Figure 1

Distribution of $R_I$ and $R_4$ for boson numbers $N_B=6$–11. For each $N_B$, we perform the calculations by 60 000 sets of the TBRE. In addition to two very sharp vibrational and rotational peaks [denoted by solid black circles in each panel, i.e., $(R_4,R_6)=(2.0,3.0)$ and (3.3, 7.0), $(R_4,R_8)=(2.0,4.0)$ and (3.3, 12.0)], three linear correlations between $R_I$ ($I=6,8$) and $R_4$ are easily noticed. We denote these new correlations by $\alpha$, $\beta ,$ and $\gamma$. The slopes of solid lines corresponding to the $\alpha$, $\beta ,$ and $\gamma$ correlations are 3, $\frac{21}{10}$, and $\frac{9}{5}$ in the ($R_4$, $R_6$) plots; and they are 6, $\frac{18}{5}$ and $\frac{138}{35}$ in the ($R_4$, $R_8$) plots. See the text for details. Another relatively smaller “peak” at $R_I\simeq 1.0$ ($I=4,6,8$) is denoted by “S” (called the seniority-type correlation in Ref. [11]) for $N_B=11$.Reuse & Permissions

Figure 2

Correlations between the $E_{4_1^{+}}$ and $E_{2_1^{+}}$, and between $B(E2,4_1^{+}\to 2_1^{+})$ and $B(E2,2_1^{+}\to 0_1^{+})$, for random samples corresponding the correlation $\alpha$. Here we include all samples with $|R_6-3R_4+3|<0.01$ and boson numbers from 6 to 20. The red line in panel (a), $E_{4_1^{+}}=2.03E_{2_1^{+}}+0.87$, and that in panel (b), $B(E2,4_1^{+}\to 2_1^{+})=1.654B(E2,2_1^{+}\to 0_1^{+})$, are obtained by $\chi {}^2$ fitting. The effective charge is taken to be 1, and $\chi$ is assumed to be a uniformed distributed random number between $-\frac{\sqrt{7}}{2}$ and $\frac{\sqrt{7}}{2}$.Reuse & Permissions

Figure 3

The probability of the correlation $\beta$ (solid squares in black) and that of spin-zero ground states (solid squares in red) obtained by the TBRE Hamiltonian for $N_B=6$–20. One sees that both probabilities exhibit staggerings of ${\delta }_{N_B}=3$.Reuse & Permissions

Figure 4

Distribution of two-body interactions $V_{0202}^{(2)}$ and $V_{2222}^{(2)}$ for random samples of the correlation $\beta$ for $N_B=6$. The histogram plot in red corresponds to the distribution of $V_{2222}^{(2)}$, that in green corresponds to the distribution of $V_{0202}^{(2)}$, and the plot in black corresponds to the TBRE. All distributions are normalized by using $\int P(V,\beta )\mathit{dV}=1$. One sees the repulsive $V_{0202}^{(2)}$ and attractive $V_{2222}^{(2)}$ in the correlation $\beta$.Reuse & Permissions

Figure 5

Distribution of $d$-boson number ($n_d$) and that of $d$-boson seniority ($\tau$) for the first $I=0$, 2, 4, 6, and 8 states satisfying the correlation $\beta$, for $N_B=6$. The probabilities, $P(n_d)$ and $P(\tau )$, are normalized to 1. One sees that $n_d\simeq N_B=6$ and that the value of $\tau$ is dominated by $\tau =n_d=6$ except $I=2$ (plot in red).Reuse & Permissions

Figure 6

Similar to Fig. 5, except for the correlation $\gamma$ with $N_B=8$. One sees that these yrast states are dominated by $d$ bosons, with $\tau =0$, 2, 2, 4, and 4 for $I=0$, 2, 4, 6, and 8. The probabilities $P(n_d)$ and $P(\tau )$ are normalized to 1.Reuse & Permissions

Figure 7

Distribution of two-body interactions $V_{0202}^{(2)}$ and $V_{2222}^{(0)}$ for random samples of the correlation $\gamma$ with $N_B=8$. The histogram plot in red corresponds to the distribution of $V_{2222}^{(0)}$, that in green corresponds to the distribution of $V_{0202}^{(2)}$, and the plot in black corresponds to the TBRE. All distributions are normalized for $\int P(V,\gamma )\mathit{dV}=1$. One sees the repulsive $V_{0202}^{(2)}$ and attractive $V_{2222}^{(0)}$ in the correlation $\gamma$.Reuse & Permissions

Figure 8

Similar to Fig. 5 except for the restriction of $R_I\simeq R_4\simeq 1$ with $N_B=11$. One sees that the ground states are dominated by $s$ bosons and the yrast states with $I=2,4,6,$ and 8 are dominated by $d$ bosons. The dominant value of $\tau \simeq 1$, 3, 3, 5 (minimal values of $\tau$), and 11 (maximum of $\tau$) in the $2_1^{+}$, $4_1^{+}$, $6_1^{+}$, and $8_1^{+}$ states, respectively. The probabilities $P(n_d)$ and $P(\tau )$ are normalized to 1.Reuse & Permissions

Figure 9

Distribution of two-body interactions $V_{0202}^{(2)}$ and $V_{0000}^{(0)}$ for random samples of $R_I\simeq R_4\simeq 1$ with $N_B=11$. The histogram plot in red corresponds to the distribution of $V_{0000}^{(0)}$, that in green corresponds to the distribution of $V_{0202}^{(2)}$, and the plot in black corresponds to the TBRE. All distributions are normalized for $\int P(V,S)\mathit{dV}=1$. One sees the repulsive $V_{0202}^{(2)}$ and attractive $V_{0000}^{(0)}$ in the seniority-type correlation.Reuse & Permissions