Woods-Saxon Potential Research Articles

Numerical Study of Fusion Cross Sections for 12C+12C and 16O+16O Reactions by Using Wong Formula

The height of the barrier between the interacting nuclei is one of the interesting topics in nuclear reaction, especially in charged-particle nuclear reactions. Wong formula is one method that can be used to perform such study, especially about fusion cross section. Therefore, a study about fusion cross sections of some light nuclei with Wong formula becomes very interesting to be performed. In this study, the fusion cross sections of 12C+12C and 16O+16O reactions have been calculated by using Wong formula at 12<=E<=32 MeV of energies. The potential of the interacting nuclei was approached by using Woods-Saxon potential. The calculations performed numerically by using both finite different and Nelder-Mead methods. The obtained results of this study have achieved a good agreement with the experimental results and the calculation results of the other researchers. Those results were indicated that Wong formula has good capability in explaining the experimental results concerning fusion cross section of light nuclei.

Manifestation of the important role of nuclear forces in the emission of photons in pion scattering off nuclei

Bremsstrahlung of photons emitted during the scattering of $\pi^{+}$-mesons off nuclei is studied for the first time. Role of interactions between $\pi^{+}$-mesons and nuclei in the formation of the bremsstrahlung emission is analyzed in details. We discover essential contribution of emitted photons from nuclear part of Johnson-Satchler potential to the full spectrum, in contrast to the optical Woods-Saxon potential. We observe unusual essential influence of the nuclear part of both potentials on the spectrum at high photon energies. This phenomenon opens a new experimental way to study and check non-Coulomb and nuclear interactions between pions and nuclei via measurements of the emitted photons. We provide predictions of the bremsstrahlung spectra for pion scattering off $^{44}{\rm Ca}$.

Alternative Potentials Analyzing the Scattering Cross Sections of 7,9,10,11,12,14Be Isotopes from a 12C target: Proximity Potentials

In this paper, alternative potentials to explain the scattering cross sections of 7,9,10,11,12,14Be isotopes by a 12C target nucleus at different energies are researched. For this purpose, fourteen different proximity potentials, such as Proximity 1966, Proximity 1976, Proximity 1977, Proximity 1979, Proximity 1984, Proximity 1988, Proximity 1995, Broglia and Winther 1991, Aage Winther, Bass 1973, Bass 1977, Bass 1980, Christensen and Winther 1976, and Ngo 1980, are used to produce the real potential within the optical model. The imaginary potential is formed by using the Woods-Saxon potential. The theoretical results are compared with both experimental data and data reported in the literature. The results are in good agreement with the data. The proximity potentials are observed to play a significant role in obtaining the scattering cross sections of 7,9,10,11,12,14Be isotopes.

Validity of the distorted-wave impulse-approximation description of Ca40(e,e′p)K39 data using only ingredients from a nonlocal dispersive optical model

The nonlocal implementation of the dispersive optical model (DOM) provides all the ingredients for distorted-wave impulse-approximation (DWIA) calculations of the $(e,e'p)$ reaction. It provides both the overlap function, including its normalization, and the outgoing proton distorted wave. This framework is applied to describe the knockout of a proton from the $0\textrm{d}\frac{3}{2}$ and $1\textrm{s}\frac{1}{2}$ orbitals in ${}^{40}$Ca with fixed normalizations of 0.71 and 0.60, respectively. Data were obtained in parallel kinematics for three outgoing proton energies: 70, 100, and 135 MeV. Agreement with the data is as good as, or better than, previous descriptions employing local optical potentials and overlap functions from Woods-Saxon potentials - both with standard nonlocality corrections - whose normalization (spectroscopic factor) and radius were fitted to the data. The present analysis suggests that slightly larger spectroscopic factors are obtained when nonlocal optical potentials are employed than those generated with local potentials. The results further suggest that the chosen kinematical window around 100 MeV proton energy provides the best and cleanest method to employ the DWIA for the analysis of this reaction. The conclusion that substantial ground-state correlations cannot be ignored when describing a closed-shell atomic nucleus is therefore confirmed in detail. To reach these conclusions, it is essential to have a complete description of the nucleon single-particle propagator that accounts for all elastic nucleon-scattering observables in a wide energy domain up to 200 MeV. The current nonlocal implementation of the DOM fulfills this requirement.

