Spherical Potential Research Articles

Description of nucleon scattering on 208Pb by a fully Lane-consistent dispersive spherical optical model potential

A fully Lane-consistent dispersive spherical optical potential is proposed to describe nucleon scattering interaction with doubly magic nucleus 208 Pb up to 200 MeV. The experimental neutron total cross sections, elastically scattered nucleon angular distributions and (p,n) data had been used to search the potential parameters. Good agreement between experiments and the calculations with this potential is observed. Meanwhile, the application of the determined optical potential with the same parameters to neighbouring near magic Pb-Bi isotopes is also examined to show the predictive power of this potential.

The effects of a geometrical size, external electric fields and impurity on the optical gain of a quantum dot laser with a semi-parabolic spherical well potential

In this paper, a GaAs/AlxGa1−xAs quantum dot laser with a semi-parabolic spherical well potential is assumed. By using Runge-Kutta method the eigenenergies and the eigenstates of valence and conduct bands are obtained. The effects of geometrical sizes, external electric fields and hydrogen impurity on the different electronic transitions of the optical gain are studied. The results show that the optical gain peak increases and red-shifts, by increasing the width of well or barrier, while more increasing of the width causes blue-shift and decreases it. The hydrogen impurity decreases the optical gain peak and blue-shifts it. Also, the increasing of the external electric fields cause to increase the peak of the optical gain, and (blue) red shift it. Finally, the optical gain for 1s–1s and 2s–1s transitions is prominent, while it is so weak for other transitions.

THE TIME EVOLUTION OF GAPS IN TIDAL STREAMS

ABSTRACT We model the time evolution of gaps in tidal streams that are caused by the impact of a dark matter subhalo, while these orbit a spherical gravitational potential. To this end, we make use of the simple behavior of orbits in action-angle space. A gap effectively results from the divergence of two nearby orbits whose initial phase-space separation is, for very cold thin streams, largely given by the impulse induced by the subhalo. We find that in a spherical potential, the size of a gap increases linearly with time for sufficiently long timescales. We have derived an analytic expression that shows how the growth rate depends on the mass of the perturbing subhalo, its scale, and its relative velocity with respect to the stream. We have verified these scalings using N-body simulations and find excellent agreement. For example, a subhalo of mass 10 8 M ⊙ directly impacting a very cold thin stream on an inclined orbit can induce a gap that may reach a size of several tens of kiloparsecs after a few gigayears. The gap size fluctuates importantly with phase on the orbit, and it is largest close to pericenter. This indicates that it may not be fully straightforward to invert the spectrum of gaps present in a stream to recover the mass spectrum of the subhalos.

Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation

Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution.

Effect of rescattering potential on the high-energy above-threshold ionization of a model-H atom

The high-energy above-threshold ionization of a model-H atom (with 1s state and the same binding energy as H atom) in a few-cycle laser pulse is investigated by using the improved strong-field approximation (ISFA), where the spherical shell potential is used as the rescattering potential. The results obtained from numerically solving time-dependent Schrodinger equation(TDSE) are regarded as the benchmark results. Our results show that the energy distributions in high-energy region obtained from ISFA calculations using the spherical shell potential may either match or be better than those from ISFA using Yukawa potential and zero-range potential in the laser with wavelengths of 800 and 1200 nm. In addition, the influence of the rescattering potential on the density of probability at different ejection angles is also discussed in this paper.

Elastic and total scattering cross section for halogen substituted fluoromethane molecules

The electron scattering elastic and total scattering cross section for CF3X and CF2X2 (where X=H, Cl, Br, and I) molecules are reported in this article for incident energies from 30 to 5000eV. These cross sections are relevant to various fields of applied interest, especially to plasma modelling. The calculations are performed using the quantum mechanical approach, where the potential representing the interaction dynamics between the incoming electron and the scattering centre of the molecule is formulated using spherical complex optical potential. This potential is then incorporated to Schrödinger equation, which is solved through partial wave analysis. The results obtained in the present calculation are compared with previously available experimental and theoretical data.

The role of shape invariance potentials in the relativistic quantum mechanics

The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We want to show that some of the non-relativistic solvable potentials with shape-invariant symmetry can be related to the radial Dirac equation. Using this method, the idea of supersymmetry and shape invariance can be expanded to the relativistic quantum mechanics. The spinor wave functions for some of the obtained four-component electromagnetic potential are given in terms of special functions such as Jacobi, generalized Laguerre and Hermite polynomials. The relativistic bound-states spectrum for each case is also calculated in terms of the bound-states spectrum of the solvable potentials.

