Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “radiation mapping”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 451 records · Page 25
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Detailed Characterization of CZT Detector Response for Improved Coded-Aperture Imaging Performance

Gamma-ray imaging is a powerful method for locating and quantifying sources of radiation. The coded-aperture technique demonstrates superior angular resolution in comparison to other methods (e.g., Compton reconstruction). In this method, a mask constructed of highly attenuating material encodes the scene as a shadow pattern on a position-sensitive detector; this pattern can then be used to recreate the origin(s) of incident radiation. This is typically done through convolution of the mask and shadow patterns. Iterative methods which attempt to reconstruct the observed shadow pattern using a weighted combination of simulated patterns may also be employed. In either case, errors in event position reconstruction due to detector imperfections alter the shadow pattern and will therefore degrade system performance and may introduce imaging artifacts. These effects can be mitigated with a detailed understanding of such errors – allowing for the generation of representative simulations that include the errors and/or correction of raw imager data to remove the errors. We present a calibration process for a commercially available cadmium zinc telluride (CZT) gamma imager which provides a comprehensive characterization of the spatial and energy dependence of event reconstruction. By illuminating a mask featuring a regular grid of pinholes with a calibration source, the localized response of the detector can be measured with fine granularity. These local responses are combined to generate a full detector response map which can be used to distort simulations in a manner that is representative of the observed detector data. Details of the calibration procedure and an assessment of the impact of its end products on the performance of iterative imaging methods will be presented.

Ziock, Klaus-Peter↗
DOE OSTI2021

Alpha Spectrometry Results for Groundwater Samples Collected in Northern Iraq and a Summary of the Environmental Setting of the Adaya Burial Site

The Radiation Protection Center (RPC) of the Iraqi Ministry of Environment continues to evaluate the potential health impacts associated with the Adaya Burial Site, which is located 33 kilometers (20.5 miles) southwest of Mosul. This report documents the radiological analyses of 16 groundwater samples collected from wells located in the vicinity of the Adaya Burial Site and at other sites in northern Iraq. The Adaya Burial Site is a high-risk dump site because a large volume of radioactive material and contaminated soil is located on an unsecure hillside above the village of Tall ar Ragrag. The uranium activities for the 16 water samples in northern Iraq are considered to be naturally occurring and do not indicate artificial (man-made) contamination. With one exception, the alpha spectrometry results for the 16 wells that were sampled in 2019 indicate that the water quality concerning the three uranium isotopes (Uranium-233/234, Uranium-235/236, and Uranium-238) was acceptable for potable purposes (drinking and cooking). However, Well 7 in Mosul had a Uranium-233/234 activity concentration that slightly exceeded the World Health Organization guidance level. Eight of the 16 wells are located in the villages of Tall ar Ragrag and Adaya and had naturally occurring uranium concentrations. Wells in the villages of Tall ar Ragrag and Adaya are located near the Adaya Burial Site and should be sampled on an annual schedule. The list of groundwater analytes should include metals, total uranium, isotopic uranium, gross alpha/beta, gamma spectroscopy, organic compounds, and standard water quality parameters. Our current understanding of the hydrogeologic setting in the vicinity of the Adaya Burial Site is solely based on villager's domestic wells, topographic maps, and satellite imagery. To better understand the hydrogeologic setting, a Groundwater Monitoring Program needs to be developed and should include the installation of twelve groundwater monitoring wells in the vicinity of Tall ar Ragrag and the Adaya Burial Site. Characterization of the limestone aquifer and overlying alluvium is needed. RPC should continue to support health assessments for the villagers in Tall ar Ragrag and Adaya. Collecting samples for surface water (storm water), airborne dust, vegetation, and washway sediment should be conducted on a routine basis. Human access to the Adaya Burial Site needs to be strictly limited. Livestock access on or near the burial site needs to be eliminated. The surface-water exposure pathway is likely a greater threat than the groundwater exposure pathway. Installation of a surface-water diversion or collection system is recommended in order to reduce the potential for humans and livestock to come in contact with contaminated water and sediment. To reduce exposure to villagers, groundwater treatment should be considered if elevated uranium or other contaminants are detected in drinking water. Installing water-treatment systems would likely be quicker to accomplish than remediation and excavation of the Adaya Burial Site. The known potential for human exposure to uranium and metals (such as arsenic, chromium, selenium, and strontium) at the Adaya Burial Site is serious. Additional characterization , mitigation, and remediation efforts should be given a high priority.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

