Dirac-Hartree-Bogoliubov calculation for spherical and deformed hot nuclei: Temperature dependence of the pairing energy and gaps, nuclear deformation, nuclear radii, excitation energy, and entropy

Abstract

Background: Unbound single-particle states become important in determining the properties of a hot nucleus as its temperature increases. We present relativistic mean field (RMF) for hot nuclei considering not only the self-consistent temperature and density dependence of the self-consistent relativistic mean fields but also the vapor phase that takes into account the unbound nucleon states. Purpose: The temperature dependence of the pairing gaps, nuclear deformation, radii, binding energies, entropy, and caloric curves of spherical and deformed nuclei are obtained in self-consistent RMF calculations up to the limit of existence of the nucleus. Method: We perform Dirac-Hartree-Bogoliubov (DHB) calculations for hot nuclei using a zero-range approximation to the relativistic pairing interaction to calculate proton-proton and neutron-neutron pairing energies and gaps. A vapor subtraction procedure is used to account for unbound states and to remove long range Coulomb repulsion between the hot nucleus and the gas as well as the contribution of the external nucleon gas. Results: We show that p-p and n-n pairing gaps in the 1 S 0 channel vanish for low critical temperatures in the range T c p ≈ 0.6-1.1 MeV for spherical nuclei such as 90 Zr, 124 Sn, and 140 Ce and for both deformed nuclei 150 Sm and 168 Er. We found that superconducting phase transition occurs at T c p = 1.03 pp (0) for 90 Zr, T c p = 1.16 pp (0) for 140 Ce, T c p = 0.92 pp (0) for 150 Sm, and T c p = 0.97 pp (0) for 168 Er. The superfluidity phase transition occurs at T c p = 0.72 nn (0) for 124 Sn, T c p = 1.22 nn (0) for 150 Sm, and T c p = 1.13 nn (0) for 168 Er. Thus, the nuclear superfluidity phase-at least for this channel-can only survive at very low nuclear temperatures and this phase transition (when the neutron gap vanishes) always occurs before the superconducting one, where the proton gap is zero. For deformed nuclei the nuclear deformation disappear at temperatures of about T c s = 2.0-4.0 MeV, well above the critical temperatures for pairing, T c p . If we associate the melting of hot nuclei into the surrounding vapor with the liquid-gas phase transition our results indicate that it occurs at temperatures around T = 8.0-10.0 MeV, somewhat higher than observed in many experimental results. Conclusions: The change of the pairing fields with the temperature is important and must be taken into account in order to define the superfluidity and superconducting phase transitions. We obtain a Hamiltonian form of the pairing field calibrated by an overall constant c pair to compensate for deficiencies of the interaction parameters and of the numerical calculation. When the pairing is not zero, the states close to the Fermi energy make the principal contribution to the anomalous density that appears in the pairing field. By including temperature through the use of the Matsubara formalism, the normal and anomalous densities are multiplied by a Fermi occupation factor. This leads to a reduction in the anomalous density and in the pairing as the temper...

Temperature effects on nuclear pseudospin symmetry in the Dirac-Hartree-Bogoliubov formalism

“…Note that beyond about T 8 MeV, the nuclear structure is almost completely dissolved since the stability of a hot nucleus depends on maintaining the balance between surface and Coulomb contributions, as discussed in Refs. [20][21][22][23].…”

“…We used the finite-temperature DiracHartree-Bogoliubov (FTDHB) formalism to obtain the mean field and Coulomb potentials in a self-consistently calculation [23]. This formalism allows us to take into account the pairing and deformation beyond the mean field and Coulomb potentials.…”

“…[18], but we consider explicitly the self-consistent temperature dependence of the relativistic pairing fields, as well as the vapor phase, to take into account the unbound nucleon states. This finite-temperature DHB (FTDHB) formalism was developed in an earlier work [23] and includes the Coulomb and meson mean fields, as well as pairing correlations, to calculate the properties of hot nuclei self-consistently. As discussed before, the Fock terms are neglected, although the most important effects of the Fock terms, due to exchange of the short-range σ , ω, and ρ mesons, can be taken into account by using adjusted Hartree terms.…”

“…(10). As a consequence, the pairing energy and gap tend to zero as the temperature increases [23]. Specifically, for axially symmetric potentials, the scalar and vector potential are independent of the azimuthal angle such that V S,V = V S,V (r ⊥ ,z).…”

“…Quite recently, we found that for small temperatures the vapor subtraction procedure is not very relevant to the change of the pairing fields with increasing temperature because the critical superfluidity and superconducting phase transitions occur at T ∼ 1 MeV. The effects of the vapor phase that takes into account the unbound nucleon states become important only at temperatures T 4 MeV, allowing the study of nuclear properties of finite nuclei from zero to high temperatures [23].…”