Spherical Potential Research Articles

An effective Dirac equation for a binary of two fermions

Effective Dirac equations are, by following the work of Pilkuhn, derived for a binary of two fermions from the original two-fermion Dirac-type Hamiltonian written in a 16-component spinor representation. For spherical interaction potential (like the static Coulomb potential) the spin dynamics in the product spin space and spatial dynamics in the particle–antiparticle state space can be fully decoupled and separated. The case of atomic hydrogen is solved again as an example, and the standard results including recoil effects through the reduced mass and fine structure induced by spin–orbit coupling are recovered for the energy spectrum.

The spatial and momentum distributions of a weakly interacting Fermi gas trapped in an external potential

The spatial and momentum distributions of a weakly interacting Fermi gas trapped in a spherical symmetric power-law potential are investigated, based on the mean field approximation. The concepts of “Fermi spheres” in both the coordinate and momentum spaces are introduced and the problem of how the sizes of the “Fermi spheres” depend on the external potential and interatomic interaction is discussed. Density profiles in both spaces are presented explicitly and the effects of the external potential, interatomic interaction and temperature on the spatial and momentum distributions are shown clearly. The results obtained are general because the power-law potential adopted here may include a variety of particular potentials corresponding to the various values of the exponents.

Electron impact calculations of total and ionization cross-sections for Germanium Hydrides (GeH X; X = 1–4) and Digermane, Ge 2H 6

In this article we report calculations of the total elastic Q el, total ionization, Q ion, and total (complete), Q T, cross-sections for the germanium hydrides GeH X ( X = 1–4) and digermane (Ge 2H 6) upon electron impact for energies from circa threshold to 2000 eV. Spherical Complex Optical Potential (SCOP) formalism is employed to evaluate Q el, Q inel and Q T. Total ionization cross-sections, Q ion, are derived from the total inelastic cross-sections, Q inel, using a semi-empirical formulation developed by us called ‘Complex Spherical Potential-ionization contribution’ (CSP-ic) method. A comparison of the various cross-sections is provided to show their relative contribution to the total cross-section Q T. Present results are compared with available experimental and theoretical data wherever possible and overall agreement is good. However, no experimental data on the Q ion are available for the germanium hydrides, GeH X , X = 1–3 or digermane.

Coherent control for the spherical symmetric box potential in short and intensive XUV laser fields

Abstract Coherent control calculations are presented for a spherically symmetric box potential for non-resonant two photon transition probabilities. With the help of a genetic algorithm (GA), the population of the excited states are maximized and minimized. The external driving field is a superposition of three intensive extreme ultraviolet (XUV) linearly polarized laser pulses with different frequencies in the femtosecond duration range. We solved the quantum mechanical problem within the dipole approximation. Our investigation clearly shows that the dynamics of the electron current has a strong correlation with the optimized and neutralizing pulse shape.

Binding energy of an off-center hydrogenic donor in a spherical Gaussian quantum dot

The energy levels of an off-center hydrogenic donor confined by a spherical Gaussian potential have been calculated as a function of the potential radius for different donor position by exact diagonalization method. The results have clearly demonstrated the so-called quantum size effect. The binding energy is dependent on the dot radius R, the impurity ion distance D, and the confining potential depth V 0 .

Formation of Central Massive Objects via Tidal Compression

For a density that is not too sharply peaked toward the center, the local tidal field becomes compressive in all three directions. Available gas can then collapse and form a cluster of stars in the center, including or even being dominated by a central black hole. We show that for a wide range of (deprojected) Sersic profiles in a spherical potential, the tidal forces are compressive within a region that encloses most of the corresponding light of observed nuclear clusters in both late- and early-type galaxies. In such models, tidal forces become disruptive nearly everywhere for relatively large Sersic indices n 3.5. We also show that the mass of a central massive object (CMO) required to remove all radial compressive tidal forces scales linearly with the mass of the host galaxy. If CMOs formed in (progenitor) galaxies with n ~ 1, we predict a mass fraction of ~0.1% to 0.5%, consistent with observations of nuclear clusters and supermassive black holes. While we find that tidal compression possibly drives the formation of CMOs in galaxies, beyond the central regions and on larger scales in clusters disruptive tidal forces might contribute to prevent gas from cooling.