AniMACCS User Guide

AniMACCS is a utility code in the MELCOR Accident Consequence Code System (MACCS) software suite that allows for certain MACCS output information to be visually displayed and overlaid onto a geospatial map background. AniMACCS was developed by Sandia National Laboratories for the U.S. Nuclear Regulatory Commission. MACCS is designed to calculate health and economic consequences following a release of radioactive material in the atmosphere. MACCS accomplishes this by modeling the atmospheric dispersion, deposition, and consequences of the release, which depend on several factors including the source term, weather, population, economic, and land-use characteristics of the impacted geographical area. From these inputs, MACCS determines the characteristics of the plume, as well as ground and air concentrations as a function of time and radionuclide.

63 RADIATION, THERMAL, AND OTHER ENVIRON. POLLUTAN↗
DOE OSTI2022

The physical origin of dark energy constraints from rubin observatory and CMB-S4 lensing tomography

ABSTRACT We seek to clarify the origin of constraints on the dark energy equation of state parameter from CMB lensing tomography, that is the combination of galaxy clustering and the cross-correlation of galaxies with CMB lensing in a number of redshift bins. We focus on the analytic understanding of the origin of the constraints. Dark energy information in these data arises from the influence of three primary relationships: distance as a function of redshift (geometry), the amplitude of the power spectrum as a function of redshift (growth), and the power spectrum as a function of wavenumber (shape). We find that the effects from geometry and growth play a significant role and partially cancel each other out, while the shape effect is unimportant. We also show that Dark Energy Task Force figure of merit forecasts from the combination of LSST galaxies and CMB-S4 lensing are comparable to the forecasts from cosmic shear in the absence of the CMB lensing map, thus providing an important independent check. Compared to the forecasts with the LSST galaxies alone, combining CMB lensing and LSST clustering information increases the FoM by roughly a factor of 3–4 in the optimistic scenario where systematics are fully under control. We caution that achieving these forecasts will likely require a full analysis of higher-order biasing, photometric redshift uncertainties, and stringent control of other systematic limitations, which are outside the scope of this work, whose primary purpose is to elucidate the physical origin of the constraints.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Magnetic Misalignment of Interstellar Dust Filaments

We present evidence for scale-independent misalignment of interstellar dust filaments and magnetic fields. We estimate the misalignment by comparing millimeter-wave dust-polarization measurements from Planck with filamentary structures identified in neutral-hydrogen (H I ) measurements from H I 4PI. We find that the misalignment angle displays a scale independence (harmonic coherence) for features larger than the H I 4PI beamwidth (16$\acute{.}$2). We additionally find a spatial coherence on angular scales of ${ \mathcal O }(1^\circ )$. We present several misalignment estimators formed from the auto- and cross-spectra of dust-polarization and H I -based maps, and we also introduce a map-space estimator. Applied to large regions of the high-Galactic-latitude sky, we find a global misalignment angle of ~2°, which is robust to a variety of masking choices. By dividing the sky into small regions, we show that the misalignment angle correlates with the parity-violating TB cross-spectrum measured in the Planck dust maps. The misalignment paradigm also predicts a dust EB signal, which is of relevance in the search for cosmic birefringence but as yet undetected; the measurements of EB are noisier than those of TB, and our correlations of EB with misalignment angle are found to be weaker and less robust to masking choices. We also introduce an H I -based dust-polarization template constructed from the Hessian matrix of the H I intensity, which is found to correlate more strongly than previous templates with Planck dust B modes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

