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At least 307 records · Page 17

Vacuum Pyrolysis of Hybrid Poplar Milled Wood Lignin with Fourier Transform-Ion Cyclotron Resonance Mass Spectrometry Analysis of Feedstock and Products for the Elucidation of Reaction Mechanisms

The pyrolysis of lignocellulosic materials is a promising technique to produce fuels and chemicals. It is well known that the most abundant products of lignin pyrolysis are oligomeric molecules, known as pyrolytic lignin (PL). The chemical composition of PL has been extensively studied; however, there is still an important debate whether these oligomers are produced directly from the lignin or from the recombination of monomeric pyrolytic products. Existing theories are unable to describe the effect of vacuum on the distribution of pyrolysis products. Hybrid poplar milled wood lignin (MWL) was initially isolated and thoroughly characterized by Fourier transform-ion cyclotron resonance mass spectrometry (FT-ICR MS). Chemical formulas were assigned to each oligomeric compound detected. The MWL was also subjected to vacuum pyrolysis in a modified pyroprobe at 250, 750, and 1000 mbar (absolute pressure), and the resulting liquid products were analyzed by FT-ICR MS. A new strategy to assign structural representations to the oligomeric PL products is proposed, based on the plausible pyrolysis reaction mechanisms of depolymerization/fragmentation applied to original MWL oligomer formulas. Our results support the hypothesis that PL is formed from the removal of moieties from primary lignin pyrolysis products with between three and five aromatic rings. This depolymerization/fragmentation allows the oligomers to reduce their molecular weights to the point where they can be removed from the reaction zone by direct vaporization. Furthermore, this phenomenon highlights the importance of pressure on removal mechanisms and their impact on the molecular weight of the resulting products from lignin pyrolysis.

09 BIOMASS FUELS↗

Stereoisomer-dependent unimolecular kinetics of 2,4-dimethyloxetanyl peroxy radicals

2,4,dimethyloxetane is an important cyclic ether intermediate that is produced from hydroperoxyalkyl (QOOH) radicals in the low-temperature combustion of n-pentane. However, the reaction mechanisms and rates of consumption pathways remain unclear. In the present work, the pressure- and temperature-dependent kinetics of seven cyclic ether peroxy radicals, which stem from 2,4,dimethyloxetane via H-abstraction and O 2 addition, were determined. The automated kinetic workflow code, KinBot, was used to model the complexity of the chemistry in a stereochemically resolved manner and solve the resulting master equations from 300–1000 K and from 0.01–100 atm. The main conclusions from the calculations include (i) diastereomeric cyclic ether peroxy radicals show significantly different reactivities, (ii) the stereochemistry of the peroxy radical determines which QOOH isomerization steps are possible, (iii) conventional QOOH decomposition pathways, such as cyclic ether formation and HO 2 elimination, compete with ring-opening reactions, which primarily produce OH radicals, the outcome of which is sensitive to stereochemistry. Ring-opening reactions lead to unique products, such as unsaturated, acyclic peroxy radicals, that form direct connections with species present in other chemical kinetics mechanisms through "cross-over" reactions that may complicate the interpretation of experimental results from combustion of n-pentane and, by extension, other alkanes. For example, one cross-over reaction involving 1-hydroperoxy-4-pentanone-2-yl produces 2-(hydroperoxymethyl)-3-butanone-1-yl, which is an iso-pentane-derived ketohydroperoxide (KHP). At atmospheric pressure, the rate of chemical reactions of all seven peroxy radicals compete with that of collisional stabilization, resulting in well-skipping reactions. However, at 100 atm, only one out of seven peroxy radicals undergoes significant well-skipping reactions. Here, the rates produced from the master equation calculations provide the first foundation for the development of detailed sub-mechanisms for cyclic ether intermediates. In addition, analysis of the complex reaction mechanisms of 2,4-dimethyloxetane-derived peroxy radicals provides insights into the effects of stereoisomers on reaction pathways and product yields.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

A parametric analysis of oilfield design factors affecting the detectability and characterization of electrically conductive hydrofracks

