Elucidating the finite temperature quasiparticle random phase approximation

In many astrophysical scenarios, such as core-collapse supernovae and neutron star mergers, as in well as heavy-ion collision experiments, transitions between thermally populated nuclear excited states have been shown to play an important role. Because of its simplicity and ability to extrapolate, the finite-temperature quasiparticle random phase approximation (FT-QRPA) is an efficient method for studying the properties of hot nuclei. The statistical ensembles in the FT-QRPA make the theory much richer than its zero-temperature counterpart, but also obscure the meaning of various physical quantities. In this work, we clarify several aspects of the FT-QRPA, including notation seen in the literature, and demonstrate how to extract physical quantities from the theory. To illustrate the correct treatment of finite-temperature transitions, we place special emphasis on the charge-exchange transitions described by the proton-neutron FT-QRPA (FT-PNQRPA). With the FT-PNQRPA built on the nuclear energy-density functional theory, we obtain solutions in a relativistic matrix approach and also in the non-relativistic finite amplitude method. We show that the proper treatment of de-excitations from thermally populated excited states causes the Ikeda sum rule to be fulfilled. In addition, we demonstrate the impact of these transitions on stellar electron capture (EC) rates in \( ^{58,78}\)Ni. While their inclusion does not affect the EC rates in \( ^{58}\)Ni, the rates in \( ^{78}\)Ni are dominated by de-excitations for temperatures \(T > 0.5\) MeV. In systems with a large negative Q-value, the inclusion of de-excitations within the FT-QRPA is necessary for a complete description of reaction rates at finite temperature.