All-epitaxial long-range surface plasmon polariton structures with integrated active materials

We demonstrate all-epitaxial structures capable of supporting short- and long-range surface plasmon polariton (SRSPP and LRSPP) modes in the long-wave infrared region of the electromagnetic spectrum. The SRSPP and LRSPP modes are bound to the interfaces of a buried heavily doped (n ++ ) semiconductor layer and surrounding quantum-engineered type-II superlattice (T2SL) materials. The surrounding T2SLs are designed to allow optical transitions across the frequency dispersion of the SPP modes. We map the SPP dispersion in our structure using grating-coupled angle- and polarization-dependent reflection and photoluminescence spectroscopy. The epitaxial structures are analytically described using a simplified three-layer system (T2SL/n ++ /T2SL) and modeled using rigorous coupled wave analysis with excellent agreement to our experimental results. The presented structures offer the potential to serve as long-range interconnects or waveguides in all-epitaxial plasmonic/optoelectronic systems operating in the long-wave infrared.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Chaos on the hypercube

We analyze the spectral properties of a d-dimensional HyperCubic (HC) lattice model originally introduced by Parisi. The U(1) gauge links of this model give rise to a magnetic flux of constant magnitude φ but random orientation through the faces of the hypercube. The HC model, which also can be written as a model of 2d interacting Majorana fermions, has a spectral flow that is reminiscent of Maldacena-Qi (MQ) model, and its spectrum at φ = 0, actually coincides with the coupling term of the MQ model. As was already shown by Parisi, at leading order in 1/d, the spectral density of this model is given by the density function of the Q-Hermite polynomials, which is also the spectral density of the double-scaled Sachdev-Ye-Kitaev model. Parisi demonstrated this by mapping the moments of the HC model to Q-weighted sums on chord diagrams. We point out that the subleading moments of the HC model can also be mapped to weighted sums on chord diagrams, in a manner that descends from the leading moments. The HC model has a magnetic inversion symmetry that depends on both the magnitude and the orientation of the magnetic flux through the faces of the hypercube. The spectrum for fixed quantum number of this symmetry exhibits a transition from regular spectra at φ = 0 to chaotic spectra with spectral statistics given by the Gaussian Unitary Ensembles (GUE) for larger values of φ. For small magnetic flux, the ground state is gapped and is close to a Thermofield Double (TFD) state.

1/N expansion↗
DOE OSTI2021

Intelligent Consequence Control by Aerial Reconnoiter Using Unmanned Systems

In a nuclear accident, or in the aftermath of an improvised nuclear device or radiological dispersal device detonation, radioactive materials may pose a severe health threat to individuals near the site of the incident. The Intelligent Consequence Control by Aerial Reconnoiter Using Unmanned Systems (ICARUS) does the dull, dirty, and dangerous (and, sometimes, deep) work of mapping and characterizing valuable information to enable a safe and appropriate response. ICARUS provides timely answers to the questions, “What is it?” “Where is it?”, and “How far does it extend?” Using commercial, off-the-shelf proven components, the integrated chemical, biological, radiological, nuclear, and explosives detection architecture for our drones provides a solution for collecting radiological, chemical, and optical information on developing complex situations, returning actionable data for determining if it is safe for further response efforts. ICARUS can determine and transmit the location, identity, and intensity of radionuclide contamination to a remote base station. It can deliver on the unexpected.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Mapping of fragmented νf 5/2 →πf 7/2 transitions in the 73 Co→ 73 Ni decay

Excited states in 73 , 75 Ni were investigated through the β decay of 73 , 75 Co in an experiment performed at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University (MSU). The experimental results extended the level scheme of Ni 73 to 3.2-MeV excitation energy and provided the experimental information on excited states in Ni 75 . The β -delayed neutron branching ratio for Co 73 was obtained. Furthermore, the experimental results are discussed in comparison with shell-model calculations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

BEYONDPLANCK VIII. Efficient sidelobe convolution and corrections through spin harmonics