Exact Solutions of Schrödinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ2/2Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ2 / 2 Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2, which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

Discy dwarf disruption and the shape of the Galactic halo

Abstract The shape of the Galactic dark halo can, in principle, be inferred through modelling of stellar tidal streams in the Milky Way halo. The brightest and the longest of these, the Sagittarius stream, reaches out to large Galactocentric distances and hence can deliver the tightest constraints on the Galaxy's potential. In this contribution, we revisit the idea that the Sagittarius Stream was formed from a rotating progenitor. We demonstrate that the angle between the disc's angular momentum and the progenitor's orbital angular momentum does not remain constant throughout the disruption. Instead, it undergoes a dramatic evolution caused, in part, by the changes in the progenitor's moment of inertia tensor. We show that, even in a spherical potential, the streams produced as a result of a discy dwarf disruption appear to be ‘precessing’. Yet, this debris plane evolution is illusory as it is solely caused by the swaying and wobbling of the progenitor's disc. Stream plane precession is therefore not an unambiguous indicator of asphericity of the dark halo.

INFERRING THE GALACTIC POTENTIAL WITH GAIA AND FRIENDS: SYNERGIES WITH OTHER SURVEYS

ABSTRACT In the coming decade, the Gaia satellite will precisely measure the positions and velocities of millions of stars in the Galactic halo, including stars in many tidal streams. These streams, the products of hierarchical accretion of satellite galaxies by the Milky Way (MW), can be used to infer the Galactic gravitational potential thanks to their initial compactness in phase space. Plans for observations to extend Gaia’s radial velocity (RV) measurements to faint stars, and to determine precise distances to RR Lyrae in streams, would further extend the power of Gaia’s kinematic catalog to characterize the MW’s potential at large Galactocentric distances. In this work I explore the impact of these extra data on the ability to fit the potential using the method of action clustering, which statistically maximizes the information content (clumpiness) of the action space of tidal streams, eliminating the need to determine stream membership for individual stars. Using a mock halo in a toy spherical potential, updated post-launch error models for Gaia, and estimates for RV and distance errors for the tracers to be followed up, I show that combining either form of additional information with the Gaia catalog greatly reduces the bias in determining the scale radius and total mass of the Galaxy compared to the use of Gaia data alone.

The orbital PDF: the dynamical state of Milky Way sized haloes and the intrinsic uncertainty in the determination of their masses

Using realistic cosmological simulations of Milky Way sized haloes, we study their dynamical state and the accuracy of inferring their mass profiles with steady-state models of dynamical tracers. We use a new method that describes the phase-space distribution of a steady-state tracer population in a spherical potential without any assumption regarding the distribution of their orbits. Applying the method to five haloes from the Aquarius $\Lambda$CDM N-body simulation, we find that dark matter particles are an accurate tracer that enables the halo mass and concentration parameters to be recovered with an accuracy of $5\%$. Assuming a potential profile of the NFW form does not significantly affect the fits in most cases, except for halo A whose density profile differs significantly from the NFW form, leading to a $30\%$ bias in the dynamically fitted parameters. The existence of substructures in the dark matter tracers only affects the fits by $\sim 1\%$. Applying the method to mock stellar haloes generated by a particle-tagging technique, we find the stars are farther from equilibrium than dark matter particles, yielding a systematic bias of $\sim 20\%$ in the inferred mass and concentration parameter. The level of systematic biases obtained from a conventional distribution function fit to stars is comparable to ours, while similar fits to DM tracers are significantly biased in contrast to our fits. In line with previous studies, the mass bias is much reduced near the tracer half-mass radius.

The orbital PDF: general inference of the gravitational potential from steady-state tracers

We develop two general methods to infer the gravitational potential of a system using steady-state tracers, i.e. tracers with a time-independent phase-space distribution. Combined with the phase-space continuity equation, the time independence implies a universal orbital probability density function (oPDF) dP(λ|orbit) ∝ dt, where λ is the coordinate of the particle along the orbit. The oPDF is equivalent to Jeans theorem, and is the key physical ingredient behind most dynamical modelling of steady-state tracers. In the case of a spherical potential, we develop a likelihood estimator that fits analytical potentials to the system and a non-parametric method (‘phase-mark’) that reconstructs the potential profile, both assuming only the oPDF. The methods involve no extra assumptions about the tracer distribution function and can be applied to tracers with any arbitrary distribution of orbits, with possible extension to non-spherical potentials. The methods are tested on Monte Carlo samples of steady-state tracers in dark matter haloes to show that they are unbiased as well as efficient. A fully documented c/python code implementing our method is freely available at a GitHub repository linked from http://icc.dur.ac.uk/data/#oPDF.