Using a quadrupole deformed generalized Woods–Saxon plus spin-orbit potential to describe the polarized interaction 7Li→ +12C at 34MeV

An overall description of the cross-sections and tensor analyzing powers corresponding to the elastic interaction [Formula: see text] at [Formula: see text] has been obtained with a coupled channel analysis using the rotational model. Based on the high electric quadrupole moment of the [Formula: see text] projectile, an optical model potential equal to the sum of a quadrupole deformed generalized Woods–Saxon potential and a standard Spin-Orbit term has been used. The deformed part of the potential induces reorientations and transitions that involve the first four energy states of the [Formula: see text] nucleus. These states are supposed to belong to an inverted [Formula: see text][Formula: see text][Formula: see text] rotational band proposed in the literature. Partial reproductions of the inelastic interaction [Formula: see text] have also been achieved by using the same deformed potential with the same parameter values that describe the elastic interaction. Comparisons are made with previous results obtained from optical model potential and continuum-discretized coupled-channel calculations using the cluster-folding model. The use of a deformed scattering potential introduces a tensor term that allows the reproduction of the tensor analyzing powers.

Study of matter density distributions, elastic charge form factors and size radii for halo 11Be, 19C and 11Li nuclei

In this work, the calculation of matter density distributions, elastic charge form factors and size radii for halo 11Be, 19C and 11Li nuclei are calculated. Each nuclide under study are divided into two parts; one for core part and the second for halo part. The core part are studied using harmonic-oscillator radial wave functions, while the halo part are studied using the radial wave functions of Woods-Saxon potential. A very good agreement are obtained with experimental data for matter density distributions and available size radii. Besides, the quadrupole moment for 11Li are generated.

Breakup threshold anomaly in the elastic scattering of the 9Be + 80Se system

Elastic scattering angular distributions for the 9Be + 80Se system were measured at eleven energies from below to above the Coulomb barrier. The experimental elastic scattering cross sections were analyzed within the framework of the optical model to study the energy dependence of the potential. Two different potentials were used: an energy-dependent phenomenological Woods–Saxon potential and a double folding São Paulo potential. For these two potentials, the energy dependence of the real and imaginary strengths shows consistent with the presence of the breakup threshold anomaly. An alternative method is proposed for calculating the dispersion relation between the real and imaginary parts of the optical potential based on bivariate random sampling obtained from the covariance matrix of the adjusted experimental angular distributions.

Investigation of fusion hindrance in a soft asymmetric system deep below the barrier

A steeper fall of fusion excitation function, compared to the predictions of coupled-channel (CC) models, at energies below the lowest barrier between the reaction partners, is termed as deep sub-barrier fusion hindrance. This phenomenon has been observed in many symmetric and nearly symmetric systems. Different physical origins of the hindrance have been proposed though a complete understanding is yet to be achieved. This work reports the measurement of the fusion (evaporation residue) cross sections for the system 19F+181Ta, from above the barrier down to the energies where fusion hindrance is expected to come into play. CC calculation with standard Woods–Saxon potential gives a fair description of the fusion excitation function down to energies ≃14% below the barrier. This is in contrast with the observation of increasing fusion hindrance in asymmetric reactions induced by increasingly heavier projectiles, viz. 6,7Li, 11B, 12C and 16O. The most likely reason for non-observation of fusion hindrance in the present system is the ‘softness’ of the collision partners. The ‘threshold’ for the onset of fusion hindrance in reactions involving well-deformed nuclei is expected to be at an energy lower than that estimated from the systematics. It is also noted that the asymmetric reactions, which have not shown any signature of fusion hindrance within the measured energy range, are induced by projectiles with low α-particle break-up threshold and have positive Q-values for most light particle pick-up channels. Since CC calculation reproduces the fusion excitation functions for these systems, probable roles of projectile break-up and particle transfer in fusion deep below the barrier are not conclusively proven. Further measurements and inclusion of break-up and transfer channels within the framework of CC formalism would be of interest.