We introduce a new formulation of the Conviqt convolution algorithm in terms of spin harmonics, and apply this to the problem of sidelobe correction for BEYONDPLANCK, the first end-to-end Bayesian Gibbs sampling framework for CMB analysis. We compare our implementation to the previous Planck LevelS implementation, and find good agreement between the two codes in terms of accuracy, but with a speed-up reaching a factor of 3–10, depending on the frequency bandlimits, l max and m max . The new algorithm is significantly simpler to implement and maintain, since all low-level calculations are handled through an external spherical harmonic transform library. We find that our mean sidelobe estimates for Planck LFI are in good agreement with previous efforts. Additionally, we present novel sidelobe rms maps that quantify the uncertainty in the sidelobe corrections due to variations in the sky model.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Design and simulation of a muon detector to characterize geological overburden

This study presents the design, construction, and simulation of a mobile muon detector tailored for geological overburden characterization. The detector employs plastic scintillator paddles with silicon photomultipliers (SiPMs) and a QuarkNet data acquisition system, offering a portable solution suitable for remote field deployment. The simulator’s modular aluminum frame allows for adjustable geometry and directional sensitivity, while its battery system supports over a week of autonomous operation. Preliminary experimental tests confirmed that its muon flux measurements were consistent with theoretical expectations. A comprehensive simulation framework using Geant4 and CORSIKA was developed to model detector response and overburden effects. Analytical and Monte Carlo methods were used to assess quadrant resolution and infer muon directionality. This work lays the foundation for future overburden mapping and supports the development of reconstruction algorithms for geological applications.

72 - PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Subventricular Accumulation of Cu in the Aging Mouse Brain Does Not Associate with Anticipated Increases in Markers of Oxidative Stress

Natural aging is associated with mild memory loss and cognitive decline, and age is the greatest risk factor for neurodegenerative diseases, such as Alzheimer’s disease. There is substantial evidence that oxidative stress is a major contributor to both natural aging and neurodegenerative disease, and coincidently, levels of redox active metals such as Fe and Cu are known to be elevated later in life. Recently, a pronounced age-related increase in Cu content has been reported to occur in mice and rats around a vital regulatory brain region, the subventricular zone of lateral ventricles. In our study herein, we have characterized lateral ventricle Cu content in a unique murine model of accelerated aging, senescence accelerated mouse-prone 8 (SAMP8) mice. Our results confirm an age-related increase in ventricle Cu content, consistent with the studies by others in wild-type mice and rats. Specifically, we observed Cu content to increase over the time frame 1 to 5 months and 5 to 9 months, but interestingly, no significant increase occurred between 9 and 12 months (although brain Cu content at 12 months was significantly elevated relative to 1 and 5 month-old animals). Despite the magnitude of Cu increase observed within the cells that comprise the subventricular zone of lateral ventricles (average 3 mM Cu, with isolated subcellular concentrations of 17 mM), we did not detect spectroscopic markers of thiol oxidation, protein aggregation, or lipid oxidation. The lack of evidence for oxidative stress in ex vivo animal tissue is in contrast to in vitro studies demonstrating that thiol, protein, and lipid oxidation is pronounced at these Cu concentrations. We suggest that our findings most likely indicate that the Cu ions in this brain region are sequestered in an unreactive form, possibly extended chains of Cu-thiolate complexes, which do not readily redox cycle in the aqueous cytosol. In conclusion, these results also appear to partially challenge the long-held view that age-related increases in brain metal content drive oxidative stress as we did not observe a concomitant association between age-related Cu increase and markers of oxidative stress, nor did we observe a net increase in Cu content between mice aged 9 and 12 months.

