Spherical Harmonic Oscillator Research Articles

Symmetry-projected variational calculations with the numerical suite TAURUS

We present the numerical code \(\textsf {TAURUS}_{\textsf {vap}}\) that solves the variation after particle-number projection equations for real general Bogoliubov quasiparticle states represented in a spherical harmonic oscillator basis. The model space considered is invariant under spatial and isospin rotations but no specific set of orbits is assumed such that the code can carry out both valence-space and no-core calculations. In addition, no number parity is assumed for the Bogoliubov quasiparticle states such that the code can be used to describe even-even, odd-even and odd-odd nuclei. The variational procedure can be performed under several simultaneous constraints on the expectation values of a variety of operators such as the multipole deformations, the pairing field or the components of the angular momentum. To demonstrate the potential and versatility of the code, we perform several example calculations using an empirical shell-model interaction as well as a chiral interaction. The ability to perform advanced variational Bogoliubov calculations offered by this code will, we hope, be beneficial to the shell model and ab initio communities.

Ladder Operators for the Spherical 3D Harmonic Oscillator

The Schrödinger equation for an isotropic three-dimensional harmonic oscillator is solved using ladder operators. The starting point is the shape invariance condition, obtained from supersymmetric quantum mechanics. Generalized ladder operators can be constructed for the three spherical spatial coordinates. Special emphasis is given to the adaptation made to each of these coordinates. The approach used is general and is indicated as an alternative method to solve the Schrödinger equation.

Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs

<p style='text-indent:20px;'>Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the same. We find that the CUDA implementation delivers speedups of 2.27x – 5.93x.

Electronic and optical properties of double ring-shaped quantum dot

The binding energy, oscillator strengths, and absorption coefficients (ACs) for double ring-shaped quantum dots (DRSQDs) are theoretically investigated within the framework of effective mass approximation. The Schrödinger equation is solved by a variational method in order to compute the energy levels of a hydrogenic impurity in a gallium aresenide DRSQD. Obtained results have revealed the dependence of the binding energy on the dot radius and the potential parameters B and C. The energy values in the absence and presence of impurity are compared with the literature for spherical harmonic oscillator. The binding energy decreases as the quantum dot radius and the potential parameters increase. Also, it is found that the oscillator strengths and the linear and nonlinear optical ACs are quite sensitive to the potential parameters. The peaks of AC exhibit blueshift with B and C increasing.

A Simplified Analytic Treatment of Shell-Effects in Metal Clusters

A simplified treatment of shell-effects in metal clusters, such as those of Na, is considered. This treatment is carried out by means of an approximate scheme based on the spherical harmonic oscillator jellium model and its advantage is that it suggests the possibility quantities of physical interest to be calculated analytically. As a result, the variation of these quantities with the number of the valence electrons of the atoms in the cluster could be given explicitly in certain cases.

The 3-Dimensional g-Deformed Harmonic Oscillator and a Unified Description of Magic Numbers of Metal Clusters

Magic numbers predicted by a 3-dimensional (/-deformed harmonic oscillator with u9(3) D soq(3) symmetry are compared to experimental data for atomic clusters of alkali metals (Li, Na, K, Rb, Cs), noble metals (Cu, Ag, Au), divalent metals (Zn, Cd), and trivalent metals (Al, In), as well as to theoretical predictions of jellium models, Woods-Saxon and wine bottle potentials, and to the classification scheme using the 3n + / pseudo quantum number. In alkali metal clusters and noble metal clusters the 3-dimensional çr-deformed harmonic oscillator correctly predicts all experimentally observed magic numbers up to 1500 (which is the expected limit of validity for theories based on the filling of electronic shells), while in addition it gives satisfactory results for the magic numbers of clusters of divalent metals and trivalent metals, thus indicating that ug(3), which is a nonlinear extension of the u(3) symmetry of the spherical (3-dimensional isotropic) harmonic oscillator, is a good candidate for being the symmetry of systems of several metal clusters.

