Pairing phase transition: A finite-temperature relativistic Hartree-Fock-Bogoliubov study

Background: The relativistic Hartree-Fock-Bogoliubov (RHFB) theory has recently been developed and it provides a unified and highly predictive description of both nuclear mean-field and pairing correlations. Ground-state properties of finite nuclei can accurately be reproduced without neglecting exchange (Fock) contributions.

Purpose: Finite-temperature RHFB (FT-RHFB) theory has not yet been developed, leaving yet unknown its predictions for phase transitions and thermal excitations in both stable and weakly bound nuclei.

Method: FT-RHFB equations are solved in a Dirac Woods-Saxon (DWS) basis considering two kinds of pairing interactions: finite or zero range. Such a model is appropriate for describing stable as well as loosely bound nuclei since the basis states have correct asymptotic behavior for large spatial distributions.

Results: Systematic FT-RH(F)B calculations are performed for several semimagic isotopic/isotonic chains comparing the predictions of a large number of Lagrangians, among which are PKA1, PKO1, and DD-ME2. It is found that the critical temperature for a pairing transition generally follows the rule $T_c=0.60\mathrm{\Delta }\left(0\right)$ for a finite-range pairing force and $T_c=0.57\mathrm{\Delta }\left(0\right)$ for a contact pairing force, where $\mathrm{\Delta }\left(0\right)$ is the pairing gap at zero temperature. Two types of pairing persistence are analyzed: type I pairing persistence occurs in closed subshell nuclei while type II pairing persistence can occur in loosely bound nuclei strongly coupled to the continuum states.

Conclusions: This FT-RHFB calculation shows very interesting features of the pairing correlations at finite temperature and in finite systems such as pairing re-entrance and pairing persistence.

Figure 1

The neutron pairing gaps in ${}^{124}\mathrm{Sn}$ as a function of temperature, calculated with PKO2 (black) and DD-ME2 (blue) corresponding to different $M_{\mathrm{NR}}^{*}$. In the pairing channel, we compare finite-range D1S (filled circles) and contact DDCI (empty circles) forces. The analytical results (dashed lines) are also shown. Notice that here we took the factor $g=1$ for the Gogny D1S force, and for the DDCI pairing, we have taken the following values for $V_0$ (in ${\mathrm{MeVfm}}^{-3}$): 335 (PKO2) and 342 (DD-ME2) for the RH(F)B and BCS calculations. For the DDCIx force used in BCS, we took for $V_0$ (in ${\mathrm{MeVfm}}^{-3}$): 526 (PKO2) and 539 (DD-ME2).

Figure 2

The ratios $T_c/{\mathrm{\Delta }}_n\left(0\right)$ in ${}^{124}\mathrm{Sn}$ as a function of the nonrelativistic effective mass $M_{\mathrm{NR}}^{*}$, calculated using the FT-RH(F)B theory with 14 parameter sets. In the pairing channel the finite-range D1S and the contact force DDCI are employed.

Figure 3

The critical temperature $T_c$ (left axes, blue circles) and the occupation number of continuum states $N_{\mathrm{con}}$ (right axes, black squares) at zero temperature in ${}^{124}\mathrm{Sn}$ as a function of the neutron pairing gap calculated from PKO1 FT-RHFB with Gogny D1S (a) and DDCI (b) pairing forces. The dashed lines correspond to the relation $T_c=0.60\left(0.57\right){\mathrm{\Delta }}_n\left(0\right)$. The strength parameter of the pairing interaction is varied to obtain different values of the pairing gap at zero temperature.

Figure 4

Contributions, in ${}^{124}\mathrm{Sn}$, of the continuum states to the pairing number ${\tilde{N}}_{\mathrm{con}}$ (a) and to the neutron number $N_{\mathrm{con}}$ (b), normalized to their value at zero temperature, as a function of the temperature $T/0.6\mathrm{\Delta }\left(0\right)$. The results correspond to PKO1 Lagrangian and D1S pairing interaction.

Figure 5

Comparison of $\mathrm{\Delta }\left(0\right)$ (black circles), $0.60\mathrm{\Delta }\left(0\right)$ (green curves), and $T_c$ (blue circles) in the even-even Ni, Sn, Pb isotopes (left panels) and $N=50$, 82, 126 isotones (right panels), calculated in FT-RHFB with PKA1 and the Gogny pairing interaction D1S.

Figure 6

The same as in Fig. 5 but calculated in the FT-RHFB theory with the effective interaction PKO1.

Figure 7

The same as in Fig. 5 but calculated in the FT-RHB theory with the effective interaction DD-ME2.

Figure 8

The neutron pairing gaps in ${}^{68}\mathrm{Ni}$ as a function of temperature, calculated by FT-RHFB with Gogny D1S and DDCI pairing forces, and by FT-RHF-BCS with DDCI pairing force. To evaluate the persistence provoked by the subshell, the analytical results are also shown. The pairing strength $V_0$ (in MeV ${\mathrm{fm}}^{-3}$) is fixed to be 326 (DDCI) and 537 (DDCIx).

Figure 9

The neutron pairing gaps in ${}^{174}\mathrm{Sn}$ as a function of temperature, calculated with PKO1 and ${\mathrm{NL}3}^{*}$, using Gogny D1S and DDCI pairing forces. The results of the analytical model are also shown. The pairing strength $V_0$ (in MeV ${\mathrm{fm}}^{-3}$) is 333 (DDCI with PKO1), 317 (DDCI with ${\mathrm{NL}3}^{*}$), and 596 (DDCIx with the PKO1 and BCS frameworks).

Figure 10

Entropy and specific heat in ${}^{120}\mathrm{Sn}$ as a function of temperature, calculated using the FT-RH(F)B and FT-RH(F) theories.

Figure 11

The same as in Fig. 10 but for ${}^{160}\mathrm{Sn}$.