Electrical responses in the vicinity of energized steel-cased well sources offer significant potential for monitoring induced fractures. However, the high complexity of well-fracture-host models spanning multiple length scales compels analysts to simplify their numerical models due to enormous computational costs. This consequently limits our understanding regarding monitoring capabilities and the limitations of electrical measurements on realistic hydraulically fracturing systems. In this paper, we use the hierarchical finite element approach to construct geoelectric models in which geometrically complex fractures and steel-cased wells are discretely represented in 3D conducting media without sacrificing the model realism and computation efficiency. We have discovered systematic numerical analyses of the electrical responses to evaluate the influences of borehole material conductivity and the source type as well as the effects of well geometry, conductivity contrast, source location, fracture growth, and fracture propagation. Furthermore, the numerical results indicate that the borehole material property has a strong control on the electrical potentials along the production and monitoring wells. The monopole source located at a steel-cased well results in a current density distribution that decays away from the source location throughout the well length, whereas the dipole source produces a current density that dominates mainly along the dipole length. Moreover, the conductivity contrast between the fractures and host does not change the overall pattern of the electrical potentials but varies its amplitude. The fracture models near different well systems indicate that the well geometry controls the entire distribution of potentials, while the characteristics of the voltage difference profiles along the wells before and after fracturing are insensitive to the well geometry and the well in which the source is located. Further, the hydraulic-fracturing models indicate that the voltage differences along the production well before and after fracturing have strong sensitivity to fracture growth and fracture set propagation.

58 GEOSCIENCES↗

Introduction to special issue: Colloidal quantum dots

The quantum confinement effect has driven the development and exploration of colloidal semiconductor nanocrystals over approximately the past three decades. These semiconductor nanostructures are typically synthesized as stable colloidal dispersions in solution-phase chemical batch reactions at relatively low temperatures. Researchers have developed a wide range of reaction conditions to produce a variety of shapes, compositions, and structures with well-controlled sizes and low polydispersities. The shape can be controlled to produce confinement in one, two, or all three spatial dimensions, the latter defining a “quantum dot.” Combining two or more semiconductors within the various nanostructures produces additional optical and electronic degrees of freedom. These mixed compositional nanostructures have properties not accessible in bulk semiconductor systems. Further, researchers have fabricated novel nanoheterostructures, such as “dot-in-rods,” quantum wells, core/shell structures, giant-core/shell tetrapods where the core is one material and the arm is another, Janus structures that consist of spherical structures with well-defined internal heterostructures, nano-dumbbells, hollow-core structures, and many other novel and interesting structures. The structural variations are seemingly only limited by the creativity of the various researchers engaged in these endeavors.

36 MATERIALS SCIENCE↗

Simulating Mixed‐Phase Open Cellular Clouds Observed During COMBLE: Evaluation of Parameterized Turbulence Closure

Marine cold-air outbreaks, or CAOs, are airmass transformations whereby relatively cold boundary layer (BL) air is transported over relatively warm water. To more deeply understand BL and mixed-phase cloud properties during CAO conditions, the Cold-Air Outbreaks in the Marine Boundary Layer Experiment (COMBLE) took place from late 2019 into early 2020. During COMBLE, the U.S. Department of Energy's first Atmospheric Radiation Measurement Mobile Facility (AMF1) was deployed to Andenes, Norway, far downstream (~1,000 km) from the Arctic pack ice. This study examines the two most intense CAOs sampled at the AMF1 site. The observed BL structures are open cellular with high (~3–5 km) and cold (–30 to –50 °C) cloud tops, and they often have pockets of high liquid water paths (LWPs; up to ~1,000 g m –2 ) associated with strong updrafts and enhanced turbulence. We use a high-resolution mesoscale model to explore how well four turbulence closure methods represent open cellular clouds. After applying a radar simulator to model outputs for direct evaluation, cloud top properties agree well with AMF1 observations (within ~10%), but radar reflectivity and LWP agreement is more variable. Results suggest that the turbulent Prandtl number may play an important role for the simulated BL and cloud properties. All simulations produce enhanced precipitation rates that are well-correlated with a cloud transition. Finally, the eddy-diffusivity/mass-flux approach produces the deepest cloud layer and therefore the largest and most coherent cellular structures. Furthermore, we recommend the use of a non-local turbulence closure approach to better capture turbulent processes in intense CAOs.

