Single-particle Potential Research Articles

Analysis of the binding energies of the Λ- particle in hypernuclei with the RHVT approach and the Gauss potential

An analysis is carried out mainly of the ground state binding energies of the Λ-particle in hypernuclei with values of the core mass number AC between 15 and 207 (included) using, as far as possible, recent experimental data.Τhe renormalized (non- relativistic) quantum mechanical hypervirial theorem (RHVT) technique is employed in the form of s- power series expansions and a Gauss single particle potential for the motion of a Λ- particle in hypernuclei is used. Not exact analytic solution is known for the Schrödinger eigenvalue problem in this case. Thus, the approximate analytic expressions (AAE) for the energy eigenvalues which are obtained with the RHVT approach and are quite useful as long as the involved dimensionless parameter s is sufficiently small, are compared only with the numerical solution. The potential parameters are determined by a least-squares fit in the framework of the rigid core model for the hypernuclei. A discussion is also made regarding the determination of the renormalization parameter χ.

Properties of the nuclear symmetry energy studied by the radioactivity of unstable nuclei

In this paper, the properties of the symmetry energy in asymmetric nuclear matter, including the quadratic and quartic symmetry energies as well as their density slopes at the saturation density, are studied by the radioactivity of unstable nuclei. The symmetry energy was initially introduced as a quantum correction energy in the Weizsacker-Bethe liquid-drop model, which was proposed to describe the binding energies of finite nuclei. In order to systematically study the properties of the symmetry energy, the concept of the nuclear matter was introduced, which consists of neutrons and protons with finite neutron and proton densities but infinite neutron and proton numbers. By comparing the properties of the isospin symmetric nuclear matter, in which the neutron density is equal to the proton density, and those of the isospin asymmetric nuclear matter, in which the two densities are unequal, the symmetry energy is found to approximately represent the difference of the energy per nucleon in them. That is to say, the symmetry energy is an evaluation of the energy cost to convert protons in symmetric nuclear matter to neutrons in asymmetric nuclear matter. Since the symmetry energy characterizes the isospin asymmetric effect in nuclear matter, it plays critical roles in not only nuclear physics but also astrophysics. For example, the symmetry energy is closely related to the nuclear masses, the structures and properties of nuclei far away from stability line and near drip lines, the mechanism of heavy-ion reactions, the structures and component of neutron stars, the masses and radii of neutron stars, the cooling process of neutron stars, and so on. Besides of the quadratic symmetry energy, the quartic symmetry energy can have obvious effects on the properties of neutron stars as well. The quartic symmetry energy can significantly affect both the proton fraction in β-stable neutron stars and the critical density for the direct Urca process, which leads to faster cooling of neutron stars. In addition, the quartic symmetry energy is found to be very important for the location of the inner edge of crusts and the core-crust transition density and pressure in neutron stars. So the quartic symmetry energy can also have an influence on the structure of neutron stars. Because of the very important roles of the symmetry energy playing in not only nuclear physics but also astrophysics, it is of great significance to study the properties of both the quadratic and quartic symmetry energies. Based on the fundamental Hugenholtz-Van Hove (HVH) theorem, the properties of the symmetry energy are found to be closely related to the isoscalar and the isovector parts of the nucleon single-particle potentials, where the latter two parts can be extracted from the nuclear radioactivity. So by linking both the symmetry energy and the nuclear radioactivity to the nucleon single-particle potentials, the symmetry energy can be directly studied by nuclear radioactivity. In this paper, both the heavy-cluster radioactivity and the proton radioactivity are employed to constrain the properties of not only the quadratic symmetry energy but also the quartic symmetry energy. Besides, the effect of both the spectroscopic factor of proton radioactivity and the deformation in the daughter nucleus on the extracted results are also discussed, which are subsequently found to be very limited.

Electromagnetic multipole of positive and negative parity states in 24Mg by elastic and inelastic electron scattering

In the present work, the nuclear shell model with Hartree–Fock (HF) calculations have been used to investigate the nuclear structure of 24Mg nucleus. Particularly, elastic and inelastic electron scattering form factors and transition probabilities have been calculated for low-lying positive and negative states. The sd and sdpf shell model spaces have been used to calculate the one-body density matrix elements (OBDM) for positive and negative parity states respectively. Skyrme-Hartree-Fock (SHF) with different parameterizations has been tested with shell model calculation as a single particle potential for reproducing the experimental data along with a harmonic oscillator (HO) and Woods-Saxon (WS) single-particle potentials. The effect of the nuclear effective charge has been implemented via using different folding models; valance, Tassie and Bohr-Mottelson. The evaluated results have been discussed and compared with available experimental data.

