Single-particle Potential Research Articles

Range-Separated Tensor Format for Many-Particle Modeling

We introduce and analyze the new range-separated (RS) canonical/Tucker tensor format which aims for numerical modeling of the 3D long-range interaction potentials in multiparticle systems. The main idea of the RS tensor format is the independent grid-based low-rank representation of the localized and global parts in the target tensor, which allows the efficient numerical approximation of $N$-particle interaction potentials. The single-particle reference potential, described by the radial basis function $p(\|x\|)$, $x \in \mathbb{R}^d$, say $p(\|x\|)=1/\|x\|$ for $d=3$, is split into a sum of localized and long-range low-rank canonical tensors represented on a fine 3D $n\times n\times n$ Cartesian grid. The smoothed long-range contribution to the total potential sum is represented on the 3D grid in $O(n)$ storage via the low-rank canonical/Tucker tensor. We prove that the Tucker rank parameters depend only logarithmically on the number of particles $N$ and the grid size $n$. Agglomeration of the short-rang...

Essentials of the macroscopic-microscopic folded-Yukawa approach and examples of its record in providing nuclear-structure data for simulations

The macroscopic-microscopic model based on the folded-Yukawa singleparticle potential and a “finite-range” macroscopic model is probably the approach that has provided the most reliable predictions of a large number of nuclear-structure properties for all nuclei between the proton and neutron drip lines. I will describe some basic features of the model and the development philosophy that may be the reason for its success. Examples of quantities modeled within the same model framework are, nuclear masses, ground-state level structure, including spins, ground-state shapes, fission barriers, heavy-ion fusion barriers, sub-barrier fusion cross sections, β-decay half-lives and delayed neutron emission probabilities, shape coexistence, and α-decay Qα energies to name a few. I will show how well it predicted various properties measured after published results. Rather than giving an incomplete model description here I will give a timeline of model development and provide references to typical applications and references that are sufficiently complete that several individuals have written computer codes based on these references, codes whose results have excellent agreement with ours.

Transverse Magnetic Electron Scattering Form Factors for Some Odd-A Nuclei Using Shell Model and Hartree–Fock Calculations

Shell model and Hartree–Fock (HF) calculations have been adopted to study the nuclear structure for some odd-A nuclei; 17O, 27Al, 39K, and 51V. The single-particle nuclear density distributions, the associated nuclear radii, and the transverse magnetic form factors were investigated. The wave functions have been utilized from the shell model using USDA and SAS two-body effective interactions for these nuclei with the sd and d3f7 shell model spaces. On the other hand, different Skyrme parameterizations have been used within the HF method to get the single-particle potential which is used to calculate the single-particle matrix elements. The calculated form factors have been compared with available experimental data.

The effective masses of nuceons in asymmetric nuclear matter in the LOCV framework

The momentum dependence of single-particle potential (SPP) and effective masses of nuclei in asymmetric nuclear matter are studied in the framework of the lowest order constrained variational (LOCV) method at zero temperature. The Av18 interactions including two-body interactions and Urbana type three-body force (TBF) are considered as the input for the nucleon-nucleon potential. We investigate the TBF effect on the momentum-dependence of neutron and proton SPP. Also, we calculate the isospin splitting of the neutron and proton effective masses in neutron-rich nuclear matter.

Stability of the wobbling motion in the triaxially deformed odd-A nucleus

In order to analyze the content of the exact solutions for particle-rotor models with both the rigid and the hydrodynamical moments of inertia (MoI), as a theoretical probe we apply the Holstein–Primakoff (HP) boson expansion method to the total angular momentum and the single-particle angular momentum . We study the competition between Coriolis force and the single-particle potential by employing the different choices of the diagonal HP boson representations for the components of and along a common coordinate axis, and along perpendicular axes. We do not find any wobbling level sequence associated with the rotation around the principal axis with the medium MoI. The staggering in the alignments of about the axis with the medium MoI is found in the limited range of I, while the vector is confined about the axis with the largest MoI.