Research on Speech Codec with Compression using Legendre Polynomials

Speech Codecs have been developed in several forms over the past 30 years and are used in Cellular digital telephony. Based on the throat excitation pulses and mouth cavity filter model, all the methods aim to encode the speech samples without much loss of clarity. The complexity of the methods using codebook searching, filtering and matrix inversion, all need heavy computations, leading to timing constraints. After describing a tutorial elucidating the essentials of the present methods, a different direct method of encoding, suggested by the spherical harmonic oscillator functions of Legendre, is introduced. Its performance and simplicity are such that the same may find its applications soon.

Beta Spectrum Generator: High precision allowed [formula omitted] spectrum shapes

Several searches for Beyond Standard Model physics rely on an accurate and highly precise theoretical description of the allowed β spectrum. Following recent theoretical advances, a C++ implementation of an analytical description of the allowed beta spectrum shape was constructed. It implements all known corrections required to give a theoretical description accurate to a few parts in 104. The remaining nuclear structure-sensitive input can optionally be calculated in an extreme single-particle approximation with a variety of nuclear potentials, or obtained through an interface with more state-of-the-art computations. Due to its relevance in modern neutrino physics, the corresponding (anti)neutrino spectra are readily available with appropriate radiative corrections. In the interest of user-friendliness, a graphical interface was developed in Python with a coupling to a variety of nuclear databases. We present several test cases and illustrate potential usage of the code. Our work can be used as the foundation for current and future high-precision experiments related to the beta decay process.Source code: https://github.com/leenderthayen/BSGDocumentation: http://bsg.readthedocs.io Program summaryProgram Title: BSGProgram Files doi:http://dx.doi.org/10.17632/gx6yrpn22x.1Licensing provisions: MITProgramming language: C++ and PythonNature of problem: The theoretical allowed β spectrum contains a large variety of corrections from different areas of physics, each of which is important in certain energy ranges. A high precision description is required for new physics searches throughout the entire nuclear chart.Solution method: We implement the analytical corrections described in recent theoretical work. Nuclear matrix elements in allowed Gamow–Teller β decay are calculated in a spherical harmonic oscillator basis. Wave functions can be calculated in an extreme single-particle approximation using different nuclear potentials, or provided by the user as the output from more sophisticated routines. Corresponding neutrino spectra are calculated with appropriate radiative corrections. A graphical user interface written in Python additionally provides connections to a variety of nuclear databases.Additional comments: CPC Library subprograms used: ABOV_v1_0

The Effect of Relativistic Quantum Corrections on the Thermal Properties of Three-Dimensional Spherical Semiconductor Quantum Dot Under a Magnetic Field

The ground state and excited states energies of 2-electron quantum dot with spherical harmonic oscillator type potential has been determined. The energy spectrum and wave functions for the quantum dot of asymmetric confinement are obtained by analytically solving the eigenvalue equation in the magnetic field. The effect of various relativistic corrections to kinetic energy, Darwin term and spin-orbit for the zero-dimensional structure to the energy eigenvalues and wave functions is also investigated. The thermal properties like internal energy, entropy and free energy are discussed graphically with radius of quantum dot and pressure and are found to have interesting dependence on the radius and pressure parameter with relativistic corrections for our model.