Ξ hyperons in the nuclear medium described by chiral NLO interactions

Properties of the baryon-baryon interactions in the strangeness $S=-2$ sector of chiral effective field theory at the next-to-leading order (NLO) level are explored by calculating $\Xi$ single-particle potentials in symmetric nuclear matter. The results are transformed to the $\Xi$ potential in finite nuclei by a local-density approximation with convolution by a Gaussian form factor to simulate finite-range effects. The $\Xi$ potential is repulsive in a central region, and attractive in a surface area when the $\Xi$ energy is low. The attractive pocket can lower the $\Xi^-$ $s$ and $p$ atomic states. The obtained binding energies in $^{12}$C and $^{14}$N are found to be conformable with those found in emulsion experiments at Japan's National Laboratory for High Energy Physics (KEK). $K^+$ spectra of $(K^-, K^+)$ $\Xi$ production inclusive processes on $^9$Be and $^{12}$C are also evaluated, using a semi-classical distorted wave method. The absolute values of the cross section are properly reproduced for $^9$Be, but the peak locates at a lower energy position than that of the experimental data. The calculated spectrum of $^{12}$C should be compared with the forthcoming result from the new experiments recently carried out at KEK with better resolution than before. The comparison would be valuable to improve the understanding of the $\Xi N$ interaction, the parametrization of which has still large uncertainties.

Lifetime of the recently identified 10+ isomeric state at 3279 keV in the Nd136 nucleus

Background: The γ softness of 136Nd makes it possible to study the shape changes induced by two-proton or two-neutron excitation. Purpose: We measure the lifetimes of two-quasiparticle states of the bands based on the 10+ states at 3296 and 3279 keV to investigate the shape change induced by the alignment of two protons or two neutrons in the h11/2 orbital. Methods: The recoil-distance Doppler shift method was used for the study of 136Nd studies, which was formed by the fusion reaction 120Sn(20Ne, 4n)136Nd, at Ebeam = 85 MeV. Calculations were performed within the microscopic-macroscopic approach, based on the deformed Woods-Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy. Results: The lifetime of the 10+ state at 3279 keV of 136Nd was measured to be T 10+1/2 = 1.63(9) ns. The lifetimes of the 2+ state at 374 keV and of the 12+ state at 3686 keV of the ground band were also measured to be T 2+1/2 = 26.5(14) ps and T 12+1/2 = 22.5(14) ps. Conclusions: The measured lifetime of 10+ the state at 3279 keV together with other observables confirm the structure change in 136Nd. A rather small reduced hindrance of the electromagnetic decay of the 10+ state at 3279 keV would be consistent with its K-mixed character.

New calculations of five-dimensional fission barriers for actinide nuclei

New calculations of fission barrier heights in a five-dimensional (5D) deformation space for actinide nuclei have been performed based on a macro-microscopic (MM) model. To calculate potential energy surface (PES), we use the 5D generalized Lawrence shape (GLS) realistic parameterization as the shape description, the Lublin-Strasbourg Drop (LSD) model for the macroscopic energy and the folded-Yukawa model as the single-particle potential. A method to use two-center oscillator basis in calculating Hamiltonian matrix on any kind of shape description was presented. For the pairing energy, we use the SBCS model. Finally, Strutinsky method is used to calculate the microscopic correction energy. To find the fission barriers and fission paths, some concepts, such as basins, edges and neighbors from geography, are extended to identify the topographical structures on the 5D PES and we assume that the fission path follows the line of steepest descent. The fission barriers and double-humped paths of U and Pu isotopes are calculated.

Constraining spectroscopic factors near the r -process path using combined measurements: Kr86 (d, p)87Kr

