the limits inferred from GW170817 showing that the onset of non-nucleonic phases, such as Δ-resonance--hyperon admixed phase and/or the quark phase(s), are favored by this data if the nuclear EOS is stiff. Also, we show that low-mass twins and especially triplets proliferate the number of combinations of possible types of stars that can undergo a merger event, the maximal number being six in the case of triplets. The prospects for uncovering the first-order phase transition(s) to and in quark matter via measurements of tidal deformabilities in merger events are discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Novel Valence Transition in Elemental Metal Europium around 80 GPa

We report valence transition could induce structural, insulator-metal, nonmagnetic-magnetic and superconducting transitions in rare-earth metals and compounds, while the underlying physics remains unclear due to the complex interaction of localized 4f electrons as well as their coupling with itinerant electrons. The valence transition in the elemental metal europium (Eu) still has remained as a matter of debate. Using resonant x-ray emission scattering and x-ray diffraction, we pressurize the states of 4f electrons in Eu and study its valence and structure transitions up to 160 GPa. We provide compelling evidence for a valence transition around 80 GPa, which coincides with a structural transition from a monoclinic (C2/c) to an orthorhombic phase ($Pnma$). We show that the valence transition occurs when the pressure-dependent energy gap between 4f and 5d electrons approaches the Coulomb interaction. Our discovery is critical for understanding the electrodynamics of Eu, including magnetism and high-pressure superconductivity.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Data-Driven Prediction and Optimization of Energy Use for Transit Fleets of Electric and ICE Vehicles

Due to the high upfront cost of electric vehicles, many public transit agencies can afford only mixed fleets of internal combustion and electric vehicles. Optimizing the operation of such mixed fleets is challenging because it requires accurate trip-level predictions of electricity and fuel use as well as efficient algorithms for assigning vehicles to transit routes. We present a novel framework for the data-driven prediction of trip-level energy use for mixed-vehicle transit fleets and for the optimization of vehicle assignments, which we evaluate using data collected from the bus fleet of CARTA, the public transit agency of Chattanooga, TN. We first introduce a data collection, storage, and processing framework for system-level and high-frequency vehicle-level transit data, including domain-specific data cleansing methods. We train and evaluate machine learning models for energy prediction, demonstrating that deep neural networks attain the highest accuracy. Based on these predictions, we formulate the problem of minimizing energy use through assigning vehicles to fixed-route transit trips. We propose an optimal integer program as well as efficient heuristic and meta-heuristic algorithms, demonstrating the scalability and performance of these algorithms numerically using the transit network of CARTA.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2023

Atomistic insight into the ferroelastic post-stishovite transition by high-pressure single-crystal X-ray diffraction

Abstract The post-stishovite transition is a classic pseudo-proper typed ferroelastic transition with a symmetry-breaking spontaneous strain. This transition has been studied using high-pressure spontaneous strains, optic modes, and elastic moduli (Cij) based on the Landau modeling, but its atomistic information and structural distortion remain poorly understood. Here we have conducted synchrotron single-crystal X-ray diffraction measurements on stishovite crystals up to 75.3 GPa in a diamond-anvil cell. Analysis of the data reveals atomic positions, bond lengths, bond angles, and variations of SiO6 octahedra across the transition at high pressure. Our results show that the O coordinates split at ~51.4 GPa, where the apical and equatorial Si-O bond lengths cross over, the SiO6 octahedral distortion vanishes, and the SiO6 octahedra start to rotate about the c axis. Moreover, distortion mode analysis shows that an in-plane stretching distortion (GM1+ mode) occurs in the stishovite structure at high pressure while a rotational distortion (GM2+ mode) becomes dominant in the post-stishovite structure. These results are used to correlate with elastic moduli and Landau parameters (symmetry-breaking strain e1–e2 and order parameter Q) to provide atomistic insight into the ferroelastic transition. When the bond lengths of two Si-O bonds are equal due to the contribution from the GM1+ stretching mode, C11 converges with C12, and the shear wave VS1[110] polarizing along [110] and propagating along [110] vanishes. Values of e1–e2 and Q are proportional to the SiO6 rotation angle from the occurrence of the GM1+ rotational mode in the post-stishovite structure. Our results on the pseudo-proper type transition are also compared with that for the proper type in albite and improper type in CaSiO3 perovskite. The symmetry-breaking strain, in all these types of transitions, arises as the primary effect from the structural angle (such as SiO6 rotation or lattice constant angle) and its relevant distortion mode in the low-symmetry ferroelastic phase.