Geometric shielding correction approach for total cross sections of electron scattering by C6H6, C6F6, C6H12, C6H14, C6F14, C8H16, C8H18 and C8F18 at 30–5000 eV

Taking into consideration the contributions of the geometric shielding effect in the molecule, an empirical fraction, which depends on the energy of the incident electrons, the target's molecular dimension and the atomic and electronic numbers in the molecule, is presented. Using this empirical fraction, a new formulation of the additivity rule is proposed. The total cross sections for electron scattering by eight larger polyatomic molecules (C6H6, C6F6, C6H12, C6H14, C8H16, C8H18, C6F14 and C8F18) are calculated using the new additivity rule at the Hartree–Fock level for impact energies ranging from 30 to 5000 eV. The quantitative total cross sections are compared with those obtained by experiments and other theories, and good agreement is attained above 100 eV or so. The new results for C6F14 and C8F18 are also presented although no experimental and theoretical data are available for comparison in the present energy region. In these calculations, the atoms are represented by the spherical complex optical potential, which is composed of static, exchange, polarization and absorption terms.

The eccentric frame decomposition of central force fields

The rosette-shaped motion of a particle in a central force field is known to be classically solvable by quadratures. We present a new approach of describing and characterizing such motion based on the eccentricity vector of the two body problem. In general, this vector is not an integral of motion. However, the orbital motion, when viewed from the nonuniformly rotating frame defined by the orientation of the eccentricity vector, can be solved analytically and will either be a closed periodic circulation or libration. The motion with respect to inertial space is then given by integrating the argument of periapsis with respect to time. Finally we will apply the decomposition to a modern central potential, the spherical Hernquist-Newton potential, which models dark matter halos of galaxies with central black holes.

Low-lying collective excitations of nuclei as a semiclassical response

For the low-lying multipole collective excitations in nuclei, the transport coefficients of the response function theory, such as the inertia and the friction, are derived within the periodic orbit theory in the lowest orders of semiclassical expansion corresponding to the extended Thomas-Fermi approach with the surface and the curvature corrections. The collective vibrations near the spherical shape are described in the mean-field approximation through the spherical edgelike potential by using a statistical averaging of the transport coefficients. It is shown that the inertia of the collective motion is significantly larger than that of irrotational hydrodynamical flow. For large enough particle numbers in the nucleus, the mean energies of quadrupole and octupole low-lying collective vibrations, their sum rule contributions, and reduced transition probabilities are in reasonable agreement with experimental data.

AN ITERATION SCHEME FOR CALCULATING TRANSPORT PROPERTIES OF MOLECULAR SYSTEMS

An iteration method of functional theory is used to solve Poisson equation self-consistently in order to determine the retarded and advanced Green's function of a molecular system. The approach is demonstrated using a system with a spherical potential. The retarded and advanced Green's functions are calculated and the determined current–voltage results compare well with experimental data for the system involving Coulomb potential. The approach is expected to find wide applications in general systems.

The Neutron Spherical Optical Model Absorption

The energy-dependent behavior of the absorption term of the spherical neutron optical potential for doubly magic 208Pb and neighboring 209Bi is examined. These considerations suggest a model with an intuitively attractive energy dependence of the imaginary potential that provides a good description of the observed neutron cross-sections and that is qualitatively consistent with theoretical concepts. It is an alternative to arbitrary assumptions involved in many conventional optical-model interpretations reported in the literature.

