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At least 1,153 records · Page 64

Diverted Negative Triangularity plasmas on DIII–D: The benefit of high confinement without the liability of an edge pedestal

Diverted discharges at negative triangularity on the DIII-D tokamak sustain normalized confinement and pressure levels typical of standard H-mode scenarios (H 98y,2 ≅1, β N ≅3) without developing an edge pressure pedestal, despite the auxiliary power far exceeding the L → H power threshold expected from conventional scaling laws. The power degradation of confinement is substantially weaker than the ITER-89P scaling, resulting in a confinement factor that improves with increasing auxiliary power. Furthermore, the absence of the edge pedestal is beneficial in several aspects, such as eliminating the need for active mitigation or suppression of edge localized modes, low impurity retention and a reconstructed scrape-off layer heat flux width at the mid-plane that exceeds the ITPA multi-machine scaling law by up to 50%. Together with technological advantages granted by placing the divertor at larger radii, plasmas at Negative Triangularity without an edge pedestal feature both core confinement and power handling characteristics that are potentially suitable for operation in future fusion reactors.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Experimental and Monte Carlo studies of fluence corrections for graphite calorimetry in low- and high-energy clinical proton beams

Purpose: The aim of this study was to determine fluence corrections necessary to convert absorbed dose to graphite, measured by graphite calorimetry, to absorbed dose to water. Fluence corrections were obtained from experiments and Monte Carlo simulations in low- and high-energy proton beams. Methods: Fluence corrections were calculated to account for the difference in fluence between water and graphite at equivalent depths. Measurements were performed with narrow proton beams. Plane-parallel-plate ionization chambers with a large collecting area compared to the beam diameter were used to intercept the whole beam. High- and low-energy proton beams were provided by a scanning and double scattering delivery system, respectively. A mathematical formalism was established to relate fluence corrections derived from Monte Carlo simulations, using the FLUKA code [A. Ferrari et al., “FLUKA: A multi-particle transport code,” in CERN 2005-10, INFN/TC 05/11, SLAC-R-773 (2005) and T. T. Böhlen et al., “The FLUKA Code: Developments and challenges for high energy and medical applications,” Nucl. Data Sheets 120, 211–214 (2014)], to partial fluence corrections measured experimentally. Results: A good agreement was found between the partial fluence corrections derived by Monte Carlo simulations and those determined experimentally. For a high-energy beam of 180 MeV, the fluence corrections from Monte Carlo simulations were found to increase from 0.99 to 1.04 with depth. In the case of a low-energy beam of 60 MeV, the magnitude of fluence corrections was approximately 0.99 at all depths when calculated in the sensitive area of the chamber used in the experiments. Fluence correction calculations were also performed for a larger area and found to increase from 0.99 at the surface to 1.01 at greater depths. Conclusions: Fluence corrections obtained experimentally are partial fluence corrections because they account for differences in the primary and part of the secondary particle fluence. A correction factor, F(d), has been established to relate fluence corrections defined theoretically to partial fluence corrections derived experimentally. The findings presented here are also relevant to water and tissue-equivalent-plastic materials given their carbon content.

60 APPLIED LIFE SCIENCES↗

Improved Regional Moment Tensor Inversion for Moderately Large Earthquakes in the Western United States Using a 3D Earth Model Based on Full Waveform Tomography

The nature of seismic sources for moderately large (moment magnitude, M w 5.0–6.5) events are commonly characterized by their moment tensor (MT) solutions and obtained by inversion of regional distance (200–1600 km) long‐period (20–50 s) waveforms. Regional MT estimates are often calculated from average plane‐layered, one‐dimensional (1D) velocity models. However, 1D model calculations can produce misfits in the arrival times and waveform shapes that introduce errors, particularly at longer distances or for shorter periods, which are necessary for analyzing lower magnitude events. Approximate Earth models (e.g., 1D) representing broad areas may be inadequate, particularly in the crust and uppermost mantle of tectonically complex regions. In this study, we show how a three‐dimensional (3D) Earth model obtained from full waveform inversion tomography can improve waveform fits and decrease phase errors. We developed a platform and workflow to perform routine 3D MT inversions and inverted MTs for 25 earthquakes in the western United States and seven nuclear explosions using an average 1D and a recent 3D Earth model, WUS256 (Rodgers et al., 2022). Using the 3D model improves waveform fits (variance reduction and phase time shifts) compared with the 1D model, and the 3D MT solutions are stable across large distances. This study shows that 3D models obtained from full waveform tomography can improve MTs and source characterization especially at far regional distances (>800 km).

