Sn Isotopic Chains Research Articles

Isoscalar Giant Monopole Resonance in Spherical Nuclei as a Nuclear Matter Incompressibility Indicator

The incompressibility of both nuclear matter and finite nuclei is estimated by the monopole compression modes in nuclei in the framework of a nonrelativistic Hartree–Fock–Bogoliyubov method and the coherent density fluctuation model. The monopole states originate from vibrations of the nuclear density. The calculations in the model for the incompressibility in finite nuclei are based on the Brueckner energy–density functional for nuclear matter. Results for the energies of the breathing vibrational states and finite nuclei incompressibilities are obtained for various nuclei and their values are compared with recent experimental data. The evolution of the isoscalar giant monopole resonance (ISGMR) along Ni, Sn, and Pb isotopic chains is discussed. This approach can be applied to analyses of neutron stars properties, such as incompressibility, symmetry energy, slope parameter, and other astrophysical quantities, as well as for modelling dynamical behaviors within stellar environments.

Energy eigenvalue spectra and applications of the sextic and the Coulomb perturbed potentials

Keeping in view the importance of the anharmonic potentials, closed-form solutions to the N-dimensional Schrödinger equation, for the Coulomb perturbed and the sextic potentials using the Nikiforov-Uvarov functional analysis and power series methods respectively, are investigated. Complete energy spectrum is obtained for both the potentials by invoking some restrictions on wave function parameters of the sextic potential and the Greene-Aldrich approximation for centrifugal term of the Coulomb perturbed potential. A relationship between energy eigenvalues of the two potentials is verified in higher dimensions. This study is further extended to find physical applications of both the potentials. The energy eigenvalues of the Coulomb perturbed potential are used to compute the mass spectra of five mesons viz , , , and (where q = u, d). Within the framework of the sextic potential, Zr and Sn isotopic chains are analyzed and 96 Zr and isotopes have been identified as critical point nuclei. The results of this study are consistent with others similar studies.

Evolution of β -decay half-lives in stellar environments

$\ensuremath{\beta}$-decay properties of nuclei are investigated within the relativistic nuclear energy density functional framework by varying the temperature and density, conditions relevant to the final stages of stellar evolution. Both thermal and nuclear pairing effects are taken into account in the description of nuclear properties and in the finite-temperature proton-neutron relativistic quasiparticle random-phase approximation (FT-PNRQRPA) to calculate the relevant allowed and first-forbidden transitions in the $\ensuremath{\beta}$ decay. The temperature and density effects are studied on the $\ensuremath{\beta}$-decay half-lives at temperatures $T=0$--$1.5\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ and at densities $\ensuremath{\rho}{Y}_{e}={10}^{7}\phantom{\rule{4pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ and ${10}^{9}\phantom{\rule{4pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$. The relevant Gamow-Teller transitions are also investigated for Ti, Fe, Cd, and Sn isotopic chains at finite-temperatures. We find that the $\ensuremath{\beta}$-decay half-lives increase with increasing density $\ensuremath{\rho}{Y}_{e}$, whereas half-lives generally decrease with increasing temperature. It is shown that the temperature effects decrease the half-lives considerably in nuclei with longer half-lives at zero temperature, while only slight changes for nuclei with short half-lives are obtained. We also show the importance of including the de-excitation transitions in the calculation of the $\ensuremath{\beta}$-decay half-lives at finite-temperatures. Comparing the FT-PNQRPA results with the shell-model calculations for $pf$-shell nuclei, a reasonable agreement is obtained for the temperature dependence of $\ensuremath{\beta}$-decay rates. Finally, large-scale calculations of $\ensuremath{\beta}$-decay half-lives are performed at temperatures ${T}_{9}(\text{K})=5$ and ${T}_{9}(\text{K})=10$ and densities $\ensuremath{\rho}{Y}_{e}={10}^{7}$ and ${10}^{9}\phantom{\rule{4pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ for even-even nuclei in the range $8\ensuremath{\le}Z\ensuremath{\le}82$, relevant for astrophysical nucleosynthesis mechanisms.