The Calculation of Single-Nucleon Energies of Nuclei by Considering Two-Body Effective Interaction, n(k,ρ), and a Hartree-Fock Inspired Scheme

The nucleon single-particle energies (SPEs) of the selected nuclei, that is, O16, Ca40, and Ni56, are obtained by using the diagonal matrix elements of two-body effective interaction, which generated through the lowest-order constrained variational (LOCV) calculations for the symmetric nuclear matter with the Aυ18 phenomenological nucleon-nucleon potential. The SPEs at the major levels of nuclei are calculated by employing a Hartree-Fock inspired scheme in the spherical harmonic oscillator basis. In the scheme, the correlation influences are taken into account by imposing the nucleon effective mass factor on the radial wave functions of the major levels. Replacing the density-dependent one-body momentum distribution functions of nucleons, n(k,ρ), with the Heaviside functions, the role of n(k,ρ) in the nucleon SPEs at the major levels of the selected closed shell nuclei is investigated. The best fit of spin-orbit splitting is taken into account when correcting the major levels of the nuclei by using the parameterized Wood-Saxon potential and the Aυ18 density-dependent mean field potential which is constructed by the LOCV method. Considering the point-like protons in the spherical Coulomb potential well, the single-proton energies are corrected. The results show the importance of including n(k,ρ), instead of the Heaviside functions, in the calculation of nucleon SPEs at the different levels, particularly the valence levels, of the closed shell nuclei.

Ab initioBogoliubov coupled cluster theory for open-shell nuclei

Ab initio many-body methods address closed-shell nuclei up to mass A ~ 130 on the basis of realistic two- and three-nucleon interactions. Several routes to address open-shell nuclei are currently under investigation, including ideas which exploit spontaneous symmetry breaking. Singly open-shell nuclei can be efficiently described via the sole breaking of $U(1)$ gauge symmetry associated with particle number conservation, to account for their superfluid character. The present work formulates and applies Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wavefunction of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in $m$-scheme, which will eventually permit the treatment of doubly open-shell nuclei. Proof-of-principle calculations in an $N_{\text{max}}=6$ spherical harmonic oscillator basis are performed for $^{16,18,20}$O, $^{18}$Ne, $^{20}$Mg in the BCCD approximation with a chiral two-nucleon interaction, comparing to results obtained in standard coupled cluster theory when applicable. The breaking of $U(1)$ symmetry is monitored by computing the variance associated with the particle-number operator. The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e. potentially to reach several hundred additional mid-mass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei.

DIRHB—A relativistic self-consistent mean-field framework for atomic nuclei

The DIRHB package consists of three Fortran computer codes for the calculation of the ground-state properties of even–even atomic nuclei using the framework of relativistic self-consistent mean-field models. Each code corresponds to a particular choice of spatial symmetry: the DIRHBS, DIRHBZ and DIRHBT codes are used to calculate nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes, respectively. Reflection symmetry is assumed in all three cases. The latest relativistic nuclear energy density functionals are implemented in the codes, thus enabling efficient and accurate calculations over the entire nuclide chart. Program summaryProgram title: DIRHB package (codes DIRHBS, DIRHBZ and DIRHBT)Catalogue identifier: AESN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: noneNo. of lines in distributed program, including test data, etc.: 44516No. of bytes in distributed program, including test data, etc.: 317161Distribution format: tar.gzProgramming language: Fortran 77.Computer: All computers.Operating system: All operating systems. The Makefiles are specific for a Unix OS and have to be modified for Windows.RAM: Depends on the imposed symmetry and number of oscillator shells. For the triaxial test case it takes 300 MB.Classification: 17.22.Nature of problem:Ground-state properties of even–even open-shell nuclei can be calculated using the framework of self-consistent mean-field models based on relativistic energy density functionals. The structure of arbitrary heavy nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes, is modeled using the latest zero- and finite-range relativistic effective interactions. The particle–particle channel of the effective inter-nucleon interaction is described by a separable finite-range pairing force.Solution method:The current implementation of the model computes the mean-field solution of the nuclear many-body problem for even–even open-shell spherical and quadrupole deformed nuclei. The codes are used to solve the stationary relativistic Hartree–Bogoliubov equations in a self-consistent iteration scheme. At each iteration the matrix elements of the equations are updated using the modified Broyden method or the linear mixing method. The single-nucleon wave functions are expanded in a basis of spherical, axially symmetric or triaxial harmonic oscillator, depending on the assumed symmetry of the nuclear shape. For calculations that constrain the shape to specific values of the deformation parameters, the augmented Lagrangian method is used.Restrictions:Time-reversal and reflection symmetries are assumed. Open-shell even–even spherical and quadrupole deformed nuclei are considered.Running time:Depends on the imposed symmetry and number of oscillator shells. For the test cases it runs from few seconds (spherical) up to few hours (triaxial).

