Pairing and nonaxial-shape correlations in $N=150$ isotones

Background: Rotational bands have been measured around ${}^{250}\mathrm{Fm}$ associated with strong deformed-shell closures. The $K^{\pi }=2^{-}$ excited band emerges systematically in $N=150$ isotones raging from plutonium to nobelium with even-$Z$ numbers, and a sharp drop in energies was observed in californium.

Purpose: I attempt to uncover the microscopic mechanism for the appearance of such a low-energy $2^{-}$ state in ${}^{248}\mathrm{Cf}$. Furthermore, I investigate the possible occurrence of the low-energy $K^{\pi }=2^{+}$ state, the $\gamma$ vibration, to elucidate the mechanism that prefers the simultaneous breaking of the reflection and axial symmetry to the breaking of the axial symmetry alone in this mass region.

Method: I employ a nuclear energy-density functional (EDF) method: the Skyrme-Kohn-Sham-Bogoliubov and the quasiparticle random-phase approximation are used to describe the ground state and the transition to excited states.

Results: The Skyrme-type SkM* and SLy4 functionals reproduce the fall in energy but not the absolute value of the $K^{\pi }=2^{-}$ state at $Z=98$ where the proton two-quasiparticle excitation $\left[633\right]7/2\otimes \left[521\right]3/2$ plays a decisive role for the peculiar isotonic dependence. I find interweaving roles by the pairing correlation of protons and the deformed-shell closure at $Z=98$. The SkM* model predicts the $K^{\pi }=2^{-}$ state appears lower in energy in ${}^{246}\mathrm{Cf}$ than in ${}^{248}\mathrm{Cf}$ as the Fermi level of neutrons is located in between the [622]5/2 and the [734]9/2 orbitals. Except for ${}^{250}\mathrm{Fm}$ in the SkM* calculation, the $K^{\pi }=2^{+}$ state is predicted to appear higher in energy than the $K^{\pi }=2^{-}$ state because the quasiproton [521]1/2 orbital is located above the [633]7/2 orbital.

Conclusions: A systematic study of low-lying collective states in heavy actinide nuclei provides a rigorous testing ground for microscopic nuclear models. The present paper shows a need for improvements in the EDFs to describe pairing correlations and shell structures in heavy nuclei, that are indispensable in predicting the heaviest nuclei.

Figure 1

Pair gaps of neutrons and protons in ${}^{244}\mathrm{Pu},{}^{246}\mathrm{Cm},{}^{248}\mathrm{Cf},{}^{250}\mathrm{Fm}$, and ${}^{252}\mathrm{No}$. The calculated gaps (filled diamonds) are compared with the experimental data (filled squares). The results obtained by reducing the pairing strength $V_0^{\nu }=-260{\mathrm{MeV}\mathrm{fm}}^3$ whereas keeping the strength for protons, and those obtained by increasing the pairing strength $V_0^{\pi }=-320{\mathrm{MeV}\mathrm{fm}}^3$ whereas keeping the strength for neutrons are also shown by open triangles for neutrons and protons, respectively. Depicted also are the results obtained by employing the SLy4 functional (filled circles).

Figure 2

Single-particle energies of protons and neutrons near the Fermi levels in ${}^{248}\mathrm{Cf}$. The results obtained by using the SkM* and SLy4 functionals are compared. The solid and dashed lines indicate the positive- and negative-parity states, respectively. Each orbital is labeled by ${\mathrm{\Omega }}^{\pi }$.

Figure 3

Excitation energies of the (a) $K^{\pi }=2^{-}$ and (b) $K^{\pi }=2^{+}$ states in the $N=150$ isotones. The calculations employing the SkM* and SLy4 functionals are compared with the measurements [18, 22, 23]. The results obtained by increasing the pairing for protons ($V_0^{\pi }=-320{\mathrm{MeV}\mathrm{fm}}^3$) and decreasing the pairing for neutrons ($V_0^{\nu }=-260{\mathrm{MeV}\mathrm{fm}}^3$) are depicted by open triangles. The SkM* results without the dynamical pairing are given by open diamonds. Shown is also the calculations using the Gogny D1M functional [19] and the Nilsson potential [18].

Figure 4

Matrix elements $M_{\alpha \beta }$ (12) of the 2qp excitations near the Fermi levels for the $K^{\pi }=2^{-}$ state as functions of the unperturbed 2qp energy $E_{\alpha }+E_{\beta }$.

Figure 5

Similar to Fig. 4 but for the $K^{\pi }=2^{+}$ state. See Table 2 for the configurations A–D.