Geosciences↗

The Measurement of Electrical and Thermal Conductivities in Warm Dense Matter

This document serves as the Final Technical Report for grant DE-SC0012333, The Measurement of Electrical and Thermal Conductivities in Warm Dense Matter . This work proposed to make the first systematic measurements of the reflectivity of warm dense matter (WDM) at multiple wavelengths by controlling and measuring the experimental factors necessary to use the reflectivity measurements to determine the electrical and thermal conductivity as a function of temperature and, to a lesser degree, density. A new technique was developed to measure the real and imaginary components of the electrical conductivity by making only scalar measurements of the relative reflectivity different angles. A series of preliminary experiments were designed to use the second, third, and fourth harmonics of the 800 nm driving laser. Reflectivity at each harmonic for two different angles would provide the scalar information needed to extract the real and imaginary parts of the index of refraction. Experiments were made using continuous wave lasers to determine the limits of accuracy on the measured reflectance for surrogate plane mirrors. It was determined, that while the technique was sound, the practical application of it was limited for the conditions on Scarlet. Due to significant challenges with the Scarlet Laser Facility at the time, progress was delayed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Blunt Impact Brain Injury using Cellular Injury Criterion

The Advanced Combat Helmet (\ACH") military specification (\mil-spec") requires a helmeted magnesium (\Mg") Department of Transportation (\DOT") headform be dropped vertically, with an impact speed of 3.1 m/s (10 ft/s), onto a steel hemispherical target. The pass/fail criteria are based on translational acceleration (150 G) alone, absent of any rotational component. Without a rotational component, the specification's injury risk application is limited to skull fracture and peripheral hematomas (subdural, subarachnoid), since this translational acceleration injury risk assessment is based on the Wayne State Tolerance Curve (\WSTC"). To provide a more comprehensive view of injury for the entire brain, an alternative approach is needed. To meet this need, we worked with a larger group called PANTHER, a collaboration between national laboratories, industry, and academia. Collaborations specific to research and results presented here come from efforts led by Mr. Ron Szalkowski and Mr. Sushant Malave, Ms. Alice Fawzi, and Dr. Christian Franck. We have developed a prototypical injury risk criterion based on the neuronal response to abrupt changes in general motion (translation, rotation, or both). The cellular-based mild traumatic brain injury (\cbmTBI") criterion utilizes both the strain and strain rate of brain tissue to account for the stretch and rate of stretch that occurs throughout the brain as a result of blunt impact to the head. We conducted physical experiments of an ACH-fitted magnesium headform, which produced repeatable headform peak accelerations. Then, we developed a simulation of the experiment, and validated the simulation output with the experimental data. We then substituted the magnesium headform with a human headform, consisting of skin, muscle, bone, gray matter, white matter, cerebral-spinal fluid, membranes, vasculature, intravertebral discs, airway and sinus. We quantified brain injury risk using the cbmTBI criterion, using the current mil-spec test and a modified test. The modified mil-spec test used an inclined anvil target that was located posterior to the crown of the helmet in the axial plane. While the current mil-spec test produced brain deformation from head translation alone, the modified test produced brain deformation from head translation and rotation, which is closer to most real world and combat theater impacts (e.g., such as occur in tertiary blast exposure). Compared to the current mil-spec test, the modified test produced elevated strains in the human digital twin. These data, mapped to the cbmTBI criterion, suggest increased injury risk for blunt impacts that cause rotation and translation, rather than just translation alone. Moreover, these data may lead to a rotational performance metric, which is rooted in the biology and pathology of the brain's response to impact and blast, and which should be used to improve next-generation helmet designs.

