Quadrupole collectivity in the two-body random ensemble

We conduct a systematic investigation of the nuclear collective dynamics that emerges in systems with two-body random interactions. We explore the development of the mean field and study its geometry. We investigate multipole collectivities in the many-body spectra and their dependence on the underlying two-body interaction Hamiltonian. The quadrupole-quadrupole interaction component appears to be dynamically dominating in the two-body random ensemble. This quadrupole coherence leads to rotational spectral features and thus suggests the formation of the deformed mean-field of a specific geometry.

Figure 1

${(19/2)}^6$. (a) The distribution of the fractional collectivity $b$. (b) The distribution the intrinsic quadru-pole moment $q$. Only realizations with the $0_{\mathrm{gs}},2_{\mathbf{1}}$ spin sequence are included in both panels. There are 10.4% of such realizations. The 7.8% of collective realizations $(b>0.7)$ are shaded.

Figure 2

${(19/2)}^6$. The distribution of the relative transition strength $s$ for the collective realizations (shaded area in Fig. 1). The quadrupole moments shown in the inset are separated into prolate ($q>0.7$) and oblate ($q<-0.7$) shapes. The resulting distributions are shaded with a pattern and a uniform color, respectively. The fraction of oblate cases is 5.2% and the fraction of prolate cases is 1.3% relative to the total number of random realizations.

Figure 3

${(19/2)}^6$. (a) The distribution of the de-excitation ratio $B_{42}$ defined in Eq. (9). (b) The distribution of the excitation energy ratio $R_{42}$ defined in Eq. (8). The distributions are composed of 13.6% of realizations that have the $0_{\mathrm{gs}},2_1,4_1$ sequence with $b>0.7$, the $2_1$ state is not higher than the fourth excited state, and $E\left(4_1\right)>E\left(2_1\right)$. The prolate and oblate cases, which appear in the ensemble with probabilities 3.3% and 7.1% respectively, are shaded with the same patterns as in Fig. 2. The values of $B_{42}$ and $R_{42}$ for the $QQ$ Hamiltonian listed in Table are marked with the vertical grid lines.

Figure 4

${(19/2)}^6$. (a) The distribution of the triaxiality angle $\gamma$. (b) The distribution of the $K$-mixing angle $\Gamma$. (c) The distribution of the triaxiality angle ${\gamma }_{\mathrm{DF}}$ from the Davydov-Filippov model. The angles are obtained from Eqs. (11), (12), and (10). We select realizations with two states of spin 2 in the spectrum and require $b>0.7$ and $q\left(2_1\right)\approx -q\left(2_2\right)$; 18.3% of realizations satisfy this set of restrictions. The realizations with prolate and oblate shapes are shaded with the same patterns as in Figs. 2 and 3. Vertical grid lines indicate the triaxiality parameters calculated from the $QQ$ Hamiltonian, which are $\gamma =9.79{}^{\circ },$$\Gamma =0.73{}^{\circ },$ and ${\gamma }_{\mathrm{DF}}=7.52{}^{\circ }$ from Table .

Figure 5

${(19/2)}^8$. The same figure as Fig. 1 but for the eight-particle system. (a) The distribution of the fractional collectivity $b$. (b) The distribution of the intrinsic quadrupole moment $q$. The histogram is composed of 7.5% of random spectra with $0_{\mathrm{gs}}$ and $2_{\mathbf{1}}$ states. Shaded areas correspond to 4.6% of collective realizations ($b>0.7$) and 1.9% of noncollective realizations ($b<0.7$). This figure is analogous to Figs. 1 and 6, and the same shading is used in these figures.

Figure 6

${(19/2)}^6$. The distributions of the fractional collectivity $b\left(\lambda \right)$ are shown in panels (a), (b), and (c). The distributions of the intrinsic multipole moments $q_{\lambda }$ are shown in panels (d), (e), and (f). The plots are organized in three rows corresponding to multipolarities with $\lambda =4,6,\text{and}8$. Here we include realizations where, in addition to the $0_{\mathrm{gs}}$ state, the first excited state is $4_{\mathbf{1}},$ $6_{\mathbf{1}},$ or $8_{\mathbf{1}}.$ The shaded areas correspond to collective and noncollective modes with $b\left(\lambda \right)>0.7$ and $b\left(\lambda \right)<0.3$, respectively. We use the same patterns as in Fig. 1.

Figure 7

${(19/2)}^8$. The same figure as Fig. 2 but for the eight-particle system. The distribution of the relative transition strength rule $s$ for the collective realizations. This figure is analogous to Fig. 2, and the same shading is used as in Figs. 2 and 3; however, only oblate shapes $(q<-0.7)$ are seen.

