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DOE OSTI2023

K 2 Co 2 TeO 6 : A layered magnet with a S = 1 2 Co 2 + honeycomb lattice

Recent observations of Kitaev interactions in d 7 electron configurations have generated research interest in Co 2+ based honeycomb magnets. Here, we report the synthesis of a K 2 Co 2 TeO 6 single crystal. X-ray diffraction and electron microscopy experiments show that K 2 Co 2 TeO 6 crystallizes in a $P$6 3 /$mcm$ space group with undistorted honeycomb layers of Co 2+ ions. Anisotropic magnetic and thermodynamic characterizations demonstrate an effective S = $\frac{1}{2}$ state of Co 2+ at low temperature, competing magnetic phases, and weak magnetic transitions that can be suppressed by applying fields. Moreover, the interlayer K disorder level is tunable by thermal treatments, and it affects the magnetic features obviously. In conclusion, our findings provide a new and promising platform for studying Kitaev physics.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS2011

Nakhla: a Martian Meteorite with Indigenous Organic Carbonaceous Features

The Nakhla meteorite possesses discrete, well defined, structurally coherent morphologies of carbonaceous phases present within iddingsite alteration zones. Based upon both isotopic measurements and analysis of organic phases the presence of pre-terrestrial organics is now recognized. Within the microcrystalline layers of Nakhla s iddingsite, discrete clusters of salt crystals are present. These salts are predominantly halite (NaCl) with minor MgCl2 crystals. Some CaSO4, likely gypsum, appears to be partially intergrown with some of the halite. EDX mapping shows discrete C-rich features are interspersed among these crystals. A hollow semi-spherical bowl structure (~ 3 m ) has been identified and analyzed after using a focused ion beam (FIB) to cut a transverse TEM thin section of the feature and the underlying iddingsite. TEM/EDX analysis reveals that the feature is primarily carbonaceous containing C with lesser amounts of Si, S, Ca, Cl, F, Na, and minor Mn and Fe; additionally a small peak consistent with N, which has been previously seen in Nakhla carbonaceous matter, is also present. Selected area electron diffraction (SAED) shows that this C-rich material is amorphous (lacking any long-range crystallographic order) and is not graphite or carbonate. Micro-Raman spectra acquired from the same surface from which the FIB section was extracted demonstrate a typical kerogen-like D and G band structure with a weak absorption peak at 1350 and a stronger peak at 1600/cm. The C-rich feature is intimately associated with both the surrounding halite and underlying iddingsite matrix. Both iddingsite and salts are interpreted as having formed as evaporate assemblages from progressive evaporation of water bodies on Mars. This assemblage, sans the carbonaceous moieties, closely resembles iddingsite alteration features previously described which were interpreted as indigenous Martian assemblages. These distinctive macromolecular carbonaceous structures in Nakhla may represent one of the sources of the high molecular weight organic material previously identified in Nakhla. While we do not speculate on the origin of these unique carbonaceous structures, we note that the significance of such observations is that it may allow us to construct a C-cycle for Mars based on the C chemistry of the Martian meteorites with obvious implications for astrobiology and the prebiotic evolution of Mars. In any case, our observations strongly suggest that organic C exists as micrometersize, discrete structures on Mars.

McKay, D. S.↗

DOE OSTI2021

The Deep Eutectic Solvent Precipitation Synthesis of Metastable Zn 4 V 2 O 9

A precipitation method involving a deep eutectic solvent (DES)–a mixture of hydrogen bond donor and acceptor–is used to synthesize a ternary metal oxide. Without toxic reagents, precipitates consisting of Zn 3 (OH) 2 V 2 O 7 ·nH 2 O and Zn 5 (OH) 6 (CO 3 ) 2 are obtained by simply introducing deionized H 2 O to the DES solution containing dissolved ZnO and V 2 O 5 . Manipulation of the synthetic conditions demonstrates high tunability in the size/morphology of the two-dimensional nanosheets precipitated during the dynamic equilibrium process. According to differential scanning calorimetry and high-temperature powder X-ray diffraction, Zn 3 V 2 O 8 and ZnO obtained by the annealing of the precipitate are intermediates in the reaction pathway toward metastable Zn 4 V 2 O 9 . Intimate mixing of the metal precursors achieved by the precipitation method allows access to the metastable zinc-rich vanadate with unusually rapid heat treatment. Furthermore, the UV–vis and surface photovoltage spectra reveal the presence of sub-band gap states, stemming from the reduced vanadium (V 4+ ) center. Photoelectrochemical measurements confirm weak photoanodic currents for water and methanol oxidation. For the first time, this work shows the synthesis of a metastable oxide with the DES-precipitation route and provides insight into the structure–property relationship of the zinc-rich vanadate.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Comparing the Effects of Side Chain Dipole–Dipole Interactions and Hydrogen Bonding on the Mechanical and Electrical Properties of Poly(3-hexylthiophene)