A novel method for determining the single particle potential directly from the measured single particle density: Application to the charge density difference between the isotones 206Pb–205Tl

We present a novel method, based on the single particle Schroedinger equation, to determine the central potential (mean-field) directly from the single particle matter density and its first and second derivatives. As an example, we consider the experimental data for the charge density difference between the isotones 206Pb–205Tl, deduced by phase shift analysis of elastic electron scattering cross-section measurements and corresponds to the shell model 3s1/2 proton orbit, and determine the corresponding single particle potential. We also present results of least-square fits to parametrized single particle potentials. The 3s1/2 wave functions of the determined potentials reproduce fairly well the experimental data within the quoted errors. More accurate experimental data, with uncertainty smaller by a factor of two or more, may answer the question how well can the data be reproduced by a calculated 3s1/2 wave function.

Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States.

We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a "proof of concept" demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.

Three-baryon interaction generated by determinant interaction of quarks

We discuss the three-baryon interaction generated by the determinant interaction of quarks, known as the Kobayashi-Maskawa-'t Hooft (KMT) interaction. The expectation value of the KMT interaction operator is calculated in fully-antisymmetrized quark-cluster model wave functions for one-, two- and three-octet baryon states. The three-baryon potential from the KMT interaction is found to be repulsive for $NN\Lambda$ and $N\Lambda\Lambda$ systems, while it is zero for the $NNN$ system. The strength and range of the three-baryon potential are found to be comparable to those for the $NNN$ three-body potential obtained in lattice QCD simulations. The contribution to the $\Lambda$ single particle potential in nuclear matter is found to be 0.28 MeV and 0.73 MeV in neutron matter and symmetric nuclear matter at normal nuclear density, respectively. These repulsive forces are not enough to solve the hyperon puzzle, but may be measured in high-precision hyperisotope experiments.

Excitonic giant-dipole potentials in cuprous oxide

In this work we predict the existence of a novel species of Wannier excitons when exposed to crossed electric and magnetic fields. In particular, we present a theory of giant-dipole excitons in $\textrm{Cu}_2\rm O$ in crossed fields. Within our theoretical approach we perform a pseudoseparation of the center-of-mass motion for the field-dressed excitonic species, thereby obtaining an effective single-particle Hamiltonian for the relative motion. For arbitrary gauge fields we exactly separate the gauge-dependent kinetic energy terms from the effective single-particle interaction potential. Depending on the applied field strengths and the specific field orientation, the potential for the relative motion of electron and hole exhibits an outer well at spatial separations up to several micrometers and depths up to $380\, \mu \rm eV$, leading to possible permanent excitonic electric dipole moments of around three million Debye.

Improvement to the gross theory of β decay by inclusion of change in parity

An improvement to the single-particle structure is made to the gross theory, which is a global $\ensuremath{\beta}$-decay model. The gross theory is based on the sum rule of the intensity of the $\ensuremath{\beta}$-decay transition and a strength function. This model provides reasonable results for $\ensuremath{\beta}$-decay rates and delayed neutrons for the entire nuclear mass region. An attempt is made to improve the gross theory of nuclear $\ensuremath{\beta}$ decay by considering the change in parity at the single-particle level of ground-state nuclei. In this treatment, the nuclear matrix elements are suppressed when the parity of the single neutron and proton levels is different for the allowed transition. The assignment of parity is performed using the Woods-Saxon--type single-particle potential. The discrepancies from experimental half-lives, which appeared in the vicinity of the magic numbers of neutrons and protons, are systematically improved in the nuclear mass region.

New and efficient method for solving the eigenvalue problem for the two-center shell model with finite-depth potentials

We propose a new method to solve the eigen-value problem with a two-center single-particle potential. This method combines the usual matrix diagonalization with the method of separable representation of a two-center potential, that is, an expansion of the two-center potential with a finite basis set. To this end, we expand the potential on a harmonic oscillator basis, while single-particle wave functions on a combined basis with a harmonic oscillator and eigen-functions of a one-dimensional two-center potential. In order to demonstrate its efficiency, we apply this method to a system with two $^{16}$O nuclei, in which the potential is given as a sum of two Woods-Saxon potentials.

