Nuclear dipole response in the finite-temperature relativistic time-blocking approximation

Background: The radiative neutron capture reaction rates of the r-process nucleosynthesis are immensely affected by the microscopic structure of the low-energy spectra of compound nuclei. The relativistic (quasiparticle) time-blocking approximation has successfully provided a good description of the low-energy strength, in particular, the strength associated with pygmy dipole resonance, describing transitions from and to the nuclear ground state. The finite-temperature generalization of this method is designed for thermally excited compound nuclei and has the potential to enrich the fine structure of the dipole strength, especially in the low-energy region.Purpose: To formulate the thermal extension of RTBA, i.e., finite-temperature relativistic time-blocking approximation (FT-RTBA) for the nuclear response, and to implement it numerically for calculations of the dipole strength in medium-light and medium-heavy nuclei.Methods: The FT-RTBA equations are derived using the Matsubara Green's function formalism. We show that with the help of a temperature-dependent projection operator on the subspace of the imaginary time it is possible to reduce the Bethe-Salpeter equation for the nuclear response function to a single frequency variable equation also at finite temperatures. The approach is implemented self-consistently in the framework of quantum hadrodynamics and keeps the ability of connecting the high-energy scale of heavy mesons and the low-energy domain of nuclear medium polarization effects in a parameter-free way.Results: The method was applied to the medium-light $^{48}\mathrm{Ca}, ^{68}\mathrm{Ni}$, and to the medium-heavy $^{100,120,132}\mathrm{Sn}$ nuclei. The excitation energies ${E}^{*}$ of the considered compound nuclei were calculated and found to increase quickly starting from temperatures $0.5\ensuremath{\le}T\ensuremath{\le}1.0$ MeV. The nucleonic single-particle energies and occupancies change accordingly because of the increasing diffuseness of the Fermi-Dirac distribution with the temperature increase. The dipole response of these nuclei was computed in the FT-RTBA and compared to the finite-temperature relativistic random-phase approximation (RPA). It was found that the giant dipole resonance (GDR) undergoes additional fragmentation (i) due to the thermal unblocking of the transitions between single-particle states located on the same side of the Fermi surface and (ii) because of the general reinforcement of the particle-vibration coupling with the temperature growth. The low-energy part of the dipole strength distribution is moderately enhanced at temperatures $T\ensuremath{\le}4.0$ MeV and increases dramatically above this temperature range. The width of the strength distribution grows rapidly with temperature at $T\ensuremath{\ge}1.0$ MeV. The energy-weighted sum rule in a wide finite-energy interval remains nearly flat as the temperature increases. The traditional view of the pygmy dipole resonance (PDR) as an oscillation of the weakly bound neutron excess against the isospin-saturated core is nearly maintained up to the temperature $T=5.0$ MeV and changes to the GDR-like pattern above this temperature. The collective behavior of the PDR disappears in the range of temperatures $1.0\ensuremath{\le}T\ensuremath{\le}5.0$ MeV and restores beyond this range which might be, however, already beyond the limit of existence of the nuclei.Conclusions: We present a consistent microscopic theory and a numerically stable and executable calculation scheme for computing the nuclear response at finite temperature taking into account the PVC spreading mechanism, in addition to the Landau damping. The presented calculations of the dipole response within a self-consistent relativistic framework reveal that, although the Landau damping plays the leading role in the temperature evolution of the strength distribution, (i) at moderate temperatures the PVC effects remain almost as strong as at $T=0$ and (ii) at high temperatures they are tremendously reinforced because of the formation of the new collective low-energy modes. In the dipole channel, the latter effect is responsible for the ``disappearance'' of the high-frequency GDR or, in other words, brings the GDR to the low-energy domain.