The evolution of streams in a time-dependent potential

We study the evolution of streams in a time-dependent spherical gravitational potential. Our goal is to establish what are the imprints of this time evolution on the properties of streams as well as their observability. To this end, we have performed a suite of test-particle experiments for a host system that doubles its mass during the integration time and for a variety of initial conditions. In these experiments we found that the most striking imprint is a misalignment of ~10° in the angular location of the apocentres of the streams compared to the static case (and to the orbit of the centre of mass), which only becomes apparent for sufficiently long streams. We have also developed an analytic model using action-angle variables which allows us to explain this behaviour and to identify the most important signature of time evolution, namely a difference in the slope defined by the distribution of particles along a stream in frequency and in angle space. Although a difference in slope can arise when the present-day potential is not correctly modelled, this shortcoming can be by-passed because in this case, streams are no longer straight lines in angle space, but depict a wiggly appearance and an implausible energy gradient. The difference in slope due to time evolution is small, typically ~10-2 and its amplitude depends on the growth rate of the potential, but nonetheless we find that it could be observable if accurate full-space information for nearby long streams is available. On the other hand, disregarding this effect may bias the determination of the present-day characteristics of the potential.

Movement of a relativistic particle in a spherical potential box

Movement of a relativistic particle in a spherical potential box

Electron elastic scattering offA@C60: The role of atomic polarization under confinement

The present paper explores possible features of electron elastic scattering off endohedral fullerenes $A$@C$_{60}$. It focuses on how dynamical polarization of the encapsulated atom $A$ by an incident electron might alter scattering off $A$@C$_{60}$ compared to the static-atom-$A$ case, as well as how the C$_{60}$ confinement modifies the impact of atomic polarization on electron scattering compared to the free-atom case. The aim is to provide researchers with a "relative frame of reference" for understanding which part of the scattering processes could be due to electron scattering off the encapsulated atom and which due to scattering off the C$_{60}$ cage. To meet the goal, the C$_{60}$ cage is modeled by an attractive spherical potential of a certain inner radius, thickness, and depth which is a model used frequently in a great variety of fullerene studies to date. Then, the Dyson equation for the self-energy part of the Green's function of an incident electron moving in the combined field of an encapsulated atom $A$ and C$_{60}$ is solved in order to account for the impact of dynamical polarization of the encaged atom upon $e + A@{\rm C_{60}}$ scattering. The Ba@C$_{60}$ endohedral is chosen as the case study. The impact is found to be significant, and its utterly different role compared to that in $e + Ba$ scattering is unraveled.

Haloes light and dark: dynamical models of the stellar halo and constraints on the mass of the Galaxy

We develop a flexible set of action-based distribution functions (DFs) for stellar haloes. The DFs have five free parameters, controlling the inner and outer density slope, break radius, flattening and anisotropy respectively. The DFs generate flattened stellar haloes with a rapidly varying logarithmic slope in density, as well as a spherically aligned velocity ellipsoid with a long axis that points towards the Galactic centre - all attributes possessed by the stellar halo of the Milky Way. We use our action-based distribution function to model the blue horizontal branch stars extracted from the Sloan Digital Sky Survey as stellar halo tracers in a spherical Galactic potential. As the selection function is hard to model, we fix the density law from earlier studies and solve for the anisotropy and gravitational potential parameters. Our best fit model has a velocity anisotropy that becomes more radially anisotropic on moving outwards. It changes from $\beta \approx 0.4$ at Galactocentric radius of 15 kpc to $\approx 0.7$ at 60 kpc. This is a gentler increase than is typically found in simulations of stellar haloes built from the mutiple accretion of smaller systems. We find the potential corresponds to an almost flat rotation curve with amplitude of $\approx 200$ kms$^{-1}$ at these distances. This implies an enclosed mass of $\approx 4.5 \times 10^{11} M_\odot$ within a spherical shell of radius 50 kpc.