Neutron-hole states in 131Sn and spin-orbit splitting in neutron-rich nuclei

In atomic nuclei, the spin-orbit interaction originates from the coupling of the orbital motion of a nucleon with its intrinsic spin. Recent experimental and theoretical works have suggested a weakening of the spin-orbit interaction in neutron-rich nuclei far from stability. To study this phenomenon, we have investigated the spin-orbit energy splittings of single-hole and single-particle valence neutron orbits of 132Sn. The spectroscopic strength of single-hole states in 131Sn was determined from the measured differential cross sections of the tritons from the neutron-removing 132Sn(d, t)131Sn reaction, which was studied in inverse kinematics at the Holifield Radioactive Ion Beam Facility at Oak Ridge National Laboratory. The spectroscopic factors of the lowest 3/2+, 1/2+ and 5/2+ states were found to be consistent with their maximal values of (2j+1), confirming the robust N=82 shell closure at 132Sn. We compared the spin-orbit splitting of neutron single-hole states in 131Sn to those of single-particle states in 133Sn determined in a recent measurement of the 132Sn(d, p)133Sn reaction. We found a significant reduction of the energy splitting of the weakly bound 3p orbits compared to the well-bound 2d orbits, and that all the observed energy splittings can be reproduced remarkably well by calculations using a one-body spin-orbit interaction and a Woods–Saxon potential of standard radius and diffuseness. The observed reduction of spin-orbit splitting can be explained by the extended radial wavefunctions of the weakly bound orbits, without invoking a weakening of the spin-orbit strength.

An investigation of the bound-state solutions of the Klein-Gordon equation for the generalized Woods-Saxon potential in spin symmetry and pseudo-spin symmetry limits

Recently, scattering of a Klein-Gordon particle in the presence of mixed scalar-vector generalized symmetric Woods-Saxon potential was investigated for the spin symmetric and the pseudo-spin symmetric limits in one spatial dimension. In this manuscript, the bound state solutions of the Klein-Gordon equation with mixed scalar-vector generalized symmetric Woods-Saxon potential are examined analytically within the framework of spin and pseudo-spin symmetry limits. We prove that the occurrence of bound state energy spectrum exists only in the spin symmetric limit, while in the pseudo-spin symmetric limit, the bound state spectrum does not exist. Besides the theoretical proof, the Newton-Raphson numerical methods are used to calculate the bound state energy spectra of a neutral Kaon particle, confined in a generalized symmetric Woods-Saxon potential, energy well constituted with repulsive or attractive surface interactions, for the spin and pseudo-spin symmetric limits, respectively. Numerical results are consistent with the non-existence of the bound state energy spectrum in the pseudo-spin symmetric limit.

Elastic and inelastic O16+C12 rainbow scattering within the coupled-channels mechanism

Angular distributions of the differential cross section of $^{16}\mathrm{O}+^{12}\mathrm{C}$ elastic and inelastic scattering in the energy range ${E}_{\mathrm{lab}}=181\ensuremath{-}1503$ MeV were fitted with the full cluster folded approach (DFC1) using the coupled-channels (CC) method by taking into account the effect of the low-lying states, ${0}^{+}$ and ${2}^{+}$ (${E}_{\mathrm{ex}}=4.44$ MeV) for the target nucleus. The cluster transition density is generated macroscopically by deforming the cluster ground state modified Gaussian density distribution of the target nucleus. The coupled-channels calculations with the fitted parameters for the elastic scattering provide a successful description of the inelastic experimental measurements. The obtained results are compared with the finding of the phenomenological Woods-Saxon potential. It was found that the characteristic features of the nuclear rainbow phenomena can be reproduced successfully by our CC calculations with both potentials but fail to reproduce the new Airy minimum of the nuclear rainbow structure. The deduced intrinsic quadrupole moment values, ${Q}_{20}$, derived from the analyses are compared with results of previous studies. The deformation lengths, ${\ensuremath{\delta}}_{20}$, obtained by CC calculations are in agreement with those obtained from the electromagnetic measurement values and those reported in previous studies.

Dipole and generalized oscillator strength derived electronic properties of an endohedral hydrogen atom embedded in a Debye-Hückel plasma

We report electronic properties of a hydrogen atom encaged by an endohedral cavity under the influence of a weak plasma interaction. We implement a finite-difference approach to solve the Schrödinger equation for a hydrogen atom embedded in an endohedral cavity modeled by the Woods-Saxon potential with well depth V0, inner radius R0, thickness Δ, and smooth parameter γ. The plasma interaction is described by a Debye-Hückel screening potential that characterizes the plasma in terms of a Debye screening length λD. The electronic properties of the endohedral hydrogen atom are reported for selected endohedral cavity well depths, V0, and screening lengths, λD, that emulate different confinement and plasma conditions. We find that for low screening lengths, the endohedral cavity potential dominates over the plasma interaction by confining the electron within the cavity. For large screening lengths, a competition between both interactions is observed. We assess and report the photo-ionization cross section, dipole polarizability, mean excitation energy, and electronic stopping cross section as function of λD and V0. We find a decrease of the Generalized Oscillator Strength (GOS) when the final excitation is to an s state as the plasma screening length decreases. For a final excitation into a p state, we find an increase in the GOS as the endohedral cavity well-depth increases. For the case of the electronic stopping cross section, we find that the plasma screening and endohedral cavity effects are larger in the low-to-intermediate projectile energies for all potential well depths considered. Our results agree well to available theoretical and experimental data and are a first step towards the understanding of dipole and generalized oscillator strength dependent properties of an atom in extreme conditions encaged by an endohedral cavity immersed in a plasma medium.