Poly(3-alkylthiophenes) are simple conjugated polymers with good electrical properties, but tend to be brittle, limiting their application. Here, we investigate how side chain modifications can improve their toughness without compromising their charge transport. Specifically, we compare a weak dipole–dipole interaction of ester groups with a stronger hydrogen-bonding interaction of the hydroxyl groups. Two copolymers with targeted 5 and 10 mol % ester-functionalized side chains were synthesized and hydrolyzed in the solid state to introduce hydroxyl groups. Both hydrolyzed polymers became insoluble in most organic solvents, providing a path to layer-by-layer solution processing. All polymers’ optical, thermal, structural, electrical, and mechanical properties were investigated. Optical spectroscopy revealed that functionalization at these low levels did not significantly affect aggregation or π–π stacking. Differential scanning calorimetry indicated reduced crystallinity in the functionalized copolymers compared to that of P3HT. Grazing-incidence wide-angle X-ray diffraction data showed that functionalization led to an increase in lamellar spacing without altering molecular orientation or π–π stacking. Mechanical testing highlighted that at the lower functionalization level, both ester and hydroxyl groups similarly enhanced toughness, suggesting that disorder, rather than the specific nature of the functional group, was responsible for these improvements. At the higher functionalization level, hydroxyl groups significantly increased the elastic modulus while also negatively impacting charge carrier mobility. This study suggests that with stronger hydrogen-bonding groups, only very small amounts can improve P3HT’s toughness. For a wider tunable range, it may be a better strategy to use weak interactions such as dipole–dipole interactions of ester groups.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Probing the atomic dynamics of ultrafast melting with femtosecond electron diffraction

Melting is an every-day phase transition that is determined by thermodynamic parameters like temperature and pressure. In contrast, ultra-fast melting is governed by the microscopic response to a rapid energy input and, thus, can reveal the strength and dynamics of atomic bonds as well as the energy flow rate to the lattice. Accurately describing these processes remains challenging and requires detailed insights into transient states encountered. Here, we present data from femtosecond electron diffraction measurements that capture the structural evolution of copper during the ultrafast solid-to-liquid phase transformations. At absorbed energy densities 2-4 times the melting threshold, melting begins at the surface slightly below the nominal melting point followed by rapid homogeneous melting throughout the volume. Molecular dynamics simulations reproduce these observations and reveal a weak electron-lattice energy transfer rate for the given experimental conditions. Both simulations and experiments show no indications of rapid lattice collapse when its temperature surpasses proposed limits of superheating, providing evidence that the inherent dynamics limits the speed of disordering in ultrafast melting of metals.

FOS: Physical sciences↗

DOE OSTI2024

Magnetic properties of the double trillium lattice antiferromagnet KBaCr 2 ⁢(PO 4 ) 3

Trillium lattices formed by corner-shared triangular units are the platform for magnetic frustration in three dimensions. Herein, we report structural and magnetic properties of the Cr-based (S = 3/2) double trillium lattice material KBaCr 2 ⁢(PO 4 ) 3 studied by x-ray diffraction, dc and ac magnetization, heat capacity, thermal conductivity, and 31 P nuclear magnetic resonance (NMR) measurements complemented by density-functional band-structure calculations. Magnetization, heat capacity, and 31 P NMR measurements reveal the magnetic transition at T N⁢1 ≃ 12.5 K in zero field followed by another transition (T N⁢2 ) at low temperatures in weak applied fields. The 31 P spin-lattice relaxation rate in the ordered state follows the T 3 behavior indicative of the two-magnon Raman process. Finally, the spin lattice of KBaCr 2 ⁢(PO 4 ) 3 comprises two crystallographically nonequivalent ferromagnetic sublattices that are coupled antiferromagnetically, thus eliminating frustration in this trillium network and leading to weak ferromagnetism in the vicinity of T N⁢1 .