Hartree-Fock Calculations of &amp;lt;sup&amp;gt;16&amp;lt;/sup&amp;gt;O and &amp;lt;sup&amp;gt;40&amp;lt;/sup&amp;gt;Ca Nuclei Using Fish-Bone Potential

Hartree-Fock calculations are carried out to describe some properties of 16O and 40Ca nuclei using the two forms of fish-bone potential (I and II). A computer simulation search program has been introduced to solve this problem. The Hilbert space was restricted to three dimensional variational space spanned by single spherical harmonic oscillator orbits. Binding energies, root mean square radii and form factors are found to have a good fit with experimental data.

Discrete Spherical Harmonic Oscillator Transforms on the Cartesian Grids Using Transformation Coefficients

The analog harmonic oscillators are well-studied in quantum physics, including their energy states, wavefunctions, orthogonal properties, and eigenfunctions of the Fourier transform. In addition, the continuous solutions in different dimension and coordinate systems are known. Some discrete equivalents of the 1D wavefunctions were also studied. However, in the 3D spherical coordinate system, the discrete equivalents of the 3D wavefunctions are not established. In this paper, we focus on the spherical harmonic oscillator wavefunctions (SHOWs) the spherical harmonic oscillator transforms (SHOTs), and their discrete implementation. The SHOWs can be synthesized by linear combinations of the Hermite Gaussian functions with proper transformation coefficients. We find that computing the coefficients can be speeded up using the fast Fourier transforms or some recursive relations. These coefficients relate the Hermite transforms with the SHOTs. Some applications of the discrete SHOWs and the discrete SHOTs are introduced. First, the SHOWs are exactly the eigenfunctions of the 3D DFT. Also, the SHOTs can be used to derive the spherical harmonic oscillator descriptor (SHOD), which is a rotational invariant descriptor. We find that the SHOD is not only compatible with the existing rotational descriptors for the spherically sampled data but also outperforms the existing rotational descriptors for 3D Cartesian sampled, bandlimited input data. Besides, the SHOTs can be used to decompose 3D signals into spherical components. Hence, 3D signal reconstruction is done using partially chosen spherical component, and 3D data compression for MRI data is demonstrated using SHOTs for medical applications.

Nuclear Structure of 12,14Be Within Deformed Quasi-Particle Random Phase Approximation (DQRPA)

We developed a deformed quasi-particle random phase approximation (DQRPA) to describe the Gamow–Teller (GT) transitions on even–even neutron-rich nuclei. To describe deformed nuclei, we exploited the deformed axially symmetric Woods–Saxon potential, the deformed BCS, and the deformed QRPA with realistic two-body interaction calculated by Brueckner G-matrix based on Bonn CD potential. The deformed single particle states are expanded in terms of the spherical harmonic oscillator basis in order to take the realistic G-matrix stored in the spherical basis. We calculated GT strength distributions, B(GT), of two nuclei 12,14Be for many different deformation parameter β 2 values as a function of the excitation energy E ex w.r.t. the ground state of a parent nucleus. Our results for 12Be predict to prefer a prolate shape and B(GT) results of 14Be turn out to be independent of the β 2 values.

Path integral treatment of a noncentral electric potential

AbstractWe present a rigorous path integral treatment of a dynamical system in the axially symmetric potential $V(r,\theta ) = V(r) + \tfrac{1} {{r^2 }}V(\theta ) $ . It is shown that the Green’s function can be calculated in spherical coordinate system for $V(\theta ) = \frac{{\hbar ^2 }} {{2\mu }}\frac{{\gamma + \beta \sin ^2 \theta + \alpha \sin ^4 \theta }} {{\sin ^2 \theta \cos ^2 \theta }} $ . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number l ≥ 0, are recovered.