54 ENVIRONMENTAL SCIENCES↗

Optimizing conformance control for gas injection EOR in unconventional reservoirs

Field observations have shown that interference between wells caused by connected fractures commonly exists in unconventional reservoirs. The well interference effect and conformance issues could lead to minimal enhanced oil recovery response (EOR) in actual operations if they are not properly controlled. A series of data analysis, modeling, and experimental activities were performed in this study to investigate the gas EOR strategies in the Bakken Formation. Employing the embedded discrete fracture modeling technology, a multiple-well, multiple-fracture model was developed to capture the well interference and conformance issues in the Bakken. The model made it possible to investigate the complex flow behavior between wells in the EOR (huff ‘n’ puff) processes. Alternative EOR and conformance control strategies using water and surfactant injections were studied to improve oil production performance based on experimental observations. Results showed that water injection could be used for conformance control in the Bakken because of its relatively low mobility compared to gas; however, water could also block oil flow around the conformance control wells because of the high interfacial tension between oil and water. Adding surfactant to the injection water is more effective for improving gas EOR performance since surfactant reduces the oil–water interfacial tension and contact angle effectively. Up to 9.4% more oil could be produced from the huff ‘n’ puff wells in 2 years of EOR operations when rich gas was injected at only 3 MMscf/d with 6000 psi of surfactant injection in the offset wells for conformance control.

42 ENGINEERING↗

Pennsylvania Department of Environmental Protection (PA DEP) 26r Detailed Produced Water Compositions (version 2.0)

A database of geochemical compositions of aqueous species in produced water reported to the PA Department of Environmental Protection (PA DEP). Samples were collected between late-2010 to late-2024. Data from publicly available PA DEP 26r reports were scraped from pdf files and cumulated into tabular spreadsheet format for >3,000 produced water streams from Marcellus Shale wells in Pennsylvania. In addition to providing the original values, the NETL NEWTS team has reformatted the dataset to allow sample streams to be easily copied into OLI Studio and Geochemist WorkBench (GWB) software for modeling the geochemistry and the recovery of critical minerals, such as lithium, from these produced water streams. ***This dataset is an updated version of the PA DEP 26r Detailed Produced Water Compositions (version 1.0) dataset, providing expanded spatial and temporal coverage.***

Aqueous Chemistry↗

Ultralight dark matter detection with levitated ferromagnets

Levitated ferromagnets act as ultraprecise magnetometers, which can exhibit high quality factors due to their excellent isolation from the environment. These instruments can be utilized in searches for ultralight dark matter candidates, such as axionlike dark matter or dark-photon dark matter. In addition to being sensitive to an axion-photon coupling or kinetic mixing, which produce physical magnetic fields, ferromagnets are also sensitive to the effective magnetic field (or “axion wind”) produced by an axion-electron coupling. While the dynamics of a levitated ferromagnet in response to a dc magnetic field have been well studied, all of these couplings would produce ac fields. In this work, we study the response of a ferromagnet to an applied ac magnetic field and use these results to project their sensitivity to axion and dark-photon dark matter. We pay special attention to the direction of motion induced by an applied ac field, in particular, whether it precesses around the applied field (similar to an electron spin) or librates in the plane of the field (similar to a compass needle). We show that existing levitated ferromagnet setups can already have comparable sensitivity to an axion-electron coupling as comagnetometer or torsion balance experiments. In addition, future setups can become sensitive probes of axion-electron coupling, dark-photon kinetic mixing, and axion-photon coupling, for ultralight dark matter masses m DM ≲ feV . Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Effects of Surface Treatments on the Outgassing of 3D-Printed 316L Stainless Steel.

Fermi National Accelerator Laboratory (Fermilab) is a world leader in designing and operating particle accelerators for nuclear and particle physics research. These accelerators rely on electric fields to propel particle beams, which are then directed and focused using magnetic fields. To ensure these beams travel without interference, they must move through an ultra-high vacuum environment, typically around 10⁻⁹ torr. Maintaining such a vacuum requires materials with exceptionally low outgassing rates. 316L stainless steel is a common material choice for vacuum systems due to its low outgassing rates, excellent corrosion resistance, non-magnetic properties, and high mechanical strength. When manufactured using traditional methods like machining, extrusion, and casting, the vacuum properties of 316L stainless steel are well understood. However, the behavior of 316L stainless steel produced through additive manufacturing, such as Direct Metal Laser Sintering (DMLS), is not as well documented. This study aims to determine the outgassing rate of 316L stainless steel fabricated using DMLS. The investigation focuses on measuring the outgassing rates of 1.5 2 vacuum reducers and evaluating the effects of surface preparation techniques, such as mechanical polishing and electropolishing, on the outgassing performance. Understanding the outgassing performance of 3D-printed 316L stainless steel could enable Fermilab engineers to adopt this fabrication method for producing complex parts designed for ultra-high vacuum environments. These advancements have the potential to benefit projects, such as the Long-Baseline Neutrino Facility (LBNF) and the Proton Improvement Plan-II (PIP-II), by improving design flexibility and fostering innovation in accelerator development.

