Non-spherical Potential Research Articles

On the calculation of irregular solutions of the Schrödinger equation for non-spherical potentials with applications to metallic alloys

The irregular solutions of the stationary Schrödinger equation are important for the fundamental formal development of scattering theory. They are also necessary for the analytical properties of the Green function, which in practice can greatly speed up calculations. Nevertheless, they are seldom considered in numerical treatments because of their divergent behavior at origin. This divergence demands high numerical precision that is difficult to achieve, particularly for non-spherical potentials which lead to different divergence rates in the coupled angular momentum channels. Based on an unconventional treatment of boundary conditions, an integral-equation method is here developed which is capable of dealing with this problem. The available precision is illustrated by electron-density calculations for NiTi in its monoclinic B19’ structure.

Three-Dimension Calculation for the Scattering Problem for Non-Spherical Potential

The interaction of the 238U with a neutron is studied. Correct accounting for the non-spherical shape of the uranium nucleus is in focus. The optical potential is used as a model. It is shown that the spherically symmetric and non-spherical potentials give different scattering patterns, in particular different resonance features of the cross-section. The possibility of using the method as an extension of the particle–rotor model of the nucleus is illustrated.

Thermal self-similar disc winds: transonic nature

ABSTRACT Thermally driven two-dimensional disc winds are examined under the fully self-similar treatment, focusing the attention on the transonic nature. In the spherical potential, such as point-mass and logarithmic ones, the usual transonic solutions cannot be constructed, because the critical conditions themselves do not exist. That is, the flow can reach the sonic point in the polar direction, while there is no gravitational force in the polar direction, and the regularity condition does not hold. In the non-spherical potential, such as the Mestel self-gravitating one, the gravity works in the polar direction, and therefore, there are critical points in the flow. However, the usual transonic solutions do not exist, since the topology of the critical points is always centre (O-type). Hence, in the fully self-similar disc winds, the usual transonic solutions passing through the critical points do not exist, at least for typical cases. Instead, the new-type wind solutions are found in the point-mass case. In contrast to the usual transonic flows, the new-type solutions are continuously accelerated along the streamline by the thermal pressure, and eventually become supersonic in the streamline direction without passing through the critical points. These new-type solutions are possible, when the density distribution is centrally concentrated, and when the initial subsonic velocity is not so large to avoid singularities.

Stream Fanning and Bifurcations: Observable Signatures of Resonances in Stellar Stream Morphology

Recent observations have revealed a trove of unexpected morphological features in many of the Milky Way’s stellar streams. Explanations for such features include time-dependent deformations of the Galactic gravitational potential, local disruptions induced by dark matter substructure, and special configurations of the streams’ progenitors. In this paper, we study how these morphologies can also arise in certain static, nonspherical gravitational potentials that host a subset of resonantly trapped orbit families. The transitions, or separatrices, between these orbit families mark abrupt discontinuities in the orbital structure of the potential. We develop a novel numerical approach for measuring the libration frequencies of resonant and near-resonant orbits and apply it to study the evolution of stellar streams on these orbits. We reveal two distinct morphological features that arise in streams on near-resonant orbits: fans, which come about due to a large spread in the libration frequencies near a separatrix, and bifurcations, which arise when a separatrix splits the orbital distribution of the stellar stream between two (or more) distinct orbit families. We demonstrate that these effects can arise in some Milky Way streams for certain choices of the dark matter halo potential and discuss how this might be used to probe and constrain the global shape of the Milky Way’s gravitational potential.