42 ENGINEERING↗

Growth Oscillatory Zoning in Erythrite, Ideally Co3(AsO4)2·8H2O: Structural Variations in Vivianite-Group Minerals

The crystal structure of an oscillatory zoned erythrite sample from Aghbar mine, Bou Azzer, Morocco, was refined using synchrotron high-resolution powder X-ray diffraction (HRPXRD) data, Rietveld refinement, space group C2/m, and Z = 2. The crystal contains two sets of oscillatory zones that appear to have developed during epitaxial growth. The unit-cell parameters obtained are a = 10.24799(3) Å, b = 13.42490(7) Å, c = 4.755885(8) Å, β = 105.1116(3)°, and V = 631.680(4) Å3. The empirical formula for erythrite, obtained with electron-probe micro-analysis (EPMA), is [Co2.78Zn0.11Ni0.07Fe0.04]Σ3.00(AsO4)2·8H2O. Erythrite belongs to the vivianite-type structure that contains M1O2(H2O)4 octahedra and M22O6(H2O)4 octahedral dimers that are linked by TO4 (T5+ = As or P) tetrahedra to form complex layers parallel to the (010) plane. These layers are connected by hydrogen bonds. The average [6] = 2.122(1) Å and average [6] = 2.088(1) Å. With space group C2/m, there are two solid solutions: M3(AsO4)2·8H2O and M3(PO4)2·8H2O where M2+ = Mg, Fe, Co, Ni, or Zn. In these As- and P-series, using data from this study and from the literature, we find that their structural parameters evolve linearly with V and in a nearly parallel manner despite of the large difference in size between P5+ (0.170 Å) and As5+ (0.355 Å) cations. Average [4], [6], and [6] distances increase linearly with V. The average distance is affected by M atoms, whereas the average distance is unaffected because it contains shorter and stronger P–O bonds. Although As- and P-series occur naturally, there is no structural reason why similar V-series vivianite-group minerals do not occur naturally or cannot be synthesized.

36 MATERIALS SCIENCE↗

Enhanced Spontaneous Polarization in Ultrathin SnTe Films with Layered Antipolar Structure

2D SnTe films with a thickness of as little as 2 atomic layers (ALs) have recently been shown to be ferroelectric with in–plane polarization. Remarkably, they exhibit transition temperatures (T c ) much higher than that of bulk SnTe. Here, combining molecular beam epitaxy, variable temperature scanning tunneling microscopy, and ab initio calculations, the underlying mechanism of the Tc enhancement is unveiled, which relies on the formation of γ–SnTe, a van der Waals orthorhombic phase with antipolar inter–layer coupling in few–AL thick SnTe films. In this phase, 4n – 2 AL (n = 1, 2, 3…) thick films are found to possess finite in–plane polarization (space group Pmn2 1 ), while 4n AL thick films have zero total polarization (space group Pnma). Above 8 AL, the γ–SnTe phase becomes metastable, and can convert irreversibly to the bulk rock salt phase as the temperature is increased. This finding unambiguously bridges experiments on ultrathin SnTe films with predictions of robust ferroelectricity in GeS–type monochalcogenide monolayers. Here, the observed high transition temperature, together with the strong spin–orbit coupling and van der Waals structure, underlines the potential of atomically thin γ–SnTe films for the development of novel spontaneous polarization–based devices.

2d ferroelectrics↗

A reaction–diffusion model for grayscale digital light processing 3D printing

We report that digital light processing (DLP) 3D printing is an additive manufacturing process that utilizes light patterns to photopolymerize a liquid resin into a solid. Due to the accuracy of modern digital micromirror devices (DMD) and recent advances in resin chemistry, it is now possible to create functionally graded structures using different light intensity values, also known as grayscale DLP (g-DLP). Different intensities of light lead to differences in the polymer crosslinking density after curing, which ultimately produces a part with gradients of material properties. However, g-DLP is a complicated process. First, the DLP printing is a highly coupled chemical and physical process that involves light propagation, chemical reactions, species diffusion, heat transfer, volume shrinkage, and changes in mechanical behaviors of the curing resin. Second, in g-DLP, light gradients create strong in plane gradients of chemical species concentrations in the curing liquid resin due to the strong dependence of light intensity on the rate of monomer crosslinking. Furthermore, light gradients through the depth create concentration gradients due to the degree of cure dependent light absorption and the use of photoabsorbers. These complex physical features of the printing process must be understood in order to properly control printing parameters such as light exposure time, printing speed, and grayscale variations to achieve accurate mechanical properties. In this paper, a photopolymerization reaction–diffusion model is developed and used in conjunction with experiments to investigate the coupled effects of light propagation, chemical reaction rates, and species diffusion during g-DLP 3D printing. The model is implemented numerically utilizing the finite difference method and simulation results are compared to experimental findings of simple printed structures. The agreement between experimental and model predictions of simple quantities of interest, such as geometric feature sizes, shows that the model can capture the overcure due to free-radical and other species diffusion during printing when grayscale patterns are employed. This model lays the groundwork for future extensions that can incorporate more complex coupled physics such as heat transfer, volume shrinkage, and material property evolution, which are critically important in utilizing g-DLP 3D printing for the fabrication of high-performance parts which excellent geometric and material property tolerances.