Lifetime measurements in the even-even Cd102–108 isotopes

The heaviest N=Z doubly-magic nucleus, $^{100}$Sn, and the neighboring nuclei offer unique opportunities to investigate the properties of nuclear interaction in extreme conditions. In particular, the Cd isotopes are expected to present features similar to those found in the Sn isotopic chain, since they have only two proton holes in the Z=50 shell. In this manuscript, the lifetime measurements of low-lying states in the even-mass $^{102-108}$Cd is presented. Thanks to the powerful detection capabilities of AGATA array and VAMOS++ spectrometer, the unusual employment of multi-nucleon transfer reactions permitted to investigate the first 2$^+$ and 4$^+$ states in all these nuclei, together with various deformed bands in $^{106}$Cd. The results were interpreted in the context of new state-of-the-art beyond-mean-field calculations, using the symmetry-conserving configuration-mixing approach. Despite the similarities in the electromagnetic properties of the low-lying states, there is a fundamental structural difference between the ground-state bands in the Z=48 and Z=50 isotopes. The comparison between experimental and theoretical results revealed a rotational character of the Cd nuclei, which have prolate-deformed ground states with $\beta_2 \approx 0.2$. At this deformation Z=48 becomes a closed-shell configuration, which is favored with respect to the spherical one.

The kinks in charge radii across N = 82 and 126 revisited

We revisit the studies of the isotopic shift in the charge radii of even–even isotopes of Sn and Pb nuclei at N = 82, and 126, respectively, within the relativistic mean-field (RMF) and relativistic-Hartree–Bogoliubov (RHB) approach. The shell model is also used to estimate isotopic shift in these nuclei, for the first time, to the best of our knowledge. The ground state single-particle energies (SPEs) are calculated for non-linear NL3 and NL3* and density-dependent DD-ME2 parameter sets compared with the experimental data, wherever available. We establish a correlation between the filling of single-particle levels and the isotopic shift in occupation probabilities. The obtained SPE from the RMF and RHB approaches are in line with those used in the shell model and experimental data for both the Sn and Pb isotopic chains. The shell model calculated isotopic shift agrees with RMF and RHB approaches that explain the experimental data quite well in case the of Pb nuclei beyond N=126.

Constraining the density dependence of the symmetry energy with nuclear data and astronomical observations in the Korea-IBS-Daegu-SKKU framework

The KIDS framework for the nuclear equation of state (EoS) and energy density functional (EDF) offers the possibility to explore symmetry-energy (SE) parameters such as J (value at saturation density), L (slope), Ksym (curvature) and so on independently of each other and of assumptions about the effective mass. Here we examine the performance of EoSs with different SE parameters in reproducing nuclear properties and astronomical observations in an effort to constrain especially L and Ksym or the droplet-model counterpart Ktau. Assuming a standard EoS for symmetric matter, we explore several points on the hyperplane of (J,L,Ksym or Ktau) values. For each point, the corresponding EDF parameters and a pairing parameter are obtained for applications in spherical even-even nuclei. This is the first application of KIDS EDFs with pairing correlations. The EoSs are tested successively on properties of closed-shell nuclei, along the Sn isotopic chain, and on astronomical observations, in a step-by-step process of elimination and correction. A small regime of best-performing parameters is determined. The results strongly suggest that Ksym is negative and no lower than -200MeV, that Ktau lies between roughly -400 and -300MeV and that L lies between 40 and 65MeV with L<55MeV more likely. Correlations between symmetry-energy parameters are critically discussed. Predictions for the position of the neutron drip line and the neutron skin thickness of selected nuclei are reported. They are only weakly affected by the choice of effective mass values. Parts of the drip line can be sensitive to the SE parameters. The results underscore the role of Ktau and of precise astronomical input. Better constraints are possible with precise fits to nuclear energies and, in the future, more-precise input from astronomy.