Determination of the Binary Diffusion Coefficients and Interaction Viscosities of the Systems Carbon Dioxide–Nitrogen and Ethane–Methane in the Dilute Gas Phase from Accurate Experimental Viscosity Data Using the Kinetic Theory of Gases

The results of viscosity measurements at moderate densities on the two gaseous mixtures carbon dioxide–nitrogen and ethane–methane including the pure gases between 253.15 K and 473.15 K, originally performed by Humberg et al. at Ruhr University Bochum, Germany, using a rotating-cylinder viscometer between 0.1 MPa and 2.0 MPa, were employed to determine the interaction viscosity, eta _{12}^{(0)}, and the product of molar density and diffusion coefficient, (rho D_{12})^{(0)}, each in the limit of zero density. The isothermal viscosity data were evaluated by those authors with density series restricted to the second order at most to derive the zero-density viscosities and initial density viscosity coefficients, eta _textrm{mix}^{(0)} and eta _textrm{mix}^{(1)}, for the mixtures, as well as, eta _i^{(0)} and eta _i^{(1)} (i=1,2), respectively, for the pure gases. Humberg et al. have already compared their eta _textrm{mix}^{(0)} and eta _i^{(0)} data for carbon dioxide–nitrogen and ethane–methane with corresponding viscosity values theoretically computed for the nonspherical potentials of the intermolecular interaction. Now we employed eta _textrm{mix}^{(0)} and eta _textrm{mix}^{(1)} as well as eta _i^{(0)} and eta _i^{(1)} in two procedures to derive eta _{12}^{(0)} values. For this, we needed A_{12}^* values (ratio between effective cross-sections of viscosity and diffusion). But the second procedure applying the initial density viscosity coefficients eta _textrm{mix}^{(1)} and eta _i^{(1)} failed to yield reasonable eta _{12}^{(0)} values. The first procedure should provide the best results when it is possible to use A_{12}^* values computed for the nonspherical potential. The effect is comparatively small if eta _{12}^{(0)} is determined. But if (rho D_{12})^{(0)} is calculated from eta _{12}^{(0)} using A_{12}^* values for the nonspherical potential, the impact is several percent. Moreover, the experimentally based eta _{12}^{(0)} and (rho D_{12})^{(0)} data agree with theoretically calculated values for the nonspherical potentials.

The Correlations of Stellar Tidal Disruption Rates with Properties of Massive Black Holes and Their Host Galaxies

Stars can be either disrupted as tidal disruption events (TDEs) or swallowed whole by massive black holes (MBHs) at galactic centers when they approach sufficiently close to these MBHs. In this work, we investigate the correlations of such stellar consumption rates with both the MBH mass M BH and the inner slope of the host-galaxy mass density distribution α. We introduce a simplified analytical power-law model with a power-law stellar-mass density distribution surrounding MBHs and separate the contributions of two-body relaxation and stellar orbital precession for the stellar orbital angular momentum evolution in nonspherical galaxy potentials. The stellar consumption rates derived from this simplified model can be well consistent with the numerical results obtained with a more realistic treatment of stellar distributions and dynamics around MBHs, providing an efficient way to estimate TDE rates. The origin of the correlations of stellar consumption rates with M BH and α is explained by the dependence of this analytical model on those MBH/host-galaxy properties and by the separation of the stellar angular momentum evolution mechanisms. We propose that the strong positive correlation between the rates of stellar consumption due to two-body relaxation and α provides one interpretation for the overrepresentation of TDEs found in some rare E+A/poststarburst galaxies. We find high TDE rates for giant stars, up to those for solar-type stars. Understanding the origin of the correlations of the stellar consumption rates will be necessary for obtaining the demographics of MBHs and their host galaxies via TDEs.

Elastic scattering of electrons by water: An ab initio study

In this work we devise a theoretical and computational method to compute the elastic scattering of electrons from a non-spherical potential, such as in the case of molecules and molecular aggregates. Its main feature is represented by the ability of calculating accurate wave functions for continuum states of polycentric systems via the solution of the Lippmann-Schwinger equation, including both the correlation effects and multi-scattering interference terms, typically neglected in widely used approaches, such as the Mott theory. Within this framework, we calculate the purely elastic scattering matrix elements. As a test case, we apply our scheme to the modelling of electron-water elastic scattering. The Dirac-Hartree-Fock self-consistent field method is used to determine the non-spherical molecular potential projected on a functional space spanned by Gaussian basis set. By adding a number of multi-centric radially-arranged s-type Gaussian functions, whose exponents are system-dependent and optimized to reproduce the properties of the continuum electron wave function in different energy regions, we are able to achieve unprecedented access to the description of the low energy range of the spectrum (0.001 < E < 10 eV) up to keV, finding a good agreement with experimental data and previous theoretical results. To show the potential of our approach, we also compute the total elastic scattering cross section of electrons impinging on clusters of water molecules and zundel cation. Our method can be extended to deal with inelastic scattering events and heavy-charged particles.