36 MATERIALS SCIENCE↗

Binary dislocation junction formation and strength in hexagonal close-packed crystals

This work examines binary dislocation interactions, junction formation and junction strengths in hexagonal close-packed ( hcp ) crystals. Through a line-tension model and dislocation dynamics (DD) simulations, the interaction and dissociation of different sets of binary junctions are investigated involving one dislocation on the (011¯0) prismatic plane and a second dislocation on one of the following planes: (0001) basal, (11¯00) prismatic, (11¯01) primary pyramidal, or (2¯112) secondary pyramidal. Varying pairs of Burgers vectors are chosen from among the common types the basal type < a > 1/3 < 112¯0 >, prismatic type < c > <0001>, and pyramidal type < a+c > 1/3 < 112¯3¯ >. For binary interaction due to dislocation intersection, both the analytical results and DD-simulations indicate a relationship between symmetry of interaction maps and the relative magnitude of the Burgers vectors that constitute the junction. Using analytical formulae, a simple regressive model is also developed to represent the junction yield surface. The equation is treated as a degenerated super elliptical equation to quantify the aspect ratio and tilting angle. Lastly, the results provide analytical insights on binary dislocation interactions that may occur in general hcp metals.

36 MATERIALS SCIENCE↗

Beyond Epitaxy: Ion Implantation as a Tool for Orbital Engineering

Manipulating electronic orbital states in quantum materials provides a powerful means of controlling their physical properties and technological functionality. Here, we demonstrate that orbital populations in strongly correlated oxide thin films can be continuously and reversibly tuned through postsynthesis He ion implantation. Using LaNiO 3 as a model system, we show that the orbital preference can be systematically adjusted from favoring in-plane d x 2 –y 2 occupation toward out-of-plane d z 2 states through precise control of ion fluence. Unlike conventional heteroepitaxial approaches that lock in orbital configurations during growth, this strain-doping technique enables continuous orbital tuning and the selective modification of specific film regions after device fabrication. We demonstrate the practical impact of this control by achieving a 7-fold enhancement in oxygen reduction reaction catalysis. This work establishes ion implantation as a powerful approach for orbital engineering that complements existing synthesis-based strategies while offering unique advantages for both basic research and device development.

Doping↗

3D localization of electrophysiology catheters from a single x-ray cone-beam projection

Purpose: X-ray images allow the visualization of percutaneous devices such as catheters in real time but inherently lack depth information. The provision of 3D localization of these devices from cone beam x-ray projections would be advantageous for interventions such as electrophysiology (EP), whereby the operator needs to return a device to the same anatomical locations during the procedure. A method to achieve real-time 3D single view localization (SVL) of an object of known geometry from a single x-ray image is presented. SVL exploits the change in the magnification of an object as its distance from the x-ray source is varied. The x-ray projection of an object of interest is compared to a synthetic x-ray projection of a model of said object as its pose is varied. Methods: SVL was tested with a 3 mm spherical marker and an electrophysiology catheter. The effect of x-ray acquisition parameters on SVL was investigated. An independent reference localization method was developed to compare results when imaging a catheter translated via a computer controlled three-axes stage. SVL was also performed on clinical fluoroscopy image sequences. A commercial navigation system was used in some clinical image sequences for comparison. Results: SVL estimates exhibited little change as x-ray acquisition parameters were varied. The reproducibility of catheter position estimates in phantoms denoted by the standard deviations, (σ{sub x}, σ{sub y}, σ{sub z}) = (0.099 mm, 0.093 mm, 2.2 mm), where x and y are parallel to the detector plane and z is the distance from the x-ray source. Position estimates (x, y, z) exhibited a 4% systematic error (underestimation) when compared to the reference method. The authors demonstrated that EP catheters can be tracked in clinical fluoroscopic images. Conclusions: It has been shown that EP catheters can be localized in real time in phantoms and clinical images at fluoroscopic exposure rates. Further work is required to characterize performance in clinical images as well as the sensitivity to clinical image quality.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