Temperature Dependence of the Symmetry Energy Components for Finite Nuclei

We investigate the temperature dependence of the volume and surface components of the nuclear symmetry energy (NSE) and their ratio in the framework of the local density approximation. The results of these quantities for finite nuclei are obtained within the coherent density fluctuation model (CDFM). The CDFM weight function is obtained using the temperature-dependent proton and neutron densities calculated through the HFBTHO code that solves the nuclear Skyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformed deformed harmonic-oscillator basis. We present and discuss the values of the volume and surface contributions to the NSE and their ratio obtained for the Ni, Sn, and Pb isotopic chains around double-magic 78Ni, 132Sn, and 208Pb nuclei. The results for the T-dependence of the considered quantities are compared with estimations made previously for zero temperature showing the behavior of the NSE components and their ratio, as well as with the available experimental data. The sensitivity of the results on various forms of the density dependence of the symmetry energy is studied. We confirm the existence of “kinks” of these quantities as functions of the mass number at T = 0 MeV for the double closed-shell nuclei 78Ni and 132Sn and the lack of “kinks” for the Pb isotopes, as well as the disappearance of these kinks as the temperature increases.

Large-scale shell-model calculations near mass region 100-130

In this work, we have presented a microscopic shell-model description of the structure and collective behavior of intermediate-mass nuclei around doubly magic 100 Sn and 132 Sn nuclei. The Sn-isotopes lie between the two doubly magic nuclei and cover a range from exotic proton-rich N=Z nuclei to exotic neutron-rich nuclei with N/Z &gt; 1.6. The results obtained using BIGSTICK code for the low-level excitation states and transition probabilities for the studied Sn and Ba isotopic chains have been discussed in the radiance of available experimental data. We have used 100 Sn as a core for all the studied isotopes with the same valence space (i.e. 1d 5/2 , 2s 1/2 , 1d 3/2 , 0g 7/2 , 0h 11/2 or ‘sdgh’) for both protons (Z) and neutrons (N) between 50 and 82. These calculations are performed by means of globally optimized monopole effective interaction for the sdgh-shell.

Evolution of nuclear structure in and around Z = 50 closed shell: Generalized seniority in Cd, Sn and Te isotopes

We study the quadrupole moments and the B(E2; 2+→0+) values for the 11/2− states and the first 2+ states, respectively, by using a multi-j generalized seniority approach in the Cd (Z=48), Sn (Z=50) and Te (Z=52) isotopic chains. The g-factor trends have also been discussed. Although, Cd and Te isotopes represent two-proton hole and two-proton particle systems, thus involving both kind of particles (protons and neutrons) in contrast to Sn (Z=50) where only neutrons play a role, we find that a similar model based on neutron valence space alone is able to explain nearly all the gross features and trends. This paper represents the first attempt to test the validity of the generalized seniority scheme away from the semi-magic region and appears to be surprisingly successful. The linearly varying quadrupole moments in Cd, Sn and Te isotopes, are described by using a consistent multi-j configuration. The asymmetric double-hump behavior of B(E2) values in Cd and Te isotopes are understood in a manner identical to that of Sn isotopes by using the generalized seniority scheme for the first time. No shell quenching is supported in the calculations; hence, the neutron magic numbers, N=50 and N=82, remain robust in these isotopic chains.

Nuclear structure investigation of even-even Sn isotopes within the covariant density functional theory

The current investigation aims to study the ground-state properties of one of the most interesting isotopic chains in the periodic table, 94−168Sn, from the proton drip line to the neutron drip line by using the covariant density functional theory, which is a modern theoretical tool for the description of nuclear structure phenomena. The physical observables of interest include the binding energy, separation energy, two-neutron shell gap, rms-radii for protons and neutrons, pairing energy and quadrupole deformation. The calculations are performed for a wide range of neutron numbers, starting from the proton-rich side up to the neutron-rich one, by using the density-dependent meson-exchange and the density dependent point-coupling effective interactions. The obtained results are discussed and compared with available experimental data and with the already existing results of relativistic Mean Field (RMF) model with NL3 functional. The shape phase transition for Sn isotopic chain is also investigated. A reasonable agreement is found between our calculated results and the available experimental data.