36 MATERIALS SCIENCE↗

NASA NTRS1984

Coupled-mode analysis of phase-locked injection laser arrays

A coupled-mode analysis has been developed to describe the output of phase-locked injection laser arrays. It is shown that an array of emitters with weak coupling can only operate in a set of discrete modes determined by the number of and the spacing of the emitters. The interaction between emitters leads to a splitting of the common frequency of operation that can be estimated from the coupling strength. The coupled-mode analysis is compared to calculations based on simple diffraction theory. A consequence of the analysis is an explanation for the commonly observed discrepancy between experimentally observed far-field lobe widths and those predicted by simple diffraction theory.

Butler, J. K.↗

NASA NTRS1989

Structures of cyano-biphenyl liquid crystals

The structures of p-alkyl- p'-cyano- bicyclohexanes, C(n)H(2n+1) (C6H10)(C6H10) CN (n-CCH), and p-alkyl- p'-cyano- biphenyls, C(n)H(2n+1) (C6H4)(C6H4) CN (n-CBP), were studied. It is convenient to use an x ray image intensification device to search for symmetric x ray diffraction patterns. Despite the similarities in molecular structures of these compounds, very different crystal structures were found. For the smectic phase of 2CCH, the structure is close to rhombohedral with threefold symmetry. In contrast, the structure is close to hexagonal close-packed with two molecules per unit cell for 4CCH. Since intermolecular forces may be quite weak for these liquid crystals systems, it appears that crystal structures change considerably when the alkyl chain length is slightly altered. Different structures were also found in the crystalline phase of n-CBP for n = 6 to 9. For n = 7 to 9, the structures are close to monclinic. The structures are reminiscent of the smectic-A liquid crystal structures with the linear molecules slightly tilted away from the c-axis. In contrast, the structure is quite different for n = 6 with the molecules nearly perpendicular to the c-axis.

Chu, Yuan-Chao↗

DOE OSTI2021

Conduction via surface states in antiferromagnetic Mott-insulating Ni S 2 single crystals

Pyrite structure transition-metal disulfides exhibit diverse ground states vs d-band filling, spanning diamagnetic semiconducting, ferromagnetic metallic, antiferromagnetic Mott insulating, and superconducting in FeS 2 , CoS 2 , NiS 2 , and CuS 2 . NiS is particularly interesting and poorly understood as its Mott insulating behavior is accompanied by complex antiferromagnetic ordering below ~38 K and perplexing weak ferromagnetism below ~30 K. Temperature-, pressure-, and composition-dependent insulator-metal transitions also occur, particularly in bandwidth-controlled NiS 2–x Se x , hole-doped Ni 1–x Co x S 2 , etc. Here, we use high-quality chemical-vapor-transport-grown NiS 2 single crystals characterized by x-ray diffraction, energy-dispersive x-ray spectroscopy, magnetometry, and extensive transport and magnetotransport measurements, to generate new insight into this system. In particular, resistivity, magnetoresistance, and Hall effect analyses vs temperature, thickness, and surface preparation, provide unequivocal evidence of surface conduction, where the more conductive surface shunts essentially all current at low temperatures. The surface transport changes from two dimensional and insulating to three dimensional and metallic as the surface preparation is varied (also displaying intriguing sensitivity to magnetic ordering), significantly clarifying literature ambiguities with respect to the electronic ground state. These results have immediate implications. First, the temperature-, pressure-, and composition-dependent insulator-metal transitions deduced in the extensive prior work on NiS 2-x Se x , Ni 1–x Co x S 2 , etc., must clearly be reexamined in light of rife metallic surface conduction, not previously taken into account. Second, NiS 2 now joins FeS 2 and CoS 2 as systems in which bulk and surface electronic behaviors are strikingly different, suggesting that metallic surface states could be a universal feature of pyrite structure transition-metal disulfides.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Noncentrosymmetric Nowotny Chimney Ladder Ferromagnet Cr 4 Ge 7 with a High Curie Temperature of ~207 K

