Ground state phase transition in the Nilsson mean-field plus standard pairing model

The ground state phase transition in Nd, Sm, and Gd isotopes is investigated by using the Nilsson mean-field plus standard pairing model based on the exact solutions obtained from the extended Heine-Stieltjes correspondence. The results of the model calculations successfully reproduce the critical phenomena observed experimentally in the odd-even mass differences, odd-even differences of two-neutron separation energy, and the $\alpha$-decay and double ${\beta }^{-}$-decay energies of these isotopes. Since the odd-even effects are the most important signatures of pairing interactions in nuclei, the model calculations yield microscopic insight into the nature of the ground state phase transition manifested by the standard pairing interaction.

Figure 1

The odd-even mass differences (in MeV) for ${}^{145–155}\mathrm{Nd}, {}^{147–157}\mathrm{Sm}$, and ${}^{149–159}\mathrm{Gd}$. Experimental values are denoted as “Exp.” and are taken from [32]; the theoretical values calculated in the the Nilsson mean-field plus standard pairing model are denoted as “Th.” for the even-even cases (upper panel) and the odd-$A$ cases (lower panel).

Figure 2

The odd-even differences of the two-neutron separation energy (in MeV) for ${}^{147–155}\mathrm{Nd}, {}^{149–157}\mathrm{Sm}$, and ${}^{151–159}\mathrm{Gd}$. Experimental values are denoted as “Exp.” and are taken from [32]; the theoretical values calculated from the present model are denoted as “Th.” for the even-even cases (upper panel) and the odd-$A$ cases (lower panel).

Figure 3

The odd-even differences of the $\alpha$-decay energy (in MeV) for ${}^{147–157}\mathrm{Sm}$ and ${}^{149–159}\mathrm{Gd}$. Experimental values are denoted as “Exp.” and are taken from [32]; theoretical values calculated from the present model are denoted as “Th.” for (upper panel) even-even nuclei and (uower panel) odd-$A$ nuclei.

Figure 4

The odd-even differences of the double ${\beta }^{-}$-decay energy (in MeV) for ${}^{145–155}\mathrm{Nd}$ and ${}^{147–157}\mathrm{Sm}$. Experimental values are denoted as “Exp.” and are taken from [32]; the theoretical values calculated from the present model are denoted as “Th.” for the even-even cases (upper panel) and the odd-$A$ cases (lower panel).

Figure 5

The absolute value of the pairing interaction strength difference $\left|\mathrm{\Delta }G\right|$ (in MeV) determined under the present model for even-even nuclei (upper panel) and the odd-$A$ nuclei (lower panel) for ${}^{153–163}\mathrm{Nd}, {}^{155–165}\mathrm{Sm}$, and ${}^{157–167}\mathrm{Gd}$; the inset shows the pairing interaction strength $G$ (in MeV) for these isotopes.

Figure 6

The information entropy $I_H$ (solid line) and its derivative $d{I}_H/dG$ (dashed line) of the ground state of even-even ${}^{150–156}\mathrm{Sm}$ as functions of $G$ (in MeV). The shadowed area indicating the width of the peak in $d{I}_H/dG$ is provided to guide to the eye.