Hindered α decays of heaviest high- K isomers

To find candidates for long-lived high-$K$ isomers in even-even $Z=106\text{--}112$ superheavy nuclei, we study dominant $\ensuremath{\alpha}$-decay channels of two- and four-quasiparticle configurations at a low excitation. Energies are calculated within the microscopic-macroscopic approach with the deformed Woods-Saxon potential. Configurations are fixed by a standard blocking procedure and their energy found by a subsequent minimization over deformations. Different excitation energies of a high-$K$ configuration in parent and daughter nucleus seem particularly important for hindrance of the $\ensuremath{\alpha}$ decay. A strong hindrance is found for some four-quasiparticle states, particularly ${K}^{\ensuremath{\pi}}={20}^{+}$ and/or ${19}^{+}$ states in $^{264\text{--}270}\mathrm{Ds}$. Contrary to what was suggested in experimental papers, it is rather a proton configuration that leads to this strong hindrance. If not shortened by the electromagnetic decay, $\ensuremath{\alpha}$ half-lives of $\ensuremath{\sim}1\phantom{\rule{0.28em}{0ex}}\mathrm{s}$ could open new possibilities for studies of chemical/atomic properties of related elements.

Surface interaction effects to a Klein–Gordon particle embedded in a Woods–Saxon potential well in terms of thermodynamic functions,

Recently, it has been investigated how the thermodynamic functions vary when the surface interactions are taken into account for a nucleon that is confined in a Woods–Saxon potential well, with a non-relativistic point of view. In this manuscript, the same problem is handled with a relativistic point of view. More precisely, the Klein–Gordon equation is solved in the presence of mixed scalar–vector generalized symmetric Woods–Saxon potential energy that is coupled to momentum and mass. Employing the continuity conditions the bound state energy spectra of an arbitrarily parameterized well are derived. It is observed that, when a term representing the surface effect is taken into account, the character of Helmholtz free energy and entropy versus temperature are modified in a similar fashion as this inclusion is done in the non-relativistic regime. Whereas it is found that this inclusion leads to different characters to internal energy and specific heat functions for relativistic and non-relativistic regimes.

Exponentially fitted two-step hybrid methods for the resonant state of the Schrödinger equation

A family of exponentially fitted two-step hybrid methods for the numerical integration of the Schrödinger equation is investigated. The asymptotic expressions of the local errors for large energies are presented. The stability of the new methods is analyzed. Numerical results are reported for the widely used Woods–Saxon potential to show the efficiency of the new methods. The error analysis is clearly tested by the resonance problem.

Neutron-proton pairing correlations and deformation for N=Z nuclei in the pf shell within the deformed BCS and Hartree-Fock-Bogoliubov approach

We investigated neutron-proton pairing correlations effects on the shell evolution of ground-state energies by the deformation for $N=Z$ nuclei in $pf$ shell, such as $^{44}\mathrm{Ti}$, $^{48}\mathrm{Cr}$, $^{52}\mathrm{Fe}$, $^{64}\mathrm{Ge}$, $^{68}\mathrm{Se}$, $^{72}\mathrm{Kr}$, and $^{76}\mathrm{Sr}$. We started from a simple shell-filling model constructed by a deformed Woods-Saxon potential with ${\ensuremath{\beta}}_{2}$ deformation, and included pairing correlations in the residual interaction, which give rise to smearing of the Fermi surface revealing interesting evolution of the Fermi energy along the shell evolution. In this work, like-pairing and unlike-pairing correlations decomposed as isovector $T=1$ and isoscalar $T=0$ components are explicitly taken into account. Finally, we estimate ground-state energies comprising the mean-field energy, the pairing energy, and the self-energy due to the pairing correlations, in terms of the deformation. The enhanced $T=0$ pairing interaction supports oblate deformations for $^{72}\mathrm{Kr}$ and $^{68}\mathrm{Se}$, whose features are different from other $pf$-shell $N=Z$ nuclei considered in this work.