Low-lying quadrupole and octupole collective excitations in the Sn112,116,118,120,122,124 isotopes

Background: In recent years, considerable interest has been focused on the study of the effect of neutron skin on the collective properties of vibrational nuclei. This can be best evidenced by exclusively determining neutron and proton transition matrix elements involved in a particular excitation by investigating the Coulomb-nuclear interference feature of inelastic scattering.Purpose: Measurement of angular distributions of the inelastic scattering cross sections for excitations to low-lying ${2}_{1}^{+}$ and ${3}_{1}^{\ensuremath{-}}$ states in $^{112,116,118,120,122,124}\mathrm{Sn}$ using $^{7}\mathrm{Li}$ beam as probe at ${E}_{\mathrm{lab}}=28\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$ and determination of neutron and proton transition matrix elements involved in each excitation.Methods: Projectilelike fragments were detected using six sets of Si-surface barrier detector telescopes to measure the cross sections for elastic and inelastic scattering channels. Optical model analysis of elastic scattering data, coupled reaction channels, and continuum discretized coupled channels calculations were performed to understand the measured differential cross sections. An attempt has been made to extract the microscopic mass and charge deformation lengths.Results: For the ${2}_{1}^{+}$ state, experimental $B(E2)$ values are in good agreement with existing results obtained by electromagnetic methods. Charge and mass quadrupole vibrations are homogeneous. Significant differences are observed for excitation to the ${3}_{1}^{\ensuremath{-}}$ state across the Sn isotopic chain. Available structural information for collective octupole vibrations could not reproduce the present data for this excitation. Results show much lower values of octupole mass deformation parameters.Conclusions: Isoscalar nature of surface vibrations for the ${2}_{1}^{+}$ state in Sn isotopes is verified, with ${M}_{n}/{M}_{p}\ensuremath{\sim}N/Z$. For the ${3}_{1}^{\ensuremath{-}}$ state, damped mass vibration is the primary observation. On comparison with existing estimates, a significant deviation from isoscalar nature is conjectured for this excitation, when probed with $^{7}\mathrm{Li}$.

Temperature dependence of the volume and surface contributions to the nuclear symmetry energy within the coherent density fluctuation model

The temperature dependence of the volume and surface components of the nuclear symmetry energy (NSE) and their ratio is investigated in the framework of the local density approximation (LDA). The results of these quantities for finite nuclei are obtained within the coherent density fluctuation model (CDFM). The CDFM weight function is obtained using the temperature-dependent proton and neutron densities calculated through the HFBTHO code that solves the nuclear Skyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformed deformed harmonic-oscillator basis. We present and discuss the values of the volume and surface contributions to the NSE and their ratio obtained for the Ni, Sn, and Pb isotopic chains around double-magic $^{78}$Ni, $^{132}$Sn, and $^{208}$Pb nuclei. The results for the $T$-dependence of the considered quantities are compared with estimations made previously for zero temperature showing the behavior of the NSE components and their ratio, as well as with the available experimental data. The sensitivity of the results on various forms of the density dependence of the symmetry energy is studied. We confirm the existence of `kinks' of these quantities as functions of the mass number at $T=0$ MeV for the double closed-shell nuclei $^{78}$Ni and $^{132}$Sn and the lack of `kinks' for the Pb isotopes, as well as the disappearance of these kinks as the temperature increases.

Symmetry Energy and Its Components in Finite Nuclei

We derive the volume and surface components of the nuclear symmetry energy (NSE) and their ratio within the coherent density fluctuation model. The estimations use the results of the model for the NSE in finite nuclei based on the Brueckner and Skyrme energy-density functionals for nuclear matter. The obtained values of the volume and surface contributions to the NSE and their ratio for the Ni, Sn, and Pb isotopic chains are compared with estimations of other approaches which have used available experimental data on binding energies, neutron-skin thicknesses, and excitation energies to isobaric analog states (IAS). Apart from the density dependence investigated in our previous works, we study also the temperature dependence of the symmetry energy in finite nuclei in the framework of the local density approximation combining it with the self-consistent Skyrme-HFB method using the cylindrical transformed deformed harmonic-oscillator basis. The results for the thermal evolution of the NSE in the interval T = 0–4 MeV show that its values decrease with temperature. The investigations of the T-dependence of the neutron and proton root-mean-square radii and the corresponding neutron skin thickness point out that the effect of temperature leads mainly to a substantial increase of the neutron radii and skins, especially in nuclei which are more rich of neutrons.