Cloud, Jaiden [Central State University]↗

Commissioning & Characterization of Magnetized Gridded Thermionic Electron Source

Collaborative efforts to design and fabricate a magnetized gridded thermionic electron source have been conducted between Xelera and Jefferson Lab. Commissioning and characterization of an electron gun fabricated by Xelera was performed to benchmark the viability of future electron source designs and capabilities. The work involved simulation, installation, trouble-shooting, modifications of the design, commissioning, characterization, and magnetization of the electron beam produced. A specially designed cavity as well as novel diagnostic tools and methods were developed, implemented, and experimentally tested. Finally, the gun was used to demonstrate a previously unachieved current of magnetized electron beam from a gridded thermionic source of this uniquely compact size. This thesis addresses the following scientific and technological challenges: building a compact gridded thermionic electron source, producing high current magnetized electron beam, and demonstrating novel characterization measurements for magnetized beam.

Stefani, Mark↗

Investigating the role of the transcriptional regulator Ure2 on the metabolism of Saccharomyces cerevisiae: a multi-omics approach

Ure2 regulates nitrogen catabolite repression in Saccharomyces cerevisiae. Deletion of URE2 induces a physiological state mimicking the nitrogen starvation and autophagic responses. Previous work has shown that deletion of URE2 increases the fermentation rate of some wine-producing strains of S. cerevisiae. In this work, we investigated the effect of URE2 deletion (ΔURE2) on the metabolism of S. cerevisiae. During growth on glucose, the ΔURE2 mutant grew less well than the wild type; however, it produced ethanol at higher rates. To better under the behavior of this mutant, we performed transcriptomics and metabolomics. Analysis of the RNA sequencing results and metabolite levels indicate that the mutant strain exhibited characteristics of both nitrogen starvation and autophagy. In addition, many pyruvate decarboxylase and alcohol dehydrogenase isoforms were expressed at higher rates than the wild type. The mutant also accumulated less trehalose and glycogen, and produced more lipids. These results suggest that URE2 may be a promising target for metabolic engineering in S. cerevisiae and potentially other strains of yeast as well for the production of lipid-based fuels and chemicals.

59 BASIC BIOLOGICAL SCIENCES↗

Corn and Switchgrass Derived Sustainable Aviation Fuel Carbon Intensity

The production and use of sustainable aviation fuel (SAF) could lower aviation's carbon footprint. Feedstock agricultural management can optimize—or potentially determine—SAF sustainability and profitability, but long-term SAF feedstock comparisons are limited. Life cycle assessments (LCA) are commonly used to estimate the carbon intensity of producing SAF from emerging feedstocks due to the limited availability of long-term, primary field data. We quantified the agricultural phase and well-to-wake (WTWa) carbon intensities of producing SAF from no-till, continuous corn (Zea mays L.) under 120 kg N ha−1 year−1 (120 N corn) and switchgrass (Panicum virgatum L.) under 60 or 120 kg N ha−1 year−1 (60 N and 120 N SWG, respectively) via LCAs based on measured data from a long-term (1998-present) field experiment on marginally productive cropland located in Eastern NE, USA. Measured data included crop productivity, long-term changes in soil organic carbon (SOC) stocks (0–150 cm), annual soil N2O emissions, and field management practices. Both 120 N corn (2078 L ha−1) and 120 N SWG (1752 L ha−1) produced more SAF than 60 N SWG (1229 L ha−1) (p < 0.0001). However, making SAF from corn had greater life cycle GHG emissions (i.e., WTWa carbon intensity) than making SAF from 120 N and 60 N SWG (61.5 > −14.5 and −27.5 g CO2e MJ−1 SAF, respectively; p = 0.0007), primarily due to SOC accrual during crop production. Here, we demonstrate that agricultural management (e.g., crop type, fertilizer use) can determine the sustainably and productivity of producing SAF by using long-term, measured field data in LCAs.

