Proton-neutron pairing energies in $N=Z$ nuclei at finite temperature

The thermal behavior of isoscalar (τ = 0) and isovector (τ = 1) proton-neutron (pn) pairing energies at finite temperature is investigated with shell-model calculations. These pn pairing energies can be estimated by double differences of “thermal” energies that are extended from the double differences of binding energies as indicators of pn pairing energies at zero temperature. We found that the delicate balance between isoscalar and isovector pn pairing energies at zero temperature disappears at a finite temperature. When the temperature rises, while the isovector pn pairing energy decreases, the isoscalar pn pairing energy rather increases. We also discuss the symmetry energy at finite temperature.

Figure 1

(Color online) $\tau =0$ and $\tau =1$ pn pairing gaps estimated from double differences of binding energies for odd-odd $N=Z$ nuclei: (a) experimental ones; (b) those of shell-model calculations. The solid and dotted curves show $122.25(1-1.67A^{-1/3})/A$ and $5.18A^{-1/3}$, respectively.

Figure 2

Calculated thermal pn pairing gaps for odd-odd $N=Z$ nuclei at temperature $T=$ 2.0 MeV. The solid circles denote the τ = 0 pn pairing gap, and the open circles denote the τ = 1 pn pairing gap.

Figure 3

Calculated thermal pn pairing gaps for odd-odd $N=Z$ nuclei as a function of temperature. The solid line denotes the τ = 0 pn pairing gap, and the dotted line the τ = 1 pn pairing gap.

Figure 4

Symmetry-energy coefficient $a_{\mathrm{sym}}$ as a function of temperature for $N\approx Z$ nuclei with mass numbers $A=20$, 24, and 28.