High-Intensity Linear Accelerator with Ribbon Beam

We propose a solution of the fundamental problem in accelerator physics: significant increase of peak beam power in linear accelerator with conservation of beam brightness. It is of primary importance for operation of existing high-power accelerators (LANSCE, Spallation Neutron Source), for development of the next generation of high-power accelerators for basic scientific research and Accelerator-Driven Systems applications (Transmutation of Waste, Nuclear Reactor-Driver), and national security applications, which are all $1B-class facilities. The concept utilizes acceleration of beam with ribbon geometry instead of traditional circular beam. High value of beam current is reached using large width of the beam. Acceleration and focusing is achieved using alternative-focusing principle, which does not require traditional focusing elements (quadrupoles, solenoids). The concept utilizes advantage of 2D plane geometry which excludes beam distortion along beam width and results in conservation of high-value of beam brightness.

43 PARTICLE ACCELERATORS↗

Modeling hydraulic fracture opening and closure with proppant transport and settlement

Hydraulic fracturing is a widely used reservoir stimulation technique for improving fluid circulation in rock formations with extremely low permeability, particularly in enhanced geothermal systems (EGS). To better understand the complex processes involved and improve hydraulic stimulation performance, we have developed ELK (ELectrical fracKing), a MOOSE-based 3D finite element application designed to model the behavior of proppant-fluid mixtures in propagating fractures. ELK integrates both the fluid and proppant components, incorporating particle-driven processes such as gravity settling, particle-particle interactions, and strong density and viscosity contracts, in addition to conventional fluid-driven fracture propagation. In this contribution, we extend ELK to model propped fracture closure, which occurs after the injection phase due to a dramatic drop in the effective stress on the fracture plane. During the shut-in, flowback, and production periods, the fracture width decreases, with the closure behavior depending on proppant concentration.. At low concentrations, closure follows a nonlinear joint law linked to the stiffness of asperities in the fracture walls. While at high concentrations, it is controlled by the properties of packed proppant bed. The extended ELK application is validated against several benchmark examples, including the propagation of an inclined frictional crack, fracture opening and sliding in response to fluid injection, and flowback analysis. We believe that ELK’s enhanced capabilities can serve as a valuable tool for the design and optimization of EGS deployment.

15 - GEOTHERMAL ENERGY↗

Coexistence of Low Damping and Strong Magnetoelastic Coupling in Epitaxial Spinel Ferrite Thin Films

Low‐loss magnetization dynamics and strong magnetoelastic coupling are generally mutually exclusive properties due to opposing dependencies on spin–orbit interactions. So far, the lack of low‐damping, magnetostrictive ferrite films has hindered the development of power‐efficient magnetoelectric and acoustic spintronic devices. Here, magnetically soft epitaxial spinel NiZnAl‐ferrite thin films with an unusually low Gilbert damping parameter (<3 × 10 −3 ), as well as strong magnetoelastic coupling evidenced by a giant strain‐induced anisotropy field (≈1 T) and a sizable magnetostriction coefficient (≈10 ppm), are reported. This exceptional combination of low intrinsic damping and substantial magnetostriction arises from the cation chemistry of NiZnAl‐ferrite. At the same time, the coherently strained film structure suppresses extrinsic damping, enables soft magnetic behavior, and generates large easy‐plane magnetoelastic anisotropy. These findings provide a foundation for a new class of low‐loss, magnetoelastic thin film materials that are promising for spin‐mechanical devices.