carbon intensity↗

Characterization of Tunnel Oxides in TOPCon Solar Cells

The 1.12 nm thickness for the tunnel oxide layer is near the optimal range described by Choi et al. This thickness should be effective at enabling quantum tunneling; however, it is slightly lower than the reported optimal range which could negatively impact the passivation of the poly-Si interface. An appropriate balance between the two functions must be met to optimize efficiency. Follow up work could focus on testing the optimal range for tunnel oxide thickness in TOPCon solar cells, as well as improving the manufacturing process to produce better control of film thickness. This work could be extended into more advanced TOPCon solar cells including double or triple stack structures, as well as experimental pinhole designs.

SOLAR ENERGY↗

Increasing Power Generation at the Patua Nevada Geothermal Field through Targeted and Adaptive EGS

In 2020, Patua Acquisition Company LLC, submitted the proposal “Increasing Power Generation at the Patua Nevada Geothermal Field through Targeted and Adaptive EGS”, in response to the DOE’s Geothermal Wells of Opportunity Funding Opportunity Announcement. The primary goal of the project was to convert an idle well, Patua 16-29, to an active producer and generate at least an additional 5 MWe at the Patua plant. The project was intended to provide an example of a reproducible methodology for well stimulation in Enhanced Geothermal Systems The project was proposed to be performed in three phases over a period of performance of four years. A first project go/no-go decision was made at the end of Phase 1 based on feasibility of the wellbore, site, and ability to realize a 5 MWe benefit from the project. Several factors were recognized that led to a no-go decision at this phase: • Electrical needs for placing the stimulated well on production were not considered in the initial scoping. Costs of pipeline costs were also underestimated in the original cost share. Easements and rights-of-way to establish a geothermal pipeline underneath or across an interstate highway were also found to be intractable on the time scale of the project. • The plant is injection limited, and no design or cost estimate was in place to handle the additional production needed to realize a 5 Mwe benefit. A new injection well would need to be drilled in addition to the other facility improvements to bring in the stimulated well. Patau Acquisition Company attempted to locate additional wells of opportunity that could realize the SOPOs goals, but was unsuccessful. As a result, this project is being terminated at the end of Phase 1.

15 GEOTHERMAL ENERGY↗

The FLOod Probability Interpolation Tool (FLOPIT): A Simple Tool to Improve Spatial Flood Probability Quantification and Communication

Understanding flood probabilities is essential to making sound decisions about flood-risk management. Many people rely on flood probability maps to inform decisions about purchasing flood insurance, buying or selling real-estate, flood-proofing a house, or managing floodplain development. Current flood probability maps typically use flood zones (for example the 1 in 100 or 1 in 500-year flood zones) to communicate flooding probabilities. However, this choice of communication format can miss important details and lead to biased risk assessments. Here we develop, test, and demonstrate the FLOod Probability Interpolation Tool (FLOPIT). FLOPIT interpolates flood probabilities between water surface elevation to produce continuous flood-probability maps. FLOPIT uses water surface elevation inundation maps for at least two return periods and creates Annual Exceedance Probability (AEP) as well as inundation maps for new return levels. Potential advantages of FLOPIT include being open-source, relatively easy to implement, capable of creating inundation maps from agencies other than FEMA, and applicable to locations where FEMA published flood inundation maps but not flood probability. Using publicly available data from the Federal Emergency Management Agency (FEMA) flood risk databases as well as state and national datasets, we produce continuous flood-probability maps at three example locations in the United States: Houston (TX), Muncy (PA), and Selinsgrove (PA). We find that the discrete flood zones generally communicate substantially lower flood probabilities than the continuous estimates.