X-ray fluorescence↗
DOE OSTI2025

Isochronous and period-doubling diagrams for symplectic maps of the plane

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more thorough understanding of the underlying qualitative aspects. This paper aims to address this gap by revisiting the foundational concepts of reversibility and associated symmetries, first explored in the early works of G.D. Birkhoff. We extend the original framework proposed by Hénon by adding a period-doubling diagram to his isochronous diagram, which allows to represents the system’s bifurcations and the groups of symmetric periodic orbits that emerge in typical bifurcations of the fixed point. A qualitative and quantitative explanation of the main features of the region of parameters with bounded motion is provided, along with the application of this technique to other symplectic mappings, including cases of multiple reversibility. Modern chaos indicators, such as the Reversibility Error Method (REM) and the Generalized Alignment Index (GALI), are employed to distinguish between various dynamical regimes in the mixed space of variables and parameters. These tools prove effective in differentiating regular and chaotic dynamics, as well as in identifying twistless orbits and their associated bifurcations. Additionally, we discuss the application of these methods to real-world problems, such as visualizing dynamic aperture in accelerator physics, where our findings have direct relevance.

43 PARTICLE ACCELERATORS↗
DOE OSTI2026

Opportunities for imaging light nuclei with a second interaction region at the Electron-Ion Collider

The upcoming Electron-Ion Collider (EIC) will address several outstanding puzzles in modern nuclear physics. Key questions—such as the partonic structure of nucleons and nuclei and the origin of their mass and spin—can be explored through high-energy electron-proton and electron-nucleus collisions. Here, to maximize its scientific reach, the EIC community has advocated for the addition of a second interaction region equipped with a detector complementary to the EIC general purpose collider detector, ePIC. The preconceptual design of this interaction region aims to provide a different configuration from the first interaction region, which enhances forward acceptance at very small scattering angles (𝜃 ∼0 mrad). This machine configuration would significantly benefit exclusive, tagging, and diffractive physics programs, complementing those of the ePIC experiment. In particular, accessing coherent diffractive processes on light nuclei by tagging of the full, intact nucleus is essential for mapping their spatial parton distributions. In this work, we present an exploratory study of the detection capabilities for light nuclei at a second EIC interaction region, with a detailed discussion of the accessible kinematic phase space and its implications for imaging.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Qu8its for quantum simulations of lattice quantum chromodynamics

We explore the utility of d = 8 qudits, qu8its, for quantum simulations of the dynamics of 1+1⁢D SU(3) lattice quantum chromodynamics, including a mapping for arbitrary number of flavors and lattice size and a reorganization of the Hamiltonian for efficient time evolution. Recent advances in parallel gate applications, along with the shorter application times of single-qudit operations compared with two-qudit operations, lead to significant projected advantages in quantum simulation fidelities and circuit depths using qu8its rather than qubits. The number of two-qudit entangling gates required for time evolution using qu8its is found to be more than a factor of 5 fewer than for qubits. Here, we anticipate that the developments presented in this work will enable improved quantum simulations to be performed using emerging quantum hardware.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Chemical Imaging of Atmospheric Biomass Burning Particles from North American Wildfires