Interference Phenomenon in Electron-Molecule Collisions

This article discusses how the pattern of elastic scattering of an electron on a pair of identical atomic centers is modified if we abandon the assumption, standard in molecular physics, that outside of some molecular sphere surrounding the centers, the wave function of the molecular continuum is atomic-like, being a linear combination of the regular and irregular solutions of the wave equation. For this purpose, the elastic scattering of slow particles by a pair of non- overlapping short-range potentials has been studied. The continuum wave function of the particle is represented as a combination of a plane wave and two spherical s-waves propagating freely throughout space. The asymptotic behavior of this function determines the amplitude of elastic particle scattering in closed form. It is demonstrated that this amplitude can be represented as a partial expansion in a set of the orthonormal functions Zλ(r) other than spherical harmonics Ylm(r). General formulas for these functions are obtained. The coefficients of the scattering amplitude expansion into a series of functions Zλ(r) and determine the scattering phases ηλ(k) for the considered two- atomic target. The special features of the S-matrix method for the case of arbitrary non-spherical potentials are discussed.

Simulation of robotic space operations with minimum base reaction manipulator

Autonomous robotic capture has been identified as a key technology for On-Orbit Servicing (OOS) and Active Debris Removal (ADR) missions. However, manoeuvring spacecraft-mounted manipulators is a challenging task since it generates disturbance torques on the satellite. To mitigate this problem several minimum reaction control strategies have been developed to reach the desired End-Effector (EE) pose while minimizing the dynamic disturbances transferred to the spacecraft by the robotic arm.This paper presents the development of a Simulation Tool in the MATLAB/Simulink environment capable of simulating the dynamics of a satellite equipped with a 7-DoF robotic arm during the target capture phase. The manipulator, as well as the spacecraft, is implemented by using Simscape Multibody and joint actuators are modelled as Brushless DC motors controlled by PID controllers. The spacecraft attitude is Nadir-Pointing, it is controlled by means of quaternion feedback and Linear-Quadratic-Regulator (LQR) and it is actuated by three Reaction Wheels (RW). Orbital perturbations such as non-spherical gravity potential (EGM2008 model) and atmospheric drag are considered. In addition, the minimum reaction control strategy called Kinetic Energy Minimization (MKE) is employed during the robotic arm manoeuvres.The goal of this study is to compare the performances obtained with the MKE method with those achieved by using the classic Inverse Kinematics (IK) in the free-flying case. The numerical results confirmed that MKE method is to be preferred since it minimizes the control torque that the Attitude Control Subsystem (ACS) must provide and reduces the EE orientation and position errors.

Features of Scattering by a Nonspherical Potential

Features of Scattering by a Nonspherical Potential

The 3D Kinematics of the Orion Nebula Cluster: NIRSPEC-AO Radial Velocities of the Core Population

Abstract The kinematics and dynamics of stellar and substellar populations within young, still-forming clusters provide valuable information for constraining theories of formation mechanisms. Using Keck II NIRSPEC+AO data, we have measured radial velocities for 56 low-mass sources within 4′ of the core of the Orion Nebula Cluster (ONC). We also remeasure radial velocities for 172 sources observed with SDSS/APOGEE. These data are combined with proper motions measured using HST ACS/WFPC2/WFC3IR and Keck II NIRC2, creating a sample of 135 sources with all three velocity components. The velocities measured are consistent with a normal distribution in all three components. We measure intrinsic velocity dispersions of ( σ v α , σ v δ , σ v r ) = (1.64 ± 0.12, 2.03 ± 0.13, 2.56 − 0.17 + 0.16 ) km s−1. Our computed intrinsic velocity dispersion profiles are consistent with the dynamical equilibrium models from Da Rio et al. (2014) in the tangential direction but not in the line-of-sight direction, possibly indicating that the core of the ONC is not yet virialized, and may require a nonspherical potential to explain the observed velocity dispersion profiles. We also observe a slight elongation along the north–south direction following the filament, which has been well studied in previous literature, and an elongation in the line-of-sight to tangential velocity direction. These 3D kinematics will help in the development of realistic models of the formation and early evolution of massive clusters.