Resonant and weakly-bound one-particle levels in quadrupole-deformed potentials

Both positive-energy and weakly-bound one-particle levels for neutrons in Y20 deformed Woods-Saxon potentials are examined in comparison with those in spherical Woods-Saxon potentials. While s1/2 levels play a unique role in spherical drip-line nuclei, the Ωπ = 1/2+ levels in Y20 deformed potentials, which always contain some amount of s1/2 component, exhibit an important role in deformed drip-line nuclei. As the potential strength becomes weaker, some weakly-bound Ωπ = 1/2+ levels continue to the positive-energy region as one-particle resonant levels, while others have no such continuation. Among an infinite number of one-particle levels at a given positive-energy and in a given deformed potential, only some selected levels expressed in terms of eigenphase are found to be important in the pair-correlated ground state of neutron-drip-line nuclei.

A spherical model with directional interactions. I. Static properties

We introduce a simple spherical model whose structural properties are similar to the ones generated by models with directional interactions, by employing a binary mixture of large and small hard spheres, with a square-well attraction acting only between particles of different sizes. The small particles provide the bonds between the large ones. With a proper choice of the interaction parameters, as well as of the relative concentration of the two species, it is possible to control the effective valence. Here we focus on a specific choice of the parameters which favors tetrahedral ordering and study the equilibrium static properties of the system in a large window of densities and temperatures. Upon lowering the temperature we observe a progressive increase in local order, accompanied by the formation of a four-coordinated network of bonds. Three different density regions are observed: At low density the system phase separates into a gas and a liquid phase; at intermediate densities a network of fully bonded particles develops; at high densities--due to the competition between excluded volume and attractive interactions--the system forms a defective network. The very same behavior has been previously observed in numerical studies of nonspherical models for molecular liquids, such as water, and in models of patchy colloidal particles. Different from these models, theoretical treatments devised for spherical potentials, e.g., integral equations and ideal mode coupling theory for the glass transition, can be applied in the present case, opening the way for a deeper understanding of the thermodynamic and dynamic behavior of low valence molecules and particles.

Viscosity and thermal conductivity of some refrigerants at moderate and high densities on the basis of Rainwater–Friend theory and the corresponding states principle

The initial density dependence of viscosity and thermal conductivity was formulated on the basis of Rainwater–Friend (RF) theory. In this work, we have first focused on the calculation of viscosity and thermal conductivity of moderately dense argon by using RF theory and an accurate ab initio potential function. This theory which was originally presented for spherical potentials have been adapted for calculation of viscosity and translational contribution of thermal conductivity of some refrigerants by introducing the corresponding states correlations for the second transport virial coefficients. Then the internal states contribution for the thermal conductivity has been determined based on the Mason–Monchick and modified Enskog theories. So, we have calculated the viscosity and thermal conductivity of some refrigerants, R32, R14, R12, R13, R22, R134a, R143a, R125, R123, R142b, at moderate densities up to about 2 mol dm −3. At high densities, beyond the validity range of RF theory, we have applied correlation expressions for the viscosity and thermal conductivity residual functions to calculate the viscosity and thermal conductivity of supercritical refrigerants and then compared with the available experimental data. In conclusion, we have shown that the RF theory in conjunction with the corresponding states residual functions present the reliable model for calculation of viscosity and thermal conductivity of refrigerants over a comparatively wide temperature and pressure range up to 65 MPa within the experimental errors.

Simulation of band structure for CrN lattices by using a 3D array of range-limited circularly symmetric attractive potential

The band structure obtention is important for establishing the different electrical and thermal properties of materials. Band structure of CrN was carried out by employing a 3D array range-limited spherical symmetric potential. For this purpose, several tools of Mathematics package were used. Initially, the unit cell is gridded in order to obtain small parts. Each grid point was well-represented by a wavefunction. The finite differences of each grid point is obtained approximately in each direction ( x , y and z ), establishing the difference between the neighboring points along the axis. The wavefunctions which are described by the Bloch functions, the Laplacian and the derivatives are included into the Schrödinger equation. Then, the equation is solved in order to obtain the eigenvalues (energies), which are plotted to obtain the band structure in (100), (110) and (111) directions.