54 ENVIRONMENTAL SCIENCES↗

The dynamic atmospheric and aeolian environment of Jezero crater, Mars

Despite the importance of sand and dust to Mars geomorphology, weather, and exploration, the processes that move sand and that raise dust to maintain Mars’ ubiquitous dust haze and to produce dust storms have not been well quantified in situ, with missions lacking either the necessary sensors or a sufficiently active aeolian environment. Perseverance rover’s novel environmental sensors and Jezero crater’s dusty environment remedy this. In Perseverance’s first 216 sols, four convective vortices raised dust locally, while, on average, four passed the rover daily, over 25% of which were significantly dusty (“dust devils”). More rarely, dust lifting by nonvortex wind gusts was produced by daytime convection cells advected over the crater by strong regional daytime upslope winds, which also control aeolian surface features. One such event covered 10 times more area than the largest dust devil, suggesting that dust devils and wind gusts could raise equal amounts of dust under nonstorm conditions.

79 ASTRONOMY AND ASTROPHYSICS↗

BEYONDPLANCK X. Planck Low Frequency Instrument frequency maps with sample-based error propagation

We present Planck Low Frequency Instrument (LFI) frequency sky maps derived within the BEYONDPLANCK framework. This framework draws samples from a global posterior distribution that includes instrumental, astrophysical, and cosmological parameters, and the main product is an entire ensemble of frequency sky map samples, each of which corresponds to one possible realization of the various modeled instrumental systematic corrections, including correlated noise, time-variable gain, as well as far sidelobe and bandpass corrections. This ensemble allows for computationally convenient end-to-end propagation of low-level instrumental uncertainties into higher-level science products, including astrophysical component maps, angular power spectra, and cosmological parameters. We show that the two dominant sources of LFI instrumental systematic uncertainties are correlated noise and gain fluctuations, and the products presented here support – for the first time – full Bayesian error propagation for these effects at full angular resolution. We compared our posterior mean maps with traditional frequency maps delivered by the Planck Collaboration, and find generally good agreement. The most important quality improvement is due to significantly lower calibration uncertainties in the new processing, as we find a fractional absolute calibration uncertainty at 70 GHz of Δg 0 /g 0 = 5 x 10 -5 , which is nominally 40 times smaller than that reported by Planck 2018. However, we also note that the original Planck 2018 estimate has a nontrivial statistical interpretation, and this further illustrates the advantage of the new framework in terms of producing self-consistent and well-defined error estimates of all involved quantities without the need of ad hoc uncertainty contributions. We describe how low-resolution data products, including dense pixel-pixel covariance matrices, may be produced from the posterior samples directly, without the need for computationally expensive analytic calculations or simulations. We conclude that posterior-based frequency map sampling provides unique capabilities in terms of low-level systematics modeling and error propagation, and may play an important role for future Cosmic Microwave Background (CMB) B-mode experiments aiming at nanokelvin precision.

79 ASTRONOMY AND ASTROPHYSICS↗

Observation of breakdown wave mechanism in avalanche ionization produced atmospheric plasma generated by a picosecond CO 2 laser

Understanding the formation and long-timescale evolution of atmospheric plasmas produced by ultrashort, long-wavelength infrared (LWIR) pulses is an important but partially understood problem. Of particular interest are plasmas produced in air with a peak laser intensity ~10 12 W/cm 2 , the so-called clamping intensity observed in LWIR atmospheric guiding experiments where tunneling and multi-photon ionization operative at near-IR or shorter wavelengths are inoperative. We find that avalanche breakdown on the surface of aerosol (dust) particles can act to seed the breakdown of air observed above the 200 GW/cm 2 threshold when a train of 3 ps 10.6 μm laser pulses separated by 18 ps is used. The breakdown first appears at the best focus but propagates backward toward the focusing optic as the plasma density approaches critical density and makes forward propagation impossible. The velocity of the backward propagating breakdown can be as high as 10 9 cm/s, an order of magnitude greater than measured with ns pulse-produced breakdown, and can be explained rather well by the so-called breakdown wave mechanism. Transverse plasma expansion with a similar velocity is assisted by UV photoionization and is observed as a secondary longitudinal breakdown mechanism in roughly 10% of the shots. When a cm-size, TW power beam is propagated, interception of aerosol particles is guaranteed and several (40 cm -3 ) breakdown sites appear, each initially producing a near-critical density plasma. On a 10 ns–1 μs timescale, shockwaves from each site expand radially and coalesce to produce a large hot gas channel. The radial velocity of the expansion agrees well with the prediction of the blast wave theory developed for ultrafast atmospheric detonations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