Dominant nucleon-pair configurations in low-lying states of deformed nuclei

In this paper we explicitly study dominant nucleon-pair configurations in low-lying states of deformed nuclei in two cases; the first is the schematic Elliott's SU(3) model with valence nucleons in the $pf$ shell, and the second is the effective $V_{\mathrm{low}-k}$ interaction with valence protons in the 50-82 shell and valence neutrons in the 82-126 shell, exemplified by ${}^{146}\mathrm{Ba}$ and ${}^{148}\mathrm{Ce}$. We find that the $SD$ nucleon-pair configurations are more predominant in low-lying states of these deformed nuclei than expected. Mixings of nucleon-pair configurations beyond the conventional $SD$-pair subspace, in particular, one-$G$ and/or one-$I$ nucleon-pair configurations, albeit small, are found to be necessary to reproduce the moment of inertia.

Figure 1

Excitation energy $E_x\left(I\right)$ and reduced electric transition probability $B(E2;I\to I-2)$ for the ground band states of six valence protons and six valence neutrons in the $pf$ shell versus spin $I$ of the states. “Elliott” denotes the results based on the Elliott's SU(3) model. “$SD$,” “$SDG$,” and “$SDGI$” denote configurations constructed by collective $SD, SDG$, and $SDGI$ nucleon pairs, respectively. “$SD1G1I$” denotes $SD$ nucleon pair subspace coupled to a space with one $G$ and/or one $I$ pair (in addition to $SD$ nucleon pairs).

Figure 2

Squared overlaps, ${\sum }_{{\sigma }_k{J}_k}{P}_{\eta {\sigma }_k{J}_k}^{\left(\alpha \right)}$ [see Eq. (4), where $\eta =\pi$ (or $\nu$), $\sigma$ specifies a nucleon-pair subspace, and $\alpha$ denotes an eigenstate], with the wave functions calculated in the full $SDGI$ pair configuration space. The results in black, red, blue, and green correspond to $0_1^{+}, 2_1^{+}, 4_1^{+}$, and $6_1^{+}$ states, respectively.

Figure 3

Excitation energy and reduced electric transition probability for the $2_1^{+}, 4_1^{+}$, and $6_1^{+}$ states versus the number of selected optimal neutron-pair basis states for systems of $N_{\pi }=N_{\nu }=3$ in the presence of the Hamiltonian of Eq. (7) and in the $pf$ shell. The exact results are derived from the Elliott's SU(3) model according to Eqs. (10) and (11).

Figure 4

Same as Fig. 1 but for the system with eight valence protons and eight valence neutrons in the $pf$ shell.

Figure 5

Same as Fig. 2 but for the system with eight valence protons and eight valence neutrons in the $pf$ shell.

Figure 6

Same as Fig. 3, but for the system with eight valence protons and eight valence neutrons in the $pf$ shell.

Figure 7

Excitation energies of the yrast states for ${}^{146}\mathrm{Ba}$ and ${}^{148}\mathrm{Ce}$. Experimental data are taken from the NNDC [46] database, and “$SD$,” “$SDG$,” “$SD1G1I$,” and “$SDGI$” correspond to calculated results in $SD, SDG, SD1G1I$, and $SDGI$ nucleon-pair configuration spaces.

Figure 8

Contribution from various nucleon-pair configurations in the yrast $0_1^{+}$ and $2_1^{+}$ states of the ${}^{148}\mathrm{Ce}$ nucleus, in wave functions calculated by using the full $SDGI$ nucleon-pair configuration space. Calculation of squared overlaps are performed via the procedure of Sec. 2b in this paper. Left panel corresponds to the $0_1^{+}$ state, and the right panel corresponds to the $2_1^{+}$ state. The results plotted in black corresponds to those with contributions below ${10}^{-6}$.

Figure 9

Squared overlap of each specific nucleon-pair configuration with the wave function obtained in the full $SDGI$-pair space for the $0_1^{+}, 2_1^{+}, 4_1^{+}$, and $6_1^{+}$ states of ${}^{146}\mathrm{Ba}$ and ${}^{148}\mathrm{Ce}$. Panels (a) and (c) correspond to proton configurations, and panels (b) and (d) correspond to neutron configurations. See Sec. 2b for the procedure to calculate the overlaps.

Figure 10

Convergence of calculated excitation energies of ${}^{146}\mathrm{Ba}$ and ${}^{148}\mathrm{Ce}$ in truncated $SDGI$ subspaces versus the number of selected, optimized nucleon-pair basis states $N_{\mathrm{ps}}$ (see Sec. 2b for details of the procedure to optimize the basis states). Panels (a) and (c) correspond to excitation energies (in units of MeV) of a few yrast states, and panels (b) and (d) present corresponding squared overlaps between the eigenstates in the truncated spaces and those in the full $SDGI$-pair space. The solid lines are exponential curves to fit data.