Confinement, correlation and relativistic effects on the photoionization of 6s subshell of Hg

A theoretical study of photoionization parameters of Hg atom confined in a spherical shell potential has been carried out using the relativistic random phase approximation. It is found that confinement, correlation and relativistic effects all bring significant and separate changes in the 6s photoionization cross-section, and also in the photoelectron angular distribution.

Calculations of elastic, ionization and total cross sections for inert gases upon electron impact: threshold to 2 keV

In this paper we report comprehensive calculations of total elastic (Qel), total ionization (Qion) and total (complete) cross sections (QT) for the impact of electrons on inert gases (He, Ne, Ar, Kr and Xe) at energies from about threshold to 2000 eV. We have employed the spherical complex optical potential (SCOP) formalism to evaluate Qel and QT and used the complex spherical potential-ionization contribution (CSP-ic) method to derive Qion. The dependence of QT on polarizability and incident energy is presented for these targets through an analytical formula. Mutual comparison of various cross sections is provided to show their relative contribution to the total cross sections QT. Comparison of QT for all these targets is carried out to present a general theoretical picture of collision processes. The present calculations also provide information, hitherto sparse, on the excitation processes of these atomic targets. These results are compared with available experimental and other theoretical data and overall good agreement is observed.

Bipolaron in a quasi-0D quantum dot

Bipolaron states in a quasi-0D quantum dot with a spherical parabolic confinement potential are investigated by applying the Feynman variational principle . The bipolaron coupling energy and self-action potential energy are found to increase with an increase in the Fröhlich electron–phonon-coupling constant. There is also a non-monotonic dependence of the bipolaron coupling energy on the quantum dot radius. With decreasing structure radius the bipolaron coupling energy increases. However, from a critical radius it starts decreasing as the radius decreases, due to the dominance of the coulomb-to-phonon mediated interaction. When electrons in the bipolaron are forcefully neighboured, the polarization of the structure is intensified and consequently there is Coulomb repulsion. The possibility of bipolaron formation depends on the strength of the direct Coulomb repulsion which, in turn, depends on the quantum dot radius. The main contribution to the bipolaron coupling energy comes from the self-action potential. This self-action potential energy influences the energy state of the bipolaron considerably. The ratio of optical-to-static dielectric constants significantly affects the bipolaron coupling energy.

A Study of a Confined Hydrogen Negative Ion

The ground and three low-excited states of the hydrogen negative ionconfined by a spherical harmonic oscillator potential are studied employingthe adiabatic hyperspherical approach method. Total energies are obtained asa function of the confined potential radii. We find that the confinement maycause accidental degeneracies between levels with different low-excitedstates and the inversion of the energy values.

Theoretical prediction of multiple fluid-fluid transitions in monocomponent fluids

The authors use the analytical equation of state obtained by the discrete perturbation theory [A. L. Benavides and A. Gil-Villegas, Mol. Phys. 97, 1225 (1999)] to study the phase diagram of fluids with discrete spherical potentials formed by a repulsive square-shoulder plus an attractive square-well interaction (SS+SW). This interaction is characterized by the usual energy and size parameters plus three dimensionless parameters: two of them measuring the widths of the SS and the SW and the third the relative height of the SS. The matter of interest is that, for certain values of the interaction parameters, the SS+SW systems exhibit more than one first-order fluid-fluid transition. The evidence that several real substances (such as water, phosphorus, carbon, and silica, among others) exhibit an extra liquid-liquid transition has drawn interest into the study of interactions responsible for this behavior. The simple SS+SW fluid is one of the systems that, in spite of being spherically symmetric, shows multiple fluid-fluid transitions. In this work the authors investigate systematically the effect on the phase diagram of varying the interaction parameters. The use of an analytical free-energy equation gives a clear thermodynamic picture of the emergence of different types of critical points, throwing new light on the phase behavior of these fluids and thus clarifying previous results obtained by other techniques. The interplay of attractive and repulsive forces with several scale lengths produces very rich phase diagrams, including cases with three critical points. The region of the interaction-parameter space where multiple critical points appear is mapped for various families of interactions.