Noncentrosymmetric magnets usually host intriguing magnetic interactions inherent in the crystal structure with broken inversion symmetry, which can give rise to rich magnetic behaviors. Here, we report herein the high-pressure synthesis, crystal structure, magnetizations, and magnetic structure of a so-called Nowotny chimney ladder compound Cr 4 Ge 7 . Our analysis on the powder neutron diffraction data revises the crystal structure as a noncentrosymmetric space group (P4̅c2, No. 116). It exhibits two magnetic orders within the temperature range of 2–400 K. The first order at ~207 K associated with a small magnetic moment of ~0.75 μ B is assigned to a commensurate ferromagnetic structure with a propagation vector k = (0, 0, 0). The weak itinerant ferromagnet nature should be caused by the complex Cr spin orders from different Wyckoff positions. The second order-like behavior at ~18 K is assumed to arise from a competition between the Dzyaloshinskii–Moriya and Heisenberg interactions. The results provide an excellent platform for the study of intricate interactions among various magnetic exchanges as well as for the exploration of high-temperature exotic magnetic properties.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Josephson detection of time-reversal symmetry broken superconductivity in SnTe nanowires

A Josephson junction (JJ) couples the supercurrent flowing between two weakly linked superconductors to the phase difference between them via a current-phase relation (CPR). While a sinusoidal CPR is expected for conventional junctions with insulating weak links, devices made from some exotic materials may give rise to unconventional CPRs and unusual Josephson effects. In this work, we present such a case: we investigate the proximity-induced superconductivity in SnTe nanowires by incorporating them as weak links in JJs and observe a deviation from the standard CPR. We report on indications of an unexpected breaking of time-reversal symmetry in these devices, detailing the unconventional characteristics that reveal this behavior. These include an asymmetric critical current in the DC Josephson effect, a prominent second harmonic in the AC Josephson effect, and a magnetic diffraction pattern with a minimum in critical current at zero magnetic field. The analysis examines how multiband effects and the experimentally visualized ferroelectric domain walls give rise to this behavior, giving insight into the Josephson effect in materials that possess ferroelectricity and/or multiband superconductivity.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

A-type antiferromagnetic order in semiconducting EuMg 2 Sb 2 single crystals

Eu-based Zintl-phase materials EuA 2 Pn 2 (A = Mg, In, Cd, Zn; Pn = Bi, Sb, As, P) have generated significant recent interest owing to the complex interplay of magnetism and band topology. Here, we investigated the crystallographic, magnetic, and electronic properties of the layered Zintlphase single crystals of EuMg 2 Sb 2 with the trigonal CaAl 2 Si 2 crystal structure (space group $P\bar{3}m1$). Electrical resistivity measurements complemented with angle-resolved photoemission spectroscopy (ARPES) studies and density functional theory (DFT) calculations find an activated behavior with intrinsic conductivity at high temperatures indicating a semiconducting electronic ground state with a narrow energy gap of 370 meV. Magnetic susceptibility and zero-field heat capacity measurements indicate that the compound undergoes antiferromagnetic (AFM) ordering at the Néel temperature T N = 8.0(2) K. Here, zero-field neutron-diffraction measurements reveal that the AFM ordering is A-type where the Eu spins (Eu 2+ , S = 7/2) arranged in ab-plane layers are aligned ferromagnetically in the ab plane and the Eu spins in adjacent layers are aligned antiferromagnetically. Eu-moment reorientation within the ab planes in the trigonal AFM domains associated with a very weak inplane magnetic anisotropy is also evident below T N at low fields < 0.05 T. Although isostructural semimetallic EuMg 2 Bi 2 is reported to host Dirac surface states, the observation of narrow-gap semiconducting behavior in EuMg 2 Sb 2 implies a strong role of spin-orbit coupling (SOC) in tuning the electronic states of these materials. Our DFT studies also suggest, besides the SOC, the more electronegative and smaller Sb than Bi shifts the low-lying conduction bands along the Γ-A direction to higher energy, resulting in an indirect bulk band gap between the Γ and M points for EuMg 2 Sb 2 .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Organo‐Functionalized Lacunary Double Cubane‐Type Oxometallates: Synthesis, Structure, and Properties of [(M II Cl) 2 (V IV O) 2 {((HOCH 2 CH 2 )(H)N(CH 2 CH 2 O))(HN(CH 2 CH 2 O) 2 )} 2 ] (M=Co, Zn)