The Deformation Structure of the Nuclei 32S and 36Ar

We applied two different approaches to investigate the deformation structures of the two nuclei S32 and Ar36 . In the first approach, we considered these nuclei as being deformed and have axes of symmetry. Accordingly, we calculated their moments of inertia by using the concept of the single-particle Schrödinger fluid as functions of the deformation parameter β. In this case we calculated also the electric quadrupole moments of the two nuclei by applying Nilsson model as functions of β. In the second approach, we used a strongly deformed nonaxial single-particle potential, depending on β and the nonaxiality parameter γ , to obtain the single-particle energies and wave functions. Accordingly, we calculated the quadrupole moments of S32 and Ar36 by filling the single-particle states corresponding to the ground- and the first excited states of these nuclei. The moments of inertia of S32 and Ar36 are then calculated by applying the nuclear superfluidity model. The obtained results are in good agreement with the corresponding experimental data.

Multichannel molecular state and rectified short-range boundary condition for spin-orbit-coupled ultracold fermions near p -wave resonances

We study the interplay of spin-orbit coupling (SOC) and strong $p$-wave interactions to the scattering property of spin-$\frac{1}{2}$ ultracold Fermi gases. Based on a two-channel square-well potential generating $p$-wave resonance, we show that the presence of an isotropic SOC, even for its length being much longer than the potential range, can greatly modify the $p$-wave short-range boundary condition (BC). As a result, the conventional $p$-wave BC cannot predict the induced molecules near $p$-wave resonances, which can be fully destroyed due to strong interference between the $s$- and $p$-wave channels. By analyzing the intrinsic reasons for the breakdown of the conventional BC, we propose a $p$-wave BC that can excellently reproduce the exact molecule solutions and also equally apply for a wide class of single-particle potentials besides SOC. This work reveals the significant effect of SOC on both the short- and long-range properties of fermions near $p$-wave resonances, paving the way for future explorations of interesting few- and many-body physics in such systems.

The Thermal Properties of Asymmetric Nuclear Matter within the Extended Brueckner–Hartree–Fock Approach

The single-particle potentials and other properties at absolute zero temperature in isospin asymmetric nuclear matter are investigated in the frame of an extended Brueckner theory. Also thermal quantities are calculated in asymmetric nuclear matter using CD-Bonn potential and the Urbana three-body forces (3BF). Also, the effects of the hole–hole contributions are investigated within the self-consistent Greens function approach. The inclusion of 3BF or the hole–hole contributions improves the predicted saturation property of symmetric nuclear matter within the Brueckner–Hartree–Fock approach and it leads to a significant stiffening of the density dependence of symmetry energy at high densities but the exact saturation point is not reproduced. This is of great importance in astrophysical calculation. A phenomenological term simulating the three-body interaction is introduced to assure the empirical saturation property. The hot properties of asymmetric nuclear matter such as the internal energy and the press...

Microscopic method for E0 transition matrix elements

We present a microscopic model for electric monopole ($E0$) transition matrix elements by combining a configuration interaction model for orbital occupations with an energy-density functional model for the single-particle potential and radial wave functions. The configuration interaction model is used to constrain the orbital occupations for the diagonal and off-diagonal matrix elements. These are used in an energy-density functional calculation to obtain a self-consistent transition density. This density contains the valence contribution, as well as the polarization of the protons by the valence protons and neutrons. We show connections between $E0$ matrix elements and isomer and isotope shifts of the charge radius. The spin-orbit correction to the charge density is important in some cases. This model accounts for a large part of the data over a wide region of the nuclear chart. It also accounts for the shape of the observed electron scattering form factors. The results depend on the Skyrme parameters used for the energy-density functional and might be used to provide new constraints for them.