Hartree-Fock Gamow basis from realistic nuclear forces

The development of the radioactive nuclear beams has made it possible to study the exotic nuclei near the drip line. The theoretical description of these nuclei which are far from the valley of stability is still a challenge in nuclear physics. These nuclei exhibit unusual properties such as halo densities and weak bindings. The coupling to the continuum is needed for the theoretical description. The Gamow shell model which extends the traditional shell model to the complex plane, can efficiently include the continuum coupling. This model unifies nuclear structure and reaction properties, and has proven to be a promising tool for the descriptions of weakly bound and unbound nuclei. The starting point of the Gamow shell model is the Berggren completeness where bound, resonant (or Gamow) and scattering states are treated on an equal footing. Employing the Berggren basis, first principle calculations can also be extended to study weak-binding nuclei and nuclear reactions. A big concern is how to obtain the single-particle Berggren basis which is used to construct the Slater determinants in many-body calculations. The single-particle basis in Gamow shell-model calculations are normally constructed from a Woods-Saxon potential with parameters fitted to experimental single-particle energies. However, in a fully microscopic approach, the single-particle basis constructed from the nucleon-nucleon interaction is preferred. In this paper, we present a method to generate the Berggren basis microscopically from realistic nucleon-nucleon forces. The obtained basis is called Hartree-Fock (HF) Gamow basis. We first performed the HF iteration in harmonic oscillator basis, and then the HF potential obtained is analytically continued to the complex- k plane. The continuation is accomplished directly by a basis transformation in which the complex Laguerre polynomials are used. By discretizing the integral of the Schrodinger equation in momentum space, the equation becomes a complex symmetric eigenvalue problem. The HF Gamow basis can be obtained by diagonalizing the complex symmetric matrix. As examples, we apply this method to 4He and 22O with the softened chiral N3LO potential. The single-particle p3/2 resonant state in 4He and d3/2 resonance state in 22O are obtained and the numerical convergence is checked. We also discuss the method to obtain the resonance state from the phase shifts of the real-energy scattering states. With the phase-shift method, the obtained 22O neutron d3/2 resonance state is consistent with the result of the complex-plane calculation. The Gamow basis obtained can be further used in many-body calculations of weakly bound nuclei.

Theoretical Study of Density Distributions and Size Radii of 8B and 17Ne

The proton, neutron and matter density distributions, the corresponding size radii and elastic electron scattering form factors of one-proton 8 B and two-proton 17 Ne halo nuclei are calculated. The theoretical technique used to fulfill calculations is by assuming that both nuclei under study are composed of two main parts; the first is the compact core and the second is the unstable halo part. The single-particle radial wavefunctions of harmonic-oscillator (HO) and Woods-Saxon (WS) potentials are used to study core and halo parts, respectively. And other approach is studied by using HO potential for both core and halo parts, but using two HO size parameters for both supposed parts. The long tail behavior which is the main characteristic of halo nuclei are well produced for both 8 B and 17 Ne. The calculated size radii are in general in good agreement with the available experimental data. The electron scattering form factors of the C0+C2 and C0 components are also calculated for 8 B and 17 Ne, respectively and compared with corresponding stable 10 B and 20 Ne nuclei. For 8 B calculations, the core-polarization (CP) effects are taken into account by using Tassie and Bohr-Mottelson models. The contribution from model-space (MS) part C2 component is taken through pwt interaction. The results of the calculated charge form factors are left for the planned electron-radioactive ion beam colliders where the study of skin or halo on the charge form factors are going to be studied.

Radiative capture of proton by C 12 $^{12}\mathrm{C}$ at low energy

Within the framework of potential cluster model, astrophysical S-factor of radiative capture reaction $^{12}\mathrm{C} (\mathrm{p},\gamma)^{13}\mathrm{N}$ has been calculated in the two body cluster model for the energy range 0–1 MeV. The nuclear interaction in the initial and final states is described by the Woods–Saxon potential. The calculated astrophysical S-factor and rates are compared with known experimental results.

Comparative analysis of proximity potentials to describe scattering of 13C projectile off 12C, 16O, 28Si and 208Pb nuclei

In this work, we examine the elastic scattering cross sections of 13C on 12C, 16O, 28Si and 208Pb target nuclei at different incident energies. For the first time, we apply six types of proximity potentials such as Broglia andWinther 1991 (BW 91), AageWinther (AW95), Christensen and Winther 1976 (CW 76), Bass 1973 (Bass 73), Bass 1977 (Bass 77) and Bass 1980 (Bass 80) in order to obtain the real part of the optical potential. The imaginary part is taken as the Woods-Saxon potential. Theoretical results are compared with each other as well as the experimental data.