Background: Modern nuclear structure models suggest that the shell structure near the valley of stability, with well-established shell closures at $N=50$, for example, changes in very neutron-rich nuclei far from stability. Single-particle properties of nuclei away from stability can be probed in single-neutron $(d,p)$ transfer reactions with beams of rare isotopes. The interpretation of these data requires reaction theories with various effective interactions. Often, approximations made to the final bound-state potential introduce a large uncertainty in the extracted single-particle properties, in particular the spectroscopic factor.Purpose: Mitigate this uncertainty using a combined measurement method to constrain the shape of the bound-state potential and to reliably extract the spectroscopic factor.Methods: The $^{2}\mathrm{H}(^{86}\mathrm{Kr},p)^{87}\mathrm{Kr}$ reaction was measured at 33 MeV/u at the National Superconducting Cyclotron Laboratory (NSCL) as a test of the combined method. The reaction protons were detected with the Oak Ridge Rutgers University Barrel Array (ORRUBA) of position sensitive silicon strip detectors, the first implementation of ORRUBA coupled to the S800 spectrograph with fast beams at NSCL.Results: These measurements at 33 MeV/u are combined with previous studies of the $^{86}\mathrm{Kr}(d,p)$ reaction at 5.5 MeV/u to demonstrate a successful case of the combined method to constrain the shape of the single-particle potential and deduce asymptotic normalization coefficients and spectroscopic factors. In particular, the single-particle asymptotic normalization coefficient for the ground state of $^{87}\mathrm{Kr}$ was constrained to ${b}_{d5/2}=6.{46}_{\ensuremath{-}0.57}^{+1.12}\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}1/2}$, and therefore the deduced spectroscopic factor is $S=0.{44}_{\ensuremath{-}0.13}^{+0.09}$ with uncertainties dominated by experimental statistics.Conclusions: By combining measurements at two very different beam energies, single-particle asymptotic normalization coefficients, at least for low angular momentum transfers, can be constrained. Therefore, spectroscopic factors can be deduced with uncertainties dominated by experimental uncertainties, rather than limited knowledge of bound-state potential parameters.

In-medium properties of a $ \Xi$ΞN interaction derived from chiral effective field theory

The in-medium properties of a baryon-baryon interaction for the strangeness $S=-2$ sector derived within chiral effective field theory up to next-to-leading order are investigated. The considered $\Xi N$ interaction is in line with available empirical constraints on the $\Lambda\Lambda$ $S$-wave scattering length and on $\Xi N$ elastic and inelastic cross sections. In particular, there are no near-threshold bound states in the $\Xi N$ system. A conventional $G$-matrix calculation for this interaction is performed and reveals that the single-particle potential of the $\Xi$-hyperon in nuclear matter is moderately attractive as suggested by recent experimental evidence for the existence of $\Xi$-hypernuclei.

Shell model and Hartree-Fock calculations of electron scattering form factors for 25Mg nucleus

Shell model and Hartree-Fock calculations have been adopted to study the elastic and inelastic electron scattering form factors for 25Mg nucleus. The wave functions for this nucleus have been utilized from the shell model using USDA two-body effective interaction for this nucleus with the sd shell model space. On the other hand, the SkXcsb Skyrme parameterization has been used within the Hartree-Fock 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.

Ground-state and stability properties of [formula omitted] isotopes based on semi-microscopic calculations

In this study, we present theoretical predictions on the stability properties of the yet unknown Og118276−293,295−308 isotopes using the Skyrme-SLy4 nucleon–nucleon interaction and the deformed Woods–Saxon single-particle potential in a semi-microscopic approach. The ground-state masses, binding energy, deformations and fission barriers are calculated from the total energy surfaces produced in a multidimensional deformation space using the dynamical differential evolution optimization method. Besides, the Q-values of the different decay modes and the nucleon separation energies for each isotope, and its α-decay (Tα) and spontaneous fission half-lives are also calculated. Our study revealed the Og290−296 elements to be the most bound Oganesson isotopes, among which 302,304,306Og elements are predicted to be spherical nuclei, and 299,300,301,303,305Og elements are shown to be almost spherical. The predicted ground-state and decay properties successfully reproduced the available data of the solely known Oganesson isotope 294Og. In addition to the principle α-decay modes, we studied the rival decay modes that challenge the stability of nuclei in the superheavy regions, namely the electron capture β+ decays and the spontaneous fission. An overall oscillating increasing behavior of Tα with the mass number is obtained, which obviously reverses the hesitating decreasing behavior of Qα and follows increasing trend of the corresponding change in the neutron-skin thickness after α-decays. The Og291,294−296,299−302 isotopes are expected to have relatively long half-lives, Tα≈ 1–45 ms, while 295Og is found to have the longest half-life between the considered nuclei. The Og291−300 isotopes exhibited relatively high fission barriers and small branching ratios for their spontaneous fission modes.

