Shape transition with temperature of the pear-shaped nuclei in covariant density functional theory

The shape evolutions of the pear-shaped nuclei ${}^{224}\mathrm{Ra}$ and even-even ${}^{144–154}\mathrm{Ba}$ with temperature are investigated by the finite-temperature relativistic mean field theory with the treatment of pairing correlations by the BCS approach. The free energy surfaces as well as the bulk properties including deformations, pairing gaps, excitation energy, and specific heat for the global minimum are studied. For ${}^{224}\mathrm{Ra}$, three discontinuities found in the specific heat curve indicate the pairing transition at temperature 0.4 MeV and two shape transitions at temperatures 0.9 and 1.0 MeV, namely one from quadrupole-octupole deformed to quadrupole deformed, and the other from quadrupole deformed to spherical. Furthermore, the gaps at $N=136$ and $Z=88$ are responsible for stabilizing the octupole-deformed global minimum at low temperatures. Similar pairing transition at $T\sim 0.5\mathrm{MeV}$ and shape transitions at $T=0.5–2.2\mathrm{MeV}$ are found for even-even ${}^{144–154}\mathrm{Ba}$. The transition temperatures are roughly proportional to the corresponding deformations at the ground states.

Figure 1

The free energy surfaces in the $({\beta }_2,{\beta }_3)$ plane at temperatures $T=0$ (a), 0.5 (b), 0.8 (c), 0.9 (d), 1.0 (e), and 1.5 (f) MeV for ${}^{224}\mathrm{Ra}$ obtained by the finite-temperature RMF+BCS calculations using the PC-PK1 energy density functional. The global minima are indicated by the solid squares. The energy separation between contour lines is 0.5 MeV.

Figure 2

The global minima deformations ${\beta }_2, {\beta }_3, {\beta }_4$ (a), excitation energies $E^{*}$ (in MeV) (b), pairing gaps ${\mathrm{\Delta }}_n, {\mathrm{\Delta }}_p$ (in MeV) (c), and the specific heat $C_{\mathrm{v}}$ (d) as functions of temperature (in MeV) for ${}^{224}\mathrm{Ra}$, obtained by the finite-temperature RMF+BCS calculations using the PC-PK1 energy density functional.

Figure 3

Neutron (a) and proton (b) single-particle levels as a function of temperature (in MeV) for the nucleus ${}^{224}\mathrm{Ra}$, obtained by constrained RMF+BCS calculations using the PC-PK1 energy density functional. The dash-dot lines denote the corresponding Fermi surfaces. The levels near the Fermi surface are labeled by Nilsson notations $\mathrm{\Omega }\pi \left[N{n}_z{m}_l\right]$ of the first leading component.

Figure 4

Same as Fig. 1, but at temperatures $T=0$ (a), 0.4 (b), 0.8 (c), 1.2 (d), 1.4 (e), and 1.6 (f) MeV for ${}^{144}\mathrm{Ba}$.

Figure 5

Same as Fig. 1, but at temperatures $T=0$ (a), 0.4 (b), 0.9 (c), 1.4 (d), 1.7 (e), 2.0 (f) MeV for ${}^{146}\mathrm{Ba}$.

Figure 6

Same as Fig. 2, but for ${}^{144}\mathrm{Ba}$.

Figure 7

Same as Fig. 2, but for ${}^{146}\mathrm{Ba}$.

Figure 8

Same as Fig. 2, but for ${}^{148}\mathrm{Ba}$.

Figure 9

Same as Fig. 2, but for ${}^{150}\mathrm{Ba}$.

Figure 10

Same as Fig. 2, but for ${}^{152}\mathrm{Ba}$.

Figure 11

Same as Fig. 2, but for ${}^{154}\mathrm{Ba}$.