Geochemistry & Geophysics↗

DOE OSTI2020

ESGF project plan for ramping down development activities and transition the project and data to a new team

This document summarizes the current software development activities carried by the ESGF team with planned delivery dates before the transition of the project to a new team. The work we started on the publishing, search and retrieval, and the backend for indexing and data management that can be deployed using Kubernetes are planned to be delivered within the next three months; however, this assumes that the current team members will continue to work on the project. The activities are listed in Table 1 by priority and in case we lose team members, development of lower priority tasks can be paused and passed to the new team with its current status. Team members can be re-assigned based on their availability and transition work needed. Table 2 below describe activities necessary to help project transition to the new team. Deadlines for these activities are before October 1 st of 2022 or the completion of a successful transition. Support during the transition period assumes that we will have 3 key ESGF members and the ESGF PI supported at the 50% level or less for the duration of the transition. The level of support is an estimate, and it depends on several factors that cannot be decided at this time. The total cost of the proposed work and support activities for fiscal year 2021 is $1,270K with $230K carry over to fiscal year 2022. The cost for the support work listed in Table 2 is the sum of the two rows “HW support” and “project support, maintenance, and transition efforts”, colored blue, in Table1 which adds up to $720K. This cost is included in the $1,270K cost in table 1.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Measurement of Thermal Conductivity in a Supercooled Hydrogel-Salt Complex Near Its Phase Transition

Solid-liquid phase transitions, i.e. solidification processes, have applications in data storage, development of novel thermoelectric materials, cooling of microelectronic substrates and air conditioning condensers. Standard analyses of solidification (Stefan problem) assume constant thermal properties of the solid and liquid sides. It is not known how these properties change across the spatial transition interface, though most studies report a discontinuity in the solid and liquid properties through the transition temperature [1]. When the phase transition releases enthalpy, recent research has shown if the phonon or electron transport time is of the same order of magnitude as the time scale of the atomic transformation, this increases the heat capacity of the solid material at temperatures near the phase transition temperature [2]. A fundamental understanding of the phase transition may help shed light on the molecular origins of supercooling and spontaneous nucleation, which will help with applications involving them. We report studies of supercooling and nucleation of sodium sulfate decahydrate, a salt hydrate which is of recent interest in thermal storage, and of a hydrogel-sodium sulfate complex which shows limited supercooling. We will also report the measurement of their thermal properties during the phase transition process. This will be done with a hot-wire setup which produces small temperature changes of the order of ~1 C.

crystallization, phase transformation, thermal con↗

DOE OSTI2020

Phase-curve Pollution of Exoplanet Transit Depths

The next generation of space telescopes will enable transformative science to understand the nature and origin of exoplanets. In particular, transit spectroscopy will reveal the chemical composition of the exoplanet atmospheres with unprecedented detail thanks to precise measurements of the visible-to-infrared transit depths down to 10 parts per million. Such a level of instrumental precision raises the challenge to obtain even more precise astrophysical models so as not to significantly influence the interpretation of the observed data. We must therefore critically revisit some of the commonly accepted assumptions that were adequate for analyzing past and current observations. A common approximation in the analysis of exoplanetary primary transits is that the planet does not contribute to the recorded flux, so-called dark planet hypothesis. In this paper, we investigate the impact of the dark planet hypothesis on the parameters obtained from the analysis of transits with particular attention to the transit depth. We develop mathematical formulae and release new software to estimate the magnitude of the potential bias. These tools will be useful in the preparation of observing proposals, as well as within the scientific consortia of the James Webb Space Telescope (JWST) and the Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL) missions. We probe the accuracy of the mathematical formulae through the analysis of synthetic observations with the JWST Mid-InfraRed Instrument. We find that self-blending from nightside emission attenuates the transit depth by >3σ for some of the known exoplanet systems, in agreement with previous work. An additional unreported effect caused by the nightside rotating into view can also impart a significant effect, but in the opposite direction (increasing the transit depth); this effect can largely be removed with conventional detrending practices, at the expense of a slight increase in noise, and mixing astrophysical variations and instrumental drifts.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1974

Comparison of techniques for determination of boundary layer transition in shock wave induced flows

Three methods of determining transition times in shock wave-induced boundary layers are compared. Experimental transition data for these comparisons were obtained in two different shock tube facilities for shock wave Mach numbers from 3.0 to 5.5. Bias in determining transition times by the conventional thin-film temperature history and difficulty in reading schlieren photographs suggest the use of a new heat-flux technique for determining transition. It is suggested that the heat-flux technique be used when possible, because of better sensitivity and a view of the entire transition process and because it reveals important information concerning departures from theoretical laminar boundary-layer development. Transition results presented extend the range of data available in the literature and are in good agreement with existing data through transition Reynolds numbers of 1,000,000.

Hall, J. L.↗

NASA NTRS1975

A numerical method for the prediction of high-speed boundary-layer transition using linear theory

A method is described of estimating the location of transition in an arbitrary laminar boundary layer on the basis of linear stability theory. After an examination of experimental evidence for the relation between linear stability theory and transition, a discussion is given of the three essential elements of a transition calculation: (1) the interaction of the external disturbances with the boundary layer; (2) the growth of the disturbances in the boundary layer; and (3) a transition criterion. The computer program which carried out these three calculations is described. The program is first tested by calculating the effect of free-stream turbulence on the transition of the Blasius boundary layer, and is then applied to the problem of transition in a supersonic wind tunnel. The effects of unit Reynolds number and Mach number on the transition of an insulated flat-plate boundary layer are calculated on the basis of experimental data on the intensity and spectrum of free-stream disturbances. Reasonable agreement with experiment is obtained in the Mach number range from 2 to 4.5.

Mack, L. M.↗