Sn Isotopic Chains Research Articles

Separable pairing force for relativistic quasiparticle random-phase approximation

We have introduced a separable pairing force, which was adjusted to reproduce the pairing properties of the Gogny force in nuclear matter. This separable pairing force is able to describe in relativistic Hartree-Bogoliubov (RHB) calculations the pairing properties in the ground state of finite nuclei on almost the same footing as the original Gogny interaction. In this work we investigate excited states using the Relativistic Quasiparticle Random-Phase Approximation (RQRPA) with the same separable pairing force. For consistency the Goldstone modes and the convergence with various cutoff parameters in this version of RQRPA are studied. The first excited ${2}^{+}$ states for the chain of Sn isotopes with $Z=50$ and the chain of isotones with $N=82$ isotones are calculated in RQRPA together with the ${3}^{\ensuremath{-}}$ states of Sn isotopes. By comparing our results with experimental data and with the results of the original Gogny force we find that this simple separable pairing interaction is very successful in depicting the pairing properties of vibrational excitations.

Relativistic Quasiparticle Random Phase Approximation with a Separable Pairing Force

In our previous work [Phys. Lett. (to be published), Chin. Phys. Lett. 23(2006) 3226], we introduced a separable pairing force for relativistic Hartree–Bogoliubov calculations. This force was adjusted to reproduce the pairing properties of the Gogny force in nuclear matter. By using the well known techniques of Talmi and Moshinsky it can be expanded in a series of separable terms and converges quickly after a few terms. It was found that the pairing properties can be depicted on almost the same footing as the original pairing interaction, not only in nuclear matter, but also in unite nuclei. In this study, we construct a relativistic quasiparticle random phase approximation (RQRPA) with this separable pairing interaction and calculate the excitation energies of the first excited 2+ states and reduced B(E2; 0+ → 2+) transition rates for a chain of Sn isotopes in RQRPA. Compared with the results of the full Gogny force, we find that this simple separable pairing interaction can describe the pairing properties of the excited vibrational states as well as the original pairing interaction.

Pygmy dipole resonances in the tin region

The evolution of the low-energy electromagnetic dipole response with the neutron excess is investigated along the Sn isotopic chain within an approach incorporating Hartree-Fock-Bogoljubov (HFB) and multiphonon quasiparticle-phonon model (QPM) theory. General aspects of the relationship of nuclear skins and dipole sum rules are discussed. Neutron and proton transition densities serve to identify the pygmy dipole resonance (PDR) as a generic mode of excitation. The PDR is distinct from the GDR by its own characteristic pattern given by a mixture of isoscalar and isovector components. Results for the $^{100}\mathrm{Sn}$-$^{132}\mathrm{Sn}$ isotopes and the several $N=82$ isotones are presented. In the heavy Sn isotopes the PDR excitations are closely related to the thickness of the neutron skin. Approaching $^{100}\mathrm{Sn}$ a gradual change from a neutron to a proton skin is found and the character of the PDR is changed correspondingly. A delicate balance between Coulomb and strong interaction effects is found. The fragmentation of the PDR strength in $^{124}\mathrm{Sn}$ is investigated by multiphonon calculations. Recent measurements of the dipole response in $^{130,132}\mathrm{Sn}$ are well reproduced.

Study of the lowest [formula omitted] excitations and B(E2) transition strengths in relativistic QRPA for Sn-, and Pb-isotopes

In a relativistic quasiparticle random phase approximation (RQRPA) we have calculated the excitation energies of the first 2+ excited states and reduced B(E2;0+→2+) transition rates for very neutron-defficient to neutron-rich long chain of Sn-isotopes with even mass numbers A=100–136. In RQRPA approach this is the first such attempt and a very satisfactory agreement with the available experimental data is obtained. Furthermore, we predict a substantial rise of the B(E2) values at neutron magic numbers 50 and 82, where the E2 energy also goes up, rather substantially, compared to that for the immediate neighbours. This feature is quite similar to the already experimentally known behaviour of E2 and B(E2) around 208Pb, which are also well reproduced in the present calculation.