Abstract Organofunctionalized tetranuclear clusters [(M II Cl) 2 (V IV O) 2 {((HOCH 2 CH 2 )(H)N(CH 2 CH 2 O))(HN(CH 2 CH 2 O) 2 )} 2 ] (1, M=Co,2: M=Zn) containing an unprecedented oxometallacyclic {M 2 V 2 Cl 2 N 4 O 8 } (M=Co, Zn) framework have been prepared by solvothermal reactions. The new oxo‐alkoxide compounds were fully characterized by spectroscopic methods, magnetic susceptibility measurement, DFT and ab initio computational methods, and complete single‐crystal X‐ray diffraction structure analysis. The isostructural clusters are formed of edge‐sharing octahedral {VO 5 N} and trigonal bipyramidal {MO 3 NCl} units. Diethanolamine ligates the bimetallic lacunary double cubane core of1and2in an unusual two‐mode fashion, unobserved previously. In the crystalline state, the clusters of1and2are joined by hydrogen bonds to form a three‐dimensional network structure. Magnetic susceptibility data indicate weakly antiferromagnetic interactions between the vanadium centers [J iso (V IV −V IV )=−5.4(1); −3.9(2) cm −1 ], and inequivalent antiferromagnetic interactions between the cobalt and vanadium centers [J iso (V IV −Co II )=−12.6 and −7.5 cm −1 ] contained in1.

Chemistry↗

DOE OSTI2020

Mapping the structural, magnetic and electronic behavior of (Eu 1-x Ca x ) 2 Ir 2 O 7 across a metal-insulator transition

In this study, we employ bulk electronic properties characterization and x-ray scattering/spectroscopy techniques to map the structural, magnetic and electronic properties of (Eu 1-x Ca x ) 2 Ir 2 O 7 as a function of Ca-doping. As expected, the metal-insulator transition temperature, T-MIT, decreases with Ca-doping until a metallic state is realized down to 2 K. In contrast, T-AFM becomes decoupled from the MIT and (likely short-range) AFM order persists into the metallic regime. This decoupling is understood as a result of the onset of an electronically phase separated state, the occurrence of which seemingly depends on both synthesis method and rare earth site magnetism. PDF analysis suggests that electronic phase separation occurs without accompanying chemical phase segregation or changes in the short-range crystallographic symmetry while synchrotron x-ray diffraction confirms that there is no change in the long-range crystallographic symmetry. X-ray absorption measurements confirm the $J_{eff}$ = 1/2 character of (Eu 1-x Ca x ) 2 Ir 2 O 7 . Surprisingly these measurements also indicate a net electron doping, rather than the expected hole doping, indicating a compensatory mechanism. Lastly, XMCD measurements show a weak Ir magnetic polarization that is largely unaffected by Ca-doping. Keywords: term, term, term.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2020

0D and 2D: The Cases of Phenylethylammonium Tin Bromide Hybrids

Tin halide perovskites and perovskite-related materials have emerged as promising lead-free hybrid materials for various optoelectronic applications. While remarkable progress has been achieved in the development of organic tin halide hybrids with diverse structures and controlled dimensionalities at the molecular level, some controversial results that have been reported recently need to be addressed. For instance, different photophysical properties have been reported for two-dimensional (2D) (PEA) 2 SnBr 4 (PEA = phenylethylammonium) by several groups with distinct emission peaks at around 468 and 550 nm. Here we report our efforts in the synthesis of phenylethylammonium tin bromide hybrids with zero-dimensional (0D) and 2D structures, and characterizations of their structural and photophysical properties. 0D [(PEA) 4 SnBr 6 ][(PEA)Br] 2 [CCl 2 H 2 ] 2 was found to exhibit strong yellow emission peak at 566 nm with a photoluminescence quantum efficiency (PLQE) of ~90%, while 2D (PEA) 2 SnBr 4 had weak emission peak at 470 nm with a PLQE of <0.1%. Interestingly, 0D [(PEA) 4 SnBr 6 ][(PEA)Br] 2 [CCl 2 H 2 ] 2 can be converted into 2D (PEA) 2 SnBr 4 upon drying, which would return to 0D [(PEA) 4 SnBr 6 ][(PEA)Br] 2 [CCl 2 H 2 ] 2 upon addition of dichloromethane. Powder X-ray diffraction results confirmed the reversible transformation between 0D and 2D structures. Density functional theory calculations showed that excitons in 0D [(PEA) 4 SnBr 6 ][(PEA)Br] 2 [CCl 2 H 2 ] 2 are highly localized, resulting in a strongly Stokes shifted broadband emission, while delocalized electronic states in 2D (PEA) 2 SnBr 4 result in weaker exciton binding, a higher exciton mobility, and a higher nonradiative decay.