Deformed explicitly correlated Gaussians.

Deformed explicitly correlated Gaussian (DECG) basis functions are introduced, and their matrix elements are calculated. All matrix elements can be calculated analytically in a closed form, except the Coulomb one, which has to be approximated by a Gaussian expansion. The DECG basis functions can be used to solve problems with nonspherical potentials. One example of such potential is the dipole self-interaction term in the Pauli-Fierz Hamiltonian. Examples are presented showing the accuracy and necessity of deformed Gaussian basis functions to accurately solve light-matter coupled systems in cavity QED.

Debye-screening effect on electron-impact excitation of helium-like Al11+ and Fe24+ ions

Debye-screening effects on the electron-impact excitation (EIE) processes for the dipole-allowed transition 1s2 1 S → 1s2p 1P in He-like Al11+ and Fe24+ ions are investigated using the fully relativistic distorted-wave methods with the Debye–Hückel (DH) model potential. Debye-screening effects on the continuum-bound (CB) interaction and target ion are discussed, both of which result in reduction of EIE cross sections. This reduction due to screening on the CB interaction is dominant. The non-spherical and spherical DH potentials are adopted for considering the screening effect on the CB interaction. It is found that the spherical DH potential could significantly overestimate the influence of plasma screening on EIE cross sections for multielectron He-like ions.

Electronic Structure and Electron Delocalization in Bare and Dressed Boron Pentamer Clusters.

The electronic structures of the lowest energy spin-states of the cationic, neutral and anionic bare boron pentamer clusters have been investigated by means of high level multiconfigurational type calculations, in view of the large static and dynamical electron correlation effects for these species. We found that resembles a singlet spin-state perfect pentagon, which bears no intra-annular chemical bonding interactions, as shown by our analysis of the electron delocalization carried out in terms of the normalized Giambiagi ring-current index, and the total and adjacent atom-pair delocalization indices. However, its lowest-energy triplet and quintet spin-state isomers have C2v symmetry, with large intra-annular chemical bonding interactions. This geometrical feature extends to both the neutral and the anionic species. Namely, the lowest-energy isomers of boron pentamer neutral and anionic clusters have peripheral and intra-annular sizable bonding interactions reflected in the delocalization of both π- and σ-type valence natural orbitals over the whole molecular plane, which impart large structural stability. In accordance to our calculations, the lowest energy triplet spin-state isomer of the anionic boron pentamer cluster has C2 symmetry, and consequently, it should show optical activity. Finally, we have studied the change of the geometrical structure of the boron pentamer clusters from planar to compact three-dimensional structures caused by the bonding of ligands to the boron atoms. Our explicit all-electron calculations have been rationalized in terms of the shell-closure of the delocalized valence orbitals of the clusters as predicted by the jellium model extended to nonspherical confinement potentials, circumscribing the role of the ligand to modulate the total number of valence electrons assigned to the core cluster.

Bound states of Dirac equation using the proper quantization rule

Using the proper quantization rule, we investigate the exact solution of Dirac equation for Hartmann and the ring-shaped non-spherical harmonic oscillator potentials under the condition of equal scalar and vector potentials. By considering the proper quantization condition within angular and radial variables, the exact relativistic energy spectra are obtained for each system. Then by the mean of suitable changes of variables, the corresponding spinor wave-functions are constructed where the normalization constants are exactly calculated. We also derived the non-relativistic limit of energy spectra.