Self-energy dispersion in the Hubbard model

We introduce the concept of self-energy dispersion as an error bound on local theories and apply it to the two-dimensional Hubbard model on the square lattice at half-filling. Since the self-energy has no single-particle analog and is not directly measurable in experiments, its general behavior as a function of momentum is an open question. In this article we benchmark the momentum dependence with the Two-Particle Self-Consistent approach together with analytical and numerical considerations and we show that through the addition of a local single-particle potential to the Hubbard model the self-energy can be flattened, such that it is essentially described by only a frequency-dependent term. We use this observation to motivate that local theories, such as the dynamical mean-field theory, should be expected to give very accurate results in the presence of a potential of this kind. Finally, we propose a simple energy argument as an estimator for the crossover from non-local to local self-energies, which can be computed even by local theories such as dynamical mean-field theory.

Incorporating self-consistent single-particle potentials into the microscopic-macroscopic method

Effective single-particle potentials obtained by self-consistent HFB calculations from the established non-relativistic and relativistic nuclear EDF approaches are incorporated into the microscopic-macroscopic method, a widely and successfully used approach for superheavy nuclei. We determine the Schrodinger-equivalent central and spin-orbit potentials incorporating effective mass effects. The method can be applied to non-relativistic and relativistic mean-fields. A parametrization in terms of the Wood-Saxon form is introduced to derive the proton and neutron potentials, appropriate for the microscopic-macroscopic method. As the first application, the extended microscopic-macroscopic approach is used to calculate the shell corrections in the heaviest nuclei. Constraints on parameters sets for central and spin-orbit potentials are derived for which the shell effects are amplified towards $ Z=120$ .

Shells in a toroidal nucleus in the intermediate-mass region

Attention is fixed on shells in toroidal nuclei in the intermediate mass region using a toroidal single-particle potential. We find that there are toroidal shells in the intermediate mass region with large single-particle energy gaps at various nucleon numbers located at different toroidal deformations characterized by the aspect ratios of toroidal major to minor radius. These toroidal shells provide extra stability at various toroidal deformations. Relative to a toroidal core, Bohr-Mottelson spin-aligning particle-hole excitations may be constructed to occupy the lowest single-particle Routhian energies to lead to toroidal high-spin isomers with different spins. Furthermore, as a nucleon in a toroidal nucleus possesses a vorticity quantum number, toroidal vortex nuclei may be constructed by making particle-hole excitations in which nucleons of one type of vorticity are promoted to populate un-occupied single-particle orbitals of the opposite vorticity. Methods for producing toroidal high-spin isomers and toroidal vortex nuclei are discussed.

Spin symmetry in the Dirac sea derived from the bare nucleon–nucleon interaction

The spin symmetry in the Dirac sea has been investigated with relativistic Brueckner–Hartree–Fock theory using the bare nucleon–nucleon interaction. Taking the nucleus 16O as an example and comparing the theoretical results with the data, the definition of the single-particle potential in the Dirac sea is studied in detail. It is found that if the single-particle states in the Dirac sea are treated as occupied states, the ground state properties are in better agreement with experimental data. Moreover, in this case, the spin symmetry in the Dirac sea is better conserved and it is more consistent with the findings using phenomenological relativistic density functionals.

Single-particle potential of the Λ hyperon in nuclear matter with chiral effective field theory NLO interactions including effects of YNN three-baryon interactions

Adopting hyperon-nucleon and hyperon-nucleon-nucleon interactions parametrized in chiral effective field theory, single-particle potentials of the $\Lambda$ and $\Sigma$ hyperons are evaluated in symmetric nuclear matter and in pure neutron matter within the framework of lowest order Bruckner theory. The chiral NLO interaction bears strong $\Lambda$N-$\Sigma$N coupling. Although the $\Lambda$ potential is repulsive if the coupling is switched off, the $\Lambda$N-$\Sigma$N correlation brings about the attraction consistent with empirical data. The $\Sigma$ potential is repulsive, which is also consistent with empirical information. The interesting result is that the $\Lambda$ potential becomes shallower beyond normal density. This provides the possibility to solve the hyperon puzzle without introducing ad hoc assumptions. The effects of the $\Lambda$NN-$\Lambda$NN and $\Lambda$NN-$\Sigma$NN three-baryon forces are considered. These three-baryon forces are first reduced to normal-ordered effective two-baryon interactions in nuclear matter and then incorporated in the $G$-matrix equation. The repulsion from the $\Lambda$NN-$\Lambda$NN interaction is of the order of 5 MeV at the normal density, and becomes larger with increasing the density. The effects of the $\Lambda$NN-$\Sigma$NN coupling compensate the repulsion at normal density. The net effect of the three-baryon interactions to the $\Lambda$ single-particle potential is repulsive at higher densities.