36 MATERIALS SCIENCE↗

DOE OSTI2022

X-ray nano-imaging of defects in thin film catalysts via cluster analysis

Functional properties of transition-metal oxides strongly depend on crystallographic defects; crystallographic lattice deviations can affect ionic diffusion and adsorbate binding energies. Scanning x-ray nanodiffraction enables imaging of local structural distortions across an extended spatial region of thin samples. Yet, localized lattice distortions remain challenging to detect and localize using nanodiffraction, due to their weak diffuse scattering. Here, in this study, we apply an unsupervised machine learning clustering algorithm to isolate the low-intensity diffuse scattering in as-grown and alkaline-treated thin epitaxially strained SrIrO 3 films. We pinpoint the defect locations, find additional strain variation in the morphology of electrochemically cycled SrIrO 3 , and interpret the defect type by analyzing the diffraction profile through clustering. Our findings demonstrate the use of a machine learning clustering algorithm for identifying and characterizing hard-to-find crystallographic defects in thin films of electrocatalysts and highlight the potential to study electrochemical reactions at defect sites in operando experiments.

42 ENGINEERING↗

DOE OSTI2025

Investigation of magnetic and electrical transport properties of YMB 4 (M = Cr, Fe, and Co) compounds

We have studied the structural, magnetic, and electrical transport properties of the YMB 4 series of compounds (M= Cr, Fe, and Co). These materials exhibit high stability and possess notable refractory and thermoelectric properties. Furthermore, recent theoretical predictions suggest that some of these compounds may display quantum magnetism and dimer formation among the magnetic ions. We have synthesized almost single phase YMB 4 compounds in the orthorhombic crystal structure (space group Pbam) using arc-melting and annealing. The Rietveld analysis of the room-temperature X-ray diffraction patterns reveals a clear change in the lattice parameter c, which correlates with the atomic radii of Cr, Fe, and Co. The temperature variation of resistivity measured between 2 K and 300 K reveals a metallic electron transport in all three YMB 4 compounds. However, no abrupt change in resistivity due to structural or magnetic phase transitions is observed. Additionally, a weak positive magnetoresistance of 1%–2% has been measured at 2.5 K, the contribution from the metallic impurity cannot be ruled out. The carrier concentration, on the order of 10 21 cm −3 , has been determined using conventional Hall measurements. The thermomagnetic curves recorded between 2 K and 350 K reveal the non-magnetic behavior of all these compounds. However, a small magnetic moment from paramagnetic/ferromagnetic impurity phases is detected at low temperatures. Above room temperature, no magnetic transition associated with the breaking of a dimer is observed.

borides↗

NASA NTRS1994

Interaction of the sonic boom with atmospheric turbulence

Theoretical research was carried out to study the effect of free-stream turbulence on sonic boom pressure fields. A new transonic small-disturbance model to analyze the interactions of random disturbances with a weak shock was developed. The model equation has an extended form of the classic small-disturbance equation for unsteady transonic aerodynamics. An alternative approach shows that the pressure field may be described by an equation that has an extended form of the classic nonlinear acoustics equation that describes the propagation of sound beams with narrow angular spectrum. The model shows that diffraction effects, nonlinear steepening effects, focusing and caustic effects and random induced vorticity fluctuations interact simultaneously to determine the development of the shock wave in space and time and the pressure field behind it. A finite-difference algorithm to solve the mixed type elliptic-hyperbolic flows around the shock wave was also developed. Numerical calculations of shock wave interactions with various deterministic and random fluctuations will be presented in a future report.

Rusak, Zvi↗