Probing the nuclear neutron skin by low-energy dipole modes

Dipole excitations below the neutron threshold in neutron-rich Sn isotopes are studied theoretically in the Quasiparticle–Phonon Model with Hartree–Fock–Bogoliubov single-particle input. Of special interest are the low-lying two-phonon 1− states and the Pygmy Dipole Resonance (PDR). The evolution of low-energy dipole excitations with neutron excess is investigated over the Sn isotopic chain including the experimentally unknown regions close to 132Sn. A dependence of the PDR strengths and centroid energies on the neutron skin thickness is found. Despite significant multi-phonon contributions to mean energies and transition strengths, the PDR states retain their one-phonon character. The fragmentation pattern is reduced with increasing neutron excess towards the N=82 shell closure which will be of advantage for future experimental work.

Spin-isospin resonances and the neutron skin of nuclei.

The Gamow-Teller resonances (GTR) and isobaric analog states (IAS) of a sequence of even-even Sn target nuclei are calculated by using the framework of the relativistic Hartree-Bogoliubov model plus proton-neutron quasiparticle random-phase approximation. The calculation reproduces the experimental data on ground-state properties, as well as the excitation energies of the isovector excitations. It is shown that the isotopic dependence of the energy spacings between the GTR and IAS provides direct information on the evolution of neutron-skin thickness along the Sn isotopic chain. A new method is suggested for determining the difference between the radii of the neutron and proton density distributions along an isotopic chain, based on measurement of the excitation energies of the GTR relative to the IAS.

Role of the Coulomb and the vector-isovector ρ potentials in the isospin asymmetry of nuclear pseudospin

We investigate the role of the Coulomb and the vector-isovector $\rho$ potentials in the asymmetry of the neutron and proton pseudospin splittings in nuclei. To this end, we solve the Dirac equation for the nucleons using central vector and scalar potentials with Woods-Saxon shape and $Z$ and $N-Z$ dependent Coulomb and $\rho$ potentials added to the vector potential. We study the effect of these potentials on the energy splittings of proton and neutron pseudospin partners along a Sn isotopic chain. We use an energy decomposition proposed in a previous work to assess the effect of a pseudospin-orbit potential on those splittings. We conclude that the effect of the Coulomb potential is quite small and the $\rho$ potential gives the main contribution to the observed isospin asymmetry of the pseudospin splittings. This isospin asymmetry results from a cancellation of the various energy terms and cannot be attributed only to the pseudospin-orbit term, confirming the dynamical character of this symmetry pointed out in previous works.

Relativistic Hartree-Bogoliubov model with density-dependent meson-nucleon couplings

The relativistic Hartree-Bogoliubov (RHB) model is extended to include density dependent meson-nucleon couplings. The effective Lagrangian is characterized by a phenomenological density dependence for the $\sigma$, $\omega$ and $\rho$ meson-nucleon vertex functions, adjusted to properties of nuclear matter and finite nuclei. Pairing correlations are described by the pairing part of the finite range Gogny interaction. The new density-dependent effective interaction DD-ME1 is tested in the analysis of the equations of state for symmetric and asymmetric nuclear matter, and of ground-state properties of the Sn and Pb isotopic chains. Results of self-consistent RHB calculations are compared with experimental data, and with results previously obtained in the RHB model with non-linear self-interactions, as well as in the density dependent relativistic hadron field (DDRH) model. Parity-violating elastic electron scattering on Pb and Sn nuclei is calculated using a relativistic optical model with inclusion of Coulomb distortion effects, and the resulting asymmetry parameters are related to the neutron ground-state density distributions.

Density dependent hadron field theory for asymmetric nuclear matter and exotic nuclei

The density dependent relativistic hadron field (DDRH) theory is applied to strongly asymmetric nuclear matter and finite nuclei far off stability. A new set of in-medium meson-nucleon vertices is derived from Dirac-Brueckner Hartree-Fock (DBHF) calculations in asymmetric matter, now accounting also for the density dependence of isovector coupling constants. The scalar-isovector $\delta$ meson is included. Nuclear matter calculations show that it is necessary to introduce a momentum correction in the extraction of coupling constants from the DBHF self-energies in order to reproduce the DBHF equation of state by DDRH mean-field calculations. The properties of DDRH vertices derived from the Groningen and the Bonn A nucleon-nucleon (NN) potentials are compared in nuclear matter calculations and for finite nuclei. Relativistic Hartree results for binding energies, charge radii, separation energies and shell gaps for the Ni and Sn isotopic chains are presented. Using the momentum corrected vertices an overall agreement to data on a level of a few percent is obtained. In the accessible range of asymmetries the $\delta$ meson contributions to the self-energies are found to be of minor importance but asymmetry dependent fluctuations may occur.