Cross section data for ionization of important cyanides

This article presents cross section calculations for interactions of important cyanides with electrons possessing energies beginning from ionization threshold of the target molecule to 5keV. These data are pursued to meet the ever increasing demand for cross sections by the relevant atomic and molecular community for modelling astrophysical, atmospheric and technological domains. The calculations have been executed using an amalgam of multi centre spherical complex optical potential (MSCOP) formalism and complex scattering potential-ionization contribution (CSP-ic) method. Cross sections are compared with experimental and theoretical data wherever available. Strong correlations are observed for the cross sections which affirms consistent and reliable cross sections. Isomeric effect has been interpreted using variation of cross section with structure and target properties. Our cross sections will be tabulated in atomic collision database for use in modelling various statistical and dynamical quantities.

Electron induced ionization cross sections for astrophysical modelling

This paper reports electron impact total ionization cross sections for C2 to C6 formates at energies varying from ionization threshold to 5keV. The ionization cross section data in such a wide energy range are necessary to complete data tables required for modelling different atmospheric and astrophysical environments. To compute these values, a modified approach called multi scattering centre spherical complex optical potential (MSCOP) formalism is employed. Using MSCOP, the total inelastic cross section is obtained, which corresponds to the total loss of incident flux into the dominant channels viz. discrete electronic excitation and continuum ionization. The ionization contribution from total inelastic cross section is estimated using a semi empirical complex scattering potential-ionization contribution (CSP-ic) method. The results show overall good agreement with previous measurements and theoretical estimates available from the literature. Besides cross sections, correlations between maximum ionization cross section versus the square root of the ratio of polarizability to ionization potential and with the number of carbon atoms in each target are investigated. Employing this correlation, the approximate polarizability volume of C4 to C6 formates has been computed for the first time.

Electron impact ionisation cross sections for atomic and molecular allotropes of phosphorous and arsenic

We report electron impact total ionisation cross sections for phosphorous (P), arsenic (As), diphosphorous (P2), diarsenic (As2), tetra phosphorous (P4) and tetra arsenic (As4) from the threshold of the target to 2000 eV. We employed spherical complex optical potential to compute total inelastic cross sections (Qinel). The total ionisation cross section is extracted from the total inelastic cross section using the complex scattering potential–ionisation contribution method. The results of most of the targets studied here compare well with the measurements and the theoretical data wherever available. The correlation between the peak of ionisation cross sections with the number of target electrons and polarisability is also reported. It is observed that the maximum ionisation cross sections depend linearly on the number of target electrons and polarisability of the target. This linear correlation is used to predict the maximum ionisation cross sections for the targets (I2, HI and PF3) where no experimental data are available.

ACTION-SPACE CLUSTERING OF TIDAL STREAMS TO INFER THE GALACTIC POTENTIAL

We present a new method for constraining the Milky Way halo gravitational potential by simultaneously fitting multiple tidal streams. This method requires full three-dimensional positions and velocities for all stars to be fit, but does not require identification of any specific stream or determination of stream membership for any star. We exploit the principle that the action distribution of stream stars is most clustered when the potential used to calculate the actions is closest to the true potential. Clustering is quantified with the Kullback-Leibler Divergence (KLD), which also provides conditional uncertainties for our parameter estimates. We show, for toy Gaia-like data in a spherical isochrone potential, that maximizing the KLD of the action distribution relative to a smoother distribution recovers the true values of the potential parameters. The precision depends on the observational errors and the number of streams in the sample; using KIII giants as tracers, we measure the enclosed mass at the average radius of the sample stars accurate to 3% and precise to 20-40%. Recovery of the scale radius is precise to 25%, and is biased 50% high by the small galactocentric distance range of stars in our mock sample (1-25 kpc, or about three scale radii, with mean 6.5 kpc). About 15 streams, with at least 100 stars per stream, are needed to obtain upper and lower bounds on the enclosed mass and scale radius when observational errors are taken into account; 20-25 streams are required to stabilize the size of the confidence interval. If radial velocities are provided for stars out to 100 kpc (10 scale radii), all parameters can be determined with 10% accuracy and 20% precision (1.3% accuracy in the case of the enclosed mass), underlining the need for ground-based spectroscopic follow-up to complete the radial velocity catalog for faint halo stars observed by Gaia.