Adiabatic fission barriers in superheavy nuclei

Using the microscopic-macroscopic model based on the deformed Woods-Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy we calculated static fission barriers $B_{f}$ for 1305 heavy and superheavy nuclei $98\leq Z \leq 126$, including even - even, odd - even, even - odd and odd - odd systems. For odd and odd-odd nuclei, adiabatic potential energy surfaces were calculated by a minimization over configurations with one blocked neutron or/and proton on a level from the 10-th below to the 10-th above the Fermi level. The parameters of the model that have been fixed previously by a fit to masses of even-even heavy nuclei were kept unchanged. A search for saddle points has been performed by the "Imaginary Water Flow" method on a basic five-dimensional deformation grid, including triaxiality. Two auxiliary grids were used for checking the effects of the mass asymmetry and hexadecapole non-axiallity. The ground states were found by energy minimization over configurations and deformations. We find that the non-axiallity significantly changes first and second fission barrier in many nuclei. The effect of the mass - asymmetry, known to lower the second, very deformed barriers in actinides, in the heaviest nuclei appears at the less deformed saddles in more than 100 nuclei. It happens for those saddles in which the triaxiallity does not play any role, what suggests a decoupling between effects of the mass-asymmetry and triaxiality. We studied also the influence of the pairing interaction strength on the staggering of $B_f$ for odd- and even-particle numbers. Finally, we provide a comparison of our results with other theoretical fission barrier evaluations and with available experimental estimates.

Fragment Mass Distributions in Low-energy Fission of $^{236}$Pu

The potential energy landscapes of Hg–U nuclei were calculated within the macroscopic–microscopic method with a Fourier nuclear shape parametrisation and the Lublin–Strasbourg Drop for the macroscopic energy. Microscopic corrections based on the Yukawa folded single-particle potential were obtained with the Strutinsky shell-correction method and the BCS approximation. The energy landscapes of even–even isotopes of Hg, Po, Ra and U were analysed in a 4-dimensional deformation space and projected onto the quadrupole–octupole plane. Extrema, ridges and valleys were localized and the electric quadrupole moments in these minima were evaluated. A comparison with the experimental data for the ground state is shown for the Hg isotopes.

Potential Energy Surfaces of Mercury up to Uranium Isotopes in the 4D Fourier Shape Parametrisation

The potential energy landscapes of Hg–U nuclei were calculated within the macroscopic–microscopic method with a Fourier nuclear shape parametrisation and the Lublin–Strasbourg Drop for the macroscopic energy. Microscopic corrections based on the Yukawa folded single-particle potential were obtained with the Strutinsky shell-correction method and the BCS approximation. The energy landscapes of even–even isotopes of Hg, Po, Ra and U were analysed in a 4-dimensional deformation space and projected onto the quadrupole–octupole plane. Extrema, ridges and valleys were localized and the electric quadrupole moments in these minima were evaluated. A comparison with the experimental data for the ground state is shown for the Hg isotopes.

Slow relaxation and sensitivity to disorder in trapped lattice fermions after a quench

We consider a system of non-interacting fermions in one dimension subject to a single-particle potential consisting of (a) a strong optical lattice, (b) a harmonic trap, and (c) uncorrelated on-site disorder. After a quench, in which the center of the harmonic trap is displaced, we study the occupation function of the fermions and the time-evolution of experimental observables. Specifically, we present numerical and analytical results for the post-quench occupation function of the fermions, and analyse the time-evolution of the real-space density profile. Unsurprisingly for a non-interacting (and therefore integrable) system, the infinite-time limit of the density profile is non-thermal. However, due to Bragg-localization of the higher-energy single-particle states, the approach to even this non-thermal state is extremely slow. We quantify this statement, and show that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localization.

Di-neutrons in neutron matter within a Brueckner-Hartree-Fock approach

We investigate the appearance of di-neutron bound states in pure neutron matter within the Brueckner-Hartree-Fock approach at zero temperature. We consider Argonne $v_{18}$ and Paris bare interactions as well as chiral two- and three-nucleon forces. Self-consistent single-particle potentials are calculated controlling explicitly singularities in the $g$ matrix associated with bound states. Di-neutrons are loosely bound, with binding energies below $1$ MeV, but are unambiguously present for Fermi momenta below $1$ fm$^{-1}$ for all interactions. Within the same framework we are able to calculate and characterize di-neutron bound states, obtaining mean radii as high as $\sim 110$ fm. The resulting equations of state and mass-radius relations for pure neutron stars are analyzed including di-neutron contributions.