Publisher's Note: Temperature dependence of the symmetry energy and neutron skins in Ni, Sn, and Pb isotopic chains [Phys. Rev. C 95 , 024314 (2017)

Publisher's Note: Temperature dependence of the symmetry energy and neutron skins in Ni, Sn, and Pb isotopic chains [Phys. Rev. C <b>95</b> , 024314 (2017)

Temperature dependence of the symmetry energy and neutron skins in Ni, Sn, and Pb isotopic chains

The temperature dependence of the symmetry energy for isotopic chains of even-even Ni, Sn, and Pb nuclei is investigated in the framework of the local density approximation (LDA). The Skyrme energy density functional with two Skyrme-class effective interactions, SkM* and SLy4, is used in the calculations. The temperature-dependent proton and neutron densities are calculated through the HFBTHO code that solves the nuclear Skyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformed deformed harmonic-oscillator basis. In addition, two other density distributions of $^{208}$Pb, namely the Fermi-type density determined within the extended Thomas-Fermi (TF) method and symmetrized-Fermi local density obtained within the rigorous density functional approach, are used. The kinetic energy densities are calculated either by the HFBTHO code or, for a comparison, by the extended TF method up to second order in temperature (with $T^{2}$ term). Alternative ways to calculate the symmetry energy coefficient within the LDA are proposed. The results for the thermal evolution of the symmetry energy coefficient in the interval $T=0-4$ MeV show that its values decrease with temperature. The temperature dependence of the neutron and proton root-mean-square radii and corresponding neutron skin thickness is also investigated, showing that the effect of temperature leads mainly to a substantial increase of the neutron radii and skins, especially in the more neutron-rich nuclei, a feature that may have consequences on astrophysical processes and neutron stars.

Reactions of exotic nuclei with the quark-meson coupling model

The nucleon-nucleon interaction is an important requirement for investigations of nuclear structure and reactions, as well as for astrophysical models such as r-process nucleosynthesis and neutron stars. The traditional approach to low-energy nuclear physics is to treat nucleons as immutable objects interacting via phenomenological forces. The use of phenomenological interactions, rather than one derived from a microscopic theory, raises questions as to the reliability of predictions for exotic regions of the nuclear chart. The quark-meson coupling (QMC) model uses a relativistic mean-field approach to provide a microscopically derived nucleon-nucleon interaction, which takes into account the quark structure of the nucleon.The Skyrme energy density functional is a popular phenomenological tool in studies of nuclear structure and reactions. In this work, the QMC density functional was used to produce a set of Skyrme parameterisations, in the hopes that they will give more reliable predictions for exotic nuclei. In conjunction with Hartree-Fock-Bogoliubov (HFB) calculations, the Skyrme-QMC (SQMC) parameterisations have been used to model the ground-state properties of individual nuclei and nucleus-nucleus potentials for Ca + Sn reactions. The SQMC parameterisation performs with an accuracy comparable to modern phenomenological functionals. From this, one can investigate the importance of the isovector terms of the nucleon-nucleon interaction, which are particularly significant for exotic, neutron-rich regions of the nuclear chart.One of the notable successes of the QMC model is its derivation of nuclear spin-orbit coupling. The isovector dependence of the spin-orbit equation of state is remarkably similar to that of the modern UNEDF1 phenomenological density functional. HFB calculations along the Sn isotopic chain reveal that the isovector properties of the spin-orbit term impact binding energies to a level that will be significant for astrophysical r-process modelling.

Volume and surface contributions to the nuclear symmetry energy within the coherent density fluctuation model

The volume and surface components of the nuclear symmetry energy (NSE) and their ratio are calculated within the coherent density fluctuation model (CDFM). The estimations use the results of the model for the NSE in finite nuclei based on the Brueckner energy-density functional for nuclear matter. In addition, we present results for the NSE and its volume and surface contributions obtained by using the Skyrme energy-density functional. The CDFM weight function is obtained using the proton and neutron densities from the self-consistent HF+BCS method with Skyrme interactions. We present and discuss the values of the volume and surface contributions to the NSE and their ratio obtained for the Ni, Sn, and Pb isotopic chains studying their isotopic sensitivity. The results are compared with estimations of other approaches which have used available experimental data on binding energies, neutron-skin thicknesses, excitation energies to isobaric analog states (IAS) and also with results of other theoretical methods.