Emori, Satoru↗

Atomically Resolved Electrically Active Intragrain Interfaces in Perovskite Semiconductors

Deciphering the atomic and electronic structures of interfaces is key to developing state-of-the-art perovskite semiconductors. However, conventional characterization techniques have limited previous studies mainly to grain-boundary interfaces, whereas the intragrain-interface microstructures and their electronic properties have been much less revealed. Herein using scanning transmission electron microscopy, we resolved the atomic-scale structural information on three prototypical intragrain interfaces, unraveling intriguing features clearly different from those from previous observations based on standalone films or nanomaterial samples. These intragrain interfaces include composition boundaries formed by heterogeneous ion distribution, stacking faults resulted from wrongly stacked crystal planes, and symmetrical twinning boundaries. The atomic-scale imaging of these intragrain interfaces enables us to build unequivocal models for the ab initio calculation of electronic properties. Our results suggest that these structure interfaces are generally electronically benign, whereas their dynamic interaction with point defects can still evoke detrimental effects. This work paves the way toward a more complete fundamental understanding of the microscopic structure-property-performance relationship in metal halide perovskites.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Strong interplay between stripe spin fluctuations, nematicity and superconductivity in FeSe

In iron-based superconductors the interactions driving the nematic order (that breaks four-fold rotational symmetry in the iron plane) may also mediate the Cooper pairing. The experimental determination of these interactions, which are believed to depend on the orbital or the spin degrees of freedom, is challenging because nematic order occurs at, or slightly above, the ordering temperature of a stripe magnetic phase. In this paper, we study FeSe—which exhibits a nematic (orthorhombic) phase transition at T s = 90 K without antiferromagnetic ordering—by neutron scattering, finding substantial stripe spin fluctuations coupled with the nematicity that are enhanced abruptly on cooling through T s . A sharp spin resonance develops in the superconducting state, whose energy (~4 meV) is consistent with an electron–boson coupling mode revealed by scanning tunnelling spectroscopy. The magnetic spectral weight in FeSe is found to be comparable to that of the iron arsenides. Finally, our results support recent theoretical proposals that both nematicity and superconductivity are driven by spin fluctuations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Hydrogen-Assisted Fracture of Stainless Steel

Symmetric four-point bending (S4PB) and anti-symmetric four-point bending (AS4PB) were applied to assess the effect of hydrogen on crack initiation and propagation in a stable, nitrogen-strengthened austenitic stainless steel (21Cr-6Ni-9Mn). Specimens of a high strength aluminum alloy (AA2219-T851), which has also been identified as hydrogen-compatible, were used for test method development prior to completing the stainless steel test matrix. Single edge notched bend (SEN(B)) specimens were extracted from forged 21Cr-6Ni-9Mn bar and AA2219 plate. The 21Cr-6Ni-9Mn specimens were then hydrogen charged with ~200 wt. ppm hydrogen; AA2219 specimens were not charged. Aluminum specimens were tested in S4PB to induced mode I (pure bending), and AS4PB to attain varying levels of mode I/II mixity and mode II (pure shear), where the ratio of mode I to mode II varies with the position of the crack plane relative to the load line. After test method troubleshooting and validation, hydrogen charged stainless steel specimens were then subjected to mode I and two ratios of mixed mode I/II. Mode II loading was not achieved due to high load limitations. Analyses of fracture profiles for both materials reveal a marked effect of loading mode mixity on initial crack propagation orientation, however a specific contribution of hydrogen was not readily identifiable. Fracture initiation toughness in the presence of hydrogen, J IH , was calculated following the J-integral approach; the mode I J IH calculated for stainless steel samples fractured in S4PB were consistent with published values determined from compact tension specimens. The peak loads required to initiate fracture during AS4PB far exceeded those required during S4PB. This is attributed to the increasing shear force applied to the specimens as the degree of mode mixity increases. Ultimately, understanding the fracture response of hydrogen exposed stainless steels subjected to mixed mode I/II loading is critical for designing hydrogen containment vessels or gas transfer systems (GTS).

36 MATERIALS SCIENCE↗

Charging the Johannsen–Psaltis spacetime

In this paper we develop the charged version of the spacetime proposed by Johannsen and Psaltis. Rotation is introduced in the deformed Reissner–Nordström spacetime by complex coordinate transformation. The event horizon and Killing horizon are studied. Killing horizons are represented graphically also. The analysis of the determinant of the metric shows that the spacetime does not have Lorentz violating regions. Similarly, from the study of the closed time-like curves, we see that no such curves exist outside the central body. Expressions for the energy and angular momentum for a particle on the equatorial plane are determined. Location of the circular photon orbits and innermost stable circular orbits is also computed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