Pairing transitions in finite-temperature relativistic Hartree-Bogoliubov theory

We formulate the finite-temperature relativistic Hartree-Bogoliubov theory for spherical nuclei based on a point-coupling functional, with the Gogny or separable pairing force. Using the functional PC-PK1, the framework is applied to the study of pairing transitions in Ca, Ni, Sn, and Pb isotopic chains. The separable pairing force reproduces the gaps calculated with the Gogny force not only at zero temperature, but also at finite temperatures. By performing a systematic calculation of the even-even Ca, Ni, Sn, and Pb isotopes, it is found that the critical temperature for a pairing transition generally follows the rule $T_c=0.6{\Delta }_n\left(0\right)$, where ${\Delta }_n\left(0\right)$ is the neutron pairing gap at zero temperature. This rule is further verified by adjusting the pairing gap at zero temperature with a strength parameter.

Figure 1

Binding energy per nucleon (a), entropy (b), neutron radius (c), and charge radius (d) as a function of temperature $T$ for the nucleus ${}^{124}$Sn, calculated with the FTRH (red open circles) and FTRHB theories (black solid circles), using the effective interaction PC-PK1.

Figure 2

The neutron pairing energy (a) and the neutron pairing gap (b) for the nucleus ${}^{124}$Sn as a function of temperature, calculated in the FTRHB theory with the effective interaction PC-PK1.

Figure 3

The neutron pairing gap as a function of temperature for selected nuclei in the Sn isotopic chain, calculated in the FTRHB theory with the Gogny pairing interaction D1S (black solid circles) and separable pairing interaction (red triangles).

Figure 4

The neutron pairing gaps at zero temperature (black solid circles) and the critical temperatures for pairing transition (red solid circles) in the even-even Sn isotopes, calculated in the FTRHB theory with the effective interaction PC-PK1 and the Gogny pairing interaction D1S. The scaled values of the neutron pairing gap at zero temperature, $0.6{\Delta }_n\left(0\right)$, $0.57{\Delta }_n\left(0\right)$, and $0.5{\Delta }_n\left(0\right)$, are denoted by the blue (dotted), green (dashed), and yellow (dash-dotted) curves, respectively.

Figure 5

Same as in the caption to Fig. 4 but for Pb (a), Ni (b), and Ca isotopes (c).

Figure 6

Contour plot for the neutron pairing gap in ${}^{124}$Sn as a function of the temperature and neutron pairing gap at zero temperature. The FTRHB calculation is carried out with the effective interaction PC-PK1 and the Gogny pairing interaction. The strength parameter of the latter is varied to obtain different values of the pairing gap at zero temperature. For each value of the zero-temperature pairing gap the critical temperature $T_c$ is denoted by a black solid circles, and the dotted line corresponds to the relation $T_c=0.6\Delta \left(0\right)$.

Figure 7

Specific heat for the nucleus ${}^{124}$Sn as a function of temperature $T$, calculated using the FTRHB with the effective interaction PC-PK1 and the Gogny pairing force.