Large-scale deformed quasiparticle random-phase approximation calculations of theγ-ray strength function using the Gogny force

Valuable theoretical predictions of nuclear dipole excitations in the whole chart are of great interest for different nuclear applications, including in particular nuclear astrophysics. Here we present large-scale calculations of the $E1$ $\gamma$-ray strength function obtained in the framework of the axially-symmetric deformed QRPA based on the finite-range Gogny force. This approach is applied to even-even nuclei, the strength function for odd nuclei being derived by interpolation. The convergence with respect to the adopted number of harmonic oscillator shells and the cut-off energy introduced in the 2-quasiparticle (2-$qp$) excitation space is analyzed. The calculations performed with two different Gogny interactions, namely D1S and D1M, are compared. A systematic energy shift of the $E1$ strength is found for D1M relative to D1S, leading to a lower energy centroid and a smaller energy-weighted sum rule for D1M. When comparing with experimental photoabsorption data, the Gogny-QRPA predictions are found to overestimate the giant dipole energy by typically $\sim$2 MeV. Despite the microscopic nature of our self-consistent Hartree-Fock-Bogoliubov plus QRPA calculation, some phenomenological corrections need to be included to take into account the effects beyond the standard 2-$qp$ QRPA excitations and the coupling between the single-particle and low-lying collective phonon degrees of freedom. For this purpose, three prescriptions of folding procedure are considered and adjusted to reproduce experimental photoabsorption data at best. All of them are shown to lead to rather similar predictions of the $E1$ strength, both at low energies and for exotic neutron-rich nuclei. Predictions of $\gamma$-ray strength functions and Maxwellian-averaged neutron capture rates for the whole Sn isotopic chain are also discussed and compared with previous theoretical calculations.

Double-step truncation procedure for large-scale shell-model calculations

We present a procedure that is helpful to reduce the computational complexity of large-scale shell-model calculations, by preserving as much as possible the role of the rejected degrees of freedom in an effective approach. Our truncation is driven first by the analysis of the effective single-particle energies of the original large-scale shell-model hamiltonian, so to locate the relevant degrees of freedom to describe a class of isotopes or isotones, namely the single-particle orbitals that will constitute a new truncated model space. The second step is to perform an unitary transformation of the original hamiltonian from its model space into the truncated one. This transformation generates a new shell-model hamiltonian, defined in a smaller model space, that retains effectively the role of the excluded single-particle orbitals. As an application of this procedure, we have chosen a realistic shell-model hamiltonian defined in a large model space, set up by seven and five proton and neutron single-particle orbitals outside 88Sr, respectively. We study the dependence of shell-model results upon different truncations of the original model space for the Zr, Mo, Ru, Pd, Cd, and Sn isotopic chains, showing the reliability of this truncation procedure.

Effect of time-odd fields on odd-even mass differences of semi-magic nuclei

The effect of time-odd fields of Skyrme interaction on neutron odd-even mass differences is studied in the framework of axially deformed Skyrme Hartree-Fock (DSHF)+BCS model. To this end, we take into account both the time-even and time-odd fields to calculate the one-neutron and two-neutron separation energies and pairing gaps of semi-magic Ca, Ni, and Sn isotopic chains. In the calculations, a surface-type pairing interaction (IS pairing) and an isospin dependent contact pairing interaction (IS+IV pairing) are adopted on top of Skyrme interactions SLy4, SLy6 and SkM*, respectively. We find that the time-odd fields have in general small effects on pairing gaps, but achieve better agreement with experimental data using SLy4 and Sly6 interactions, respectively. It is also shown that the calculations with IS+IV pairing reproduce the one-neutron separation energies of Sn isotopes better than those with the IS pairing interaction when the contributions of the time-odd fields are included.