Nonrotational states in isotonic chains of heavy nuclei

The ground-state deformations of heavy nuclei are explored with the microscopic-macroscopic approach using the single-particle Woods-Saxon potential of the quasiparticle-phonon model. The calculations of the energies of low-lying nonrotational states take into account the residual pairing and phonon-quasiparticle interactions. The spectra of these states are presented for the $N=147--161$ isotones. The sensitivity of the calculated results to the parameters of the model is studied. A rather good description of the available experimental data is demonstrated.

Effect of short-range correlations on the single proton 3s1/2 wave function in 206Pb

We consider the experimental data for difference, Δρc(r), between the charge density distributions of the isotones 206Pb - 205Tl, deduced by analysis of elastic electron scattering measurements and corresponds to the shell model 3s1/2 proton orbit. We investigate the effects of two-body short-range correlations. This is done by: (a) Determining the corresponding single particle potential (mean-field), employing a novel method, directly from the single particle proton density and its first and second derivatives. We also carried out least-square fits to parametrized single particle potentials; (b) Determining the short-range correlations effect by employing the Jastrow correlated many-body wave function to derive a correlation factor for the single particle density distribution. The 3s1/2 wave functions of the determined potentials reproduce fairly well the experimental data within the quoted errors. The calculated charge density difference, Δρc(r), obtained with the inclusion of the short-range correlation effect does not reproduce the experimental data.

Effects of chiral three-nucleon forces on 4He-nucleus scattering in a wide range of incident energies

It is a current important subject to clarify properties of chiral three-nucleon forces (3NFs) not only in nuclear matter but also in scattering between finite-size nuclei. Particularly for the elastic scattering, this study has just started and the properties are not understood in a wide range of incident energies ($E_{\rm in}$). We investigate basic properties of chiral 3NFs in nuclear matter with positive energies by using the Brueckner-Hartree-Fock method with chiral two-nucleon forces of N$^{3}$LO and 3NFs of NNLO, and analyze effects of chiral 3NFs on $^{4}$He elastic scattering from targets $^{208}$Pb, $^{58}$Ni and $^{40}$Ca over a wide range of $30 \lesssim E_{\rm in}/A_{\rm P} \lesssim 200$ MeV by using the $g$-matrix folding model, where $A_{\rm P}$ is the mass number of projectile. In symmetric nuclear matter with positive energies, chiral 3NFs make the single-particle potential less attractive and more absorptive. The effects mainly come from the Fujita-Miyazawa 2$\pi$-exchange 3NF and slightly become larger as $E_{\rm in}$ increases. These effects persist in the optical potentials of $^{4}$He scattering. As for the differential cross sections of $^{4}$He scattering, chiral-3NF effects are large in $E_{\rm in}/A_{\rm P} \gtrsim 60$ MeV and improve the agreement of the theoretical results with the measured ones. Particularly in $E_{\rm in}/A_{\rm P} \gtrsim 100$ MeV, the folding model reproduces measured differential cross sections pretty well. Cutoff ($\Lambda$) dependence is investigated for both nuclear matter and $^{4}$He scattering by considering two cases of $\Lambda=450$ and $550$ MeV. The uncertainty coming from the dependence is smaller than chiral-3NF effects even at $E_{\rm in}/A_{\rm P}=175$ MeV.

Fission fragment mass and total kinetic energy distributions of spontaneously fissioning plutonium isotopes

The fission-fragment mass and total kinetic energy (TKE) distributions are evaluated in a quantum mechanical framework using elongation, mass asymmetry, neck degree of freedom as the relevant collective parameters in the Fourier shape parametrization recently developed by us. The potential energy surfaces (PES) are calculated within the macroscopic-microscopic model based on the Lublin-Strasbourg Drop (LSD), the Yukawa-folded (YF) single-particle potential and a monopole pairing force. The PES are presented and analysed in detail for even-even Plutonium isotopes with A = 236–246. They reveal deep asymmetric valleys. The fission-fragment mass and TKE distributions are obtained from the ground state of a collective Hamiltonian computed within the Born-Oppenheimer approximation, in the WKB approach by introducing a neck-dependent fission probability. The calculated mass and total kinetic energy distributions are found in good agreement with the data.

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...