Effect of confining potential on information entropy measures in hydrogen atom: extensive and non-extensive entropy

A spherically confined hydrogen atom has been considered using two different confined potentials such as modified Kratzer and non-spherical oscillator potentials. First, the Schrodinger equation in $$ r $$ -space has been solved and the exact analytical wave functions and energy levels have been obtained. Then, the Renyi entropy and Shannon entropy in $$ r $$ -space have been calculated. In addition, we have numerically calculated the wave functions in $$ p $$ -space by performing Fourier transform. The calculations have been done for different quantum numbers $$ n $$ and $$ l $$ and also quantum size $$ r_{0} $$ . The Bialynicki–Birula–Mycielski inequality is also tested. It is found that this inequality depends on non-extensive parameter $$ q $$ .

Matrix Elements of One Dimensional Explicitly Correlated Gaussian Basis Functions

Kinetic, potential and overlap matrix elements of one dimensional correlated Gaussians multiplied by polynomial factors are presented. These matrix elements can be used to calculate energies of one dimensional cold atom systems, or to construct a tensorial product to calculate energies in 2 or 3 dimensional systems with a nonspherical potential.

The impact of baryonic discs on the shapes and profiles of self-interacting dark matter haloes

We employ isolated N-body simulations to study the response of self-interacting dark matter (SIDM) halos in the presence of the baryonic potentials. Dark matter self-interactions lead to kinematic thermalization in the inner halo, resulting in a tight correlation between the dark matter and baryon distributions. A deep baryonic potential shortens the phase of SIDM core expansion and triggers core contraction. This effect can be further enhanced by a large self-scattering cross section. We find the final SIDM density profile is sensitive to the baryonic concentration and the strength of dark matter self-interactions. Assuming a spherical initial halo, we also study evolution of the SIDM halo shape together with the density profile. The halo shape at later epochs deviates from spherical symmetry due to the influence of the non-spherical disc potential, and its significance depends on the baryonic contribution to the total gravitational potential, relative to the dark matter one. In addition, we construct a multi-component model for the Milky Way, including an SIDM halo, a stellar disc and a bulge, and show it is consistent with observations from stellar kinematics and streams.

Time-Resolved Photoelectron Imaging of Molecular Coherent Excitation and Charge Migration by Ultrashort Laser Pulses

Charge migration is a fundamental and important process in the photochemistry of molecules and has been explored by time-resolved photoelectron angular distributions. A scheme based on UV pump and polarized soft X-ray probe techniques shows that photoelectron diffraction effects enable us to reconstruct electronic coherences encoding the information of the charge migration with extreme time resolutions. We discuss how probe pulse helicity influences the probing photoelectron spectra in the presence of molecular nonspherical Coulomb potentials. This phenomenon is analyzed theoretically and simulated via ab initio calculations for the molecular hydrogen ion, offering a reliable approach for measurements of charge migration and for the exploration of molecular structure in attosecond science.

Huygens–Fresnel picture for electron-molecule elastic scattering

The elastic scattering cross sections for a slow electron by C2 and H2 molecules have been calculated within the framework of the non-overlapping atomic potential model. For the amplitudes of the multiple electron scattering by a target the wave function of the molecular continuum is represented as a combination of a plane wave and two spherical waves generated by the centers of atomic spheres. This wave function obeys the Huygens–Fresnel principle according to which the electron wave scattering by a system of two centers is accompanied by generation of two spherical waves; their interaction creates a diffraction pattern far from the target. Each of the Huygens waves, in turn, is a superposition of the partial spherical waves with different orbital angular momenta l and their projections m. The amplitudes of these partial waves are defined by the corresponding phases of electron elastic scattering by an isolated atomic potential. In numerical calculations the s- and p-phase shifts are taken into account. So the number of interfering electron waves is equal to eight: two of which are the s-type waves and the remaining six waves are of the p-type with different m values. The calculation of the scattering amplitudes in closed form (rather than in the form of S-matrix expansion) is reduced to solving a system of eight inhomogeneous algebraic equations. The differential and total cross sections of electron scattering by fixed-in-space molecules and randomly oriented ones have been calculated as well. We conclude by discussing the special features of the S-matrix method for the case of arbitrary non-spherical potentials.