Correspondence between Magic Numbers and Electron Degeneracy in Alkali Clusters

Correspondence between Magic Numbers and Electron Degeneracy in Alkali Clusters

Pulsed EPR characterization of encapsulated atomic hydrogen in octasilsesquioxane cages

Hydrogen atoms encapsulated in molecular cages are potential candidates for quantum computing applications. They provide the simplest two-spin system where the 1s electron spin, S = 1/2, is hyperfine-coupled to the proton nuclear spin, I = 1/2, with a large isotropic hyperfine coupling (A = 1420.40575 MHz for a free atom). While hydrogen atoms can be trapped in many matrices at cryogenic temperatures, it has been found that they are exceptionally stable in octasilsesquioxane cages even at room temperature [Sasamori et al., Science, 1994, 256, 1691]. Here we present a detailed spin-lattice and spin-spin relaxation study of atomic hydrogen encapsulated in Si(8)O(12)(OSiMe(2)H)(8) using X-band pulsed EPR spectroscopy. The spin-lattice relaxation times T(1) range between 1.2 s at 20 K and 41.8 μs at room temperature. The temperature dependence of the relaxation rate shows that for T < 60 K the spin-lattice relaxation is best described by a Raman process with a Debye temperature of θ(D) = 135 K, whereas for T > 100 K a thermally activated process with activation energy E(a) = 753 K (523 cm(-1)) prevails. The phase memory time T(M) = 13.9 μs remains practically constant between 200 and 300 K and is determined by nuclear spin diffusion. At lower temperatures T(M) decreases by an order of magnitude and exhibits two minima at T = 140 K and T = 60 K. The temperature dependence of T(M) between 20 and 200 K is attributed to dynamic processes that average inequivalent hyperfine couplings, e.g. rotation of the methyl groups of the cage organic substituents. The hyperfine couplings of the encapsulated proton and the cage (29)Si nuclei are obtained through numerical simulations of field-swept FID-detected EPR spectra and HYSCORE experiments, respectively. The results are discussed in terms of existing phenomenological models based on the spherical harmonic oscillator and compared to those of endohedral fullerenes.

Efficient Method to Perform Quantum Number Projection and Configuration Mixing for Most General Mean-Field States

Combining several techniques, we propose an efficient and numerically reliable method to perform the quantum number projection and configuration mixing for most general mean-field states, i.e., the Hartree-Fock-Bogoliubov (HFB) type product states without symmetry restrictions. As for example of calculations, we show the results of the simultaneous parity, number and angular-momentum projection from HFB type states generated from the cranked Woods-Saxon mean-field with a very large basis that is composed of Nmax=20 spherical harmonic oscillator shells.

Phase separation of Bose-Einstein condensates of trapped atoms and molecules with a homonuclear Feshbach resonance

We investigate phase separation of Bose-Einstein condensates (BECs) of two-component atoms and one-component molecules with a homonuclear Feshbach resonance. We develop a full model for dilute atomic and molecular gases including correlation of the Feshbach resonance and all kinds of interparticle interactions, and numerically calculate order parameters of the BECs in spherical harmonic oscillator traps at zero temperature with the Bogoliubov's classical field approximation. As a result, we find out that the Feshbach resonance can induce two types of phase separation. The actual phase structures and density profiles of the trapped gases are predicted in the whole parameter region, from the atom dominant regime to the molecule dominant regime. We focus on the role of the molecules in the phase separation. Especially in the atom dominant regime, the role of the molecules is described through effective interactions derived from our model. Furthermore we show that a perturbative and semi-classical limit of our model reproduces the conventional atomic BEC (single-channel) model.