The effects of biomass burning aerosols (BBA) on radiative forcing and cloud formation depend on chemical composition and the internal structures of individual particles within smoke plumes. To improve our understanding of the chemical and physical properties of BBA emitted at different times of the day and their evolution during atmospheric aging, we conducted a study as a part of the Fire Influence on Regional to Global Environments and Air Quality field campaign. Particle samples were collected onboard a research aircraft from smoke plumes from a wildfire in eastern Oregon during late afternoon and nighttime flights on August 28, 2019. A time-resolved aerosol collector was used to collect samples on substrates for offline spectromicroscopic imaging to investigate the single-particle characteristics of BBA particles. Approximately 20,400 individual particles from 10 selected samples were analyzed using computer-controlled scanning electron microscopy coupled with energy-dispersive X-ray microanalysis, revealing their elemental composition, morphology, and viscosity. Elemental microanalysis indicated that aged potassium is likely found in the form of K 2 SO 4 , KNO 3 , and possible K-organic salts. Further chemical speciation and carbon bonding mapping within individual particles were conducted using synchrotron-based scanning transmission X-ray microscopy (STXM) coupled with near edge X-ray absorption fine structure (NEXAFS) spectroscopy. Real-time, water-soluble light absorption measurements were acquired using a particle-into-liquid sampler instrument coupled to a liquid waveguide capillary cell and total organic analyzer. In the late afternoon samples, 65% of the total particle number population consisted entirely of organic components, compared to 46% in the nighttime particles. These differences were attributed to discrepancies in composition at the time of emission and to the daytime condensation and accumulation of photochemically formed secondary organic material on existing BBA particles, a process that halts at night. Microscopy images indicated that particle viscosity was lower in the nighttime particles (<10 1 Pa·s), likely due to increased relative humidity and a higher contribution from hygroscopic inorganic components. The chemical heterogeneity of individual particles was quantified using STXM-derived mixing state parameters. The nature of carbon bonding within individual particles was inferred from the extent of carbon sp 2 hybridization derived from NEXAFS spectra. Average percentages of sp 2 hybridization range between 40% and 60%, with no noticeable differences between late afternoon and nighttime flights. These findings were compared with the online optical properties of both late afternoon and nighttime smoke plumes, providing valuable insights into the complex relationship between chemical composition and optical properties of BBA particles at different times of the day.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Northern Hemisphere Winter Air Temperature Patterns and Their Associated Atmospheric and Ocean Conditions

The Northern Hemisphere (NH) has experienced winter Arctic warming and continental cooling in recent decades, but the dominant patterns in winter surface air temperature (SAT) are not well understood. Here, a self-organizing map (SOM) analysis is performed to identify the leading patterns in winter daily SAT fields from 1979 to 2018, and their associated atmospheric and ocean conditions are also examined. Three distinct winter SAT patterns with two phases of nearly opposite signs and a time scale of 7–12 days are found: one pattern exhibits concurrent SAT anomalies of the same sign over North America (NA) and northern Eurasia, while the other two patterns show SAT anomalies of opposite signs between, respectively, NA and the Bering Sea, and the Kara Sea and East Asia (EA). Winter SAT variations may arise from changes in the SOM frequencies. Specifically, the observed increasing trends of winter cold extremes over NA, central Eurasia, and EA during 1998–2013 can be understood as a result of the increasing occurrences of some specific SAT patterns. These SOMs are closely related to poleward advection of midlatitude warm air and equatorward movements of polar cold airmass. These meridional displacements of cold and warm airmasses cause concurrent anomalies over different regions not only in SAT but also in water vapor and surface downward longwave radiation. Anomalous sea surface temperatures in the tropical Pacific, midlatitude North Pacific, and North Atlantic and anomalous Arctic sea ice concentrations also concur to support and maintain the anomalous atmospheric circulation that causes the SAT anomalies.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Dynamics of McMillan mappings II. axially symmetric map

Here, in this article, we investigate the transverse dynamics of a single particle in a model integrable accelerator lattice, based on a McMillan axially-symmetric electron lens. Although the McMillan e-lens has been considered as a device potentially capable of mitigating collective space charge forces, some of its fundamental properties have not been described yet. The main goal of our work is to close this gap and understand the limitations and potentials of this device. It is worth mentioning that the McMillan axially symmetric map provides the first-order approximations of dynamics for a general linear lattice plus an arbitrary thin lens with motion separable in polar coordinates. Therefore, advancements in its understanding should give us a better picture of more generic and not necessarily integrable round beams. In the first part of the article, we classify all possible regimes with stable trajectories and find the canonical action-angle variables. This provides an evaluation of the dynamical aperture, Poincaré rotation numbers as functions of amplitudes, and thus determines the spread in nonlinear tunes. Also, we provide a parameterization of invariant curves, allowing for the immediate determination of the map image forward and backward in time. The second part investigates the particle dynamics as a function of system parameters. We show that there are three fundamentally different configurations of the accelerator optics causing different regimes of nonlinear oscillations. Each regime is considered in great detail, including the limiting cases of large and small amplitudes. In addition, we analyze the dynamics in Cartesian coordinates and provide a description of observable variables and corresponding spectra.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