Figure 8

${(19/2)}^8$. Probabilities to observe a certain ground-state spin $J_{\mathrm{gs}}$ for three random ensembles: (a) the TBRE, (b) the TBRE without a $J^2$ term [the $\mathcal{K}=1$ term in Eq. (13) is removed], and (c) the TBRE without both, $J^2$ and $QQ$ terms [the $\mathcal{K}=1$ and $\mathcal{K}=2$ terms in Eq. (13) are removed].

Figure 9

${(19/2)}^8$. The distribution of the fractional collectivity $b$ for the same three random ensembles as in Fig. 8, namely (a) the traditional TBRE, (b) the TBRE without a $J^2$ term, and (c) the TBRE without both $J^2$ and $QQ$ terms. We select realizations with the $0_{\mathrm{gs}}$ state followed by the first excited state $2_{\mathbf{1}}$. The fraction of such cases for ensembles (a), (b), and (c) is 7.6, 8.2, and 4.7% respectively.

Figure 10

${(13/2^{+},13/2^{+})}^6$. (a) The distribution of the fractional collectivity $b$. (b) The distribution of the intrinsic quadrupole moment $q$. The $4.1\%$ of samples have the $0_{\mathrm{gs}},2_{\mathbf{1}}$ sequence. Shaded areas correspond to 1.2% of collective realization and to 1.8% of noncollective realizations. This figure is analogous to Figs. 1, 6, and 5, and the same shading is used in these figures.

Figure 11

${(13/2^{+},13/2^{-})}^6$. (a) The distribution of the fractional collectivity $b$. (b) The distribution of the intrinsic quadrupole moment $q$. The $6.9\%$ of samples have the $0_{\mathrm{gs}}$ state and the first state $2_1$, both of positive parity. The $2.4\%$ of collective and $2.6\%$ of noncollective realizations are shaded with patterns. This figure is analogous to Figs. 1, 6, 5, and 10, and the same shading is used in these figures.

Figure 12

${(13/2^{+},13/2^{+})}^{22}$. (a) The distribution of the fractional collectivity $b$. (b) The distribution of the intrinsic quadrupole moment $q$. The system is particle-hole conjugated to that in Fig. 10. The percentage of samples with the $0_{\mathrm{gs}},2_{\mathbf{1}}$ sequence is $8.1\%$ which includes $4.8\%$ of collective and $1.7\%$ of noncollective. This figure is analogous to Figs. 1, 6, 5, 10, and 11, and the same shading is used in these figures.

Figure 13

${(0f_{7/2},1p_{3/2})}^8$. (a) The distribution of the fractional collectivity $b$. (b) The distribution of the intrinsic quadrupole moment $q$. The solid black line outlines the probability distribution for 31% of realizations with the $0_{\mathrm{gs}}$ state followed by the $2_{\mathbf{1}}$ first excited state, both states with isospin $T=0.$ The 8.8% of realizations are collective and the 12.8% are noncollective. This figure is analogous to Figs. 1, 6, 5, 10, 11, 12, and the same shading is used in these figures.

Figure 14

${(0f_{7/2},1p_{3/2})}^8$. The distribution of the relative transition strength $s$ for the collective realizations (shaded with uniform red in Fig. 13). The 3.6% of prolate cases and 1.0% of oblate are identified with shades of color and pattern (see the inset). This figure is analogous to Fig. 2, and the same shading is used as in Figs. 2, 3, 4.

Figure 15

${(0f_{7/2},1p_{3/2})}^8$. (a) The distribution of the de-excitation ratio $B_{42}$ defined in Eq. (9). (b) The distribution of the excitation energy ratio $R_{42}$ defined in Eq. (8). Collective realization discussed in Fig. 13 are selected and, in addition, we require that the second excited state has spin 4. The fraction of such cases is 4.2%, with 2.4% being prolate and 0.6% being oblate, they are shaded separately with the same patterns as in Fig. 14. The values for $B_{42}$ and $R_{42}$ from the $QQ$ Hamiltonian listed in Table are shown with the vertical grid lines. This figure is analogous to Fig. 3, and the same shading is used as in Figs. 2, 3, 4 and 14.

Figure 16

${(0f_{7/2},1p_{3/2})}^8$ with $\delta \epsilon =7$. (a) The distribution of the fractional collectivity $b$. (b) The distribution of the intrinsic quadrupole moment $q$. The system is the same as in Fig. 13, but the single-particles energies are nondegenerate, with $\delta \epsilon \equiv {\epsilon }_{p3/2}-{\epsilon }_{f7/2}=7.$ The solid black line outlines the probability distribution for 15.7% of realizations with the $0_{\mathrm{gs}}$ state followed by the $2_{\mathbf{1}}$ first excited state, both states with isospin $T=0.$ The 2.9% of realizations are collective and the 8.1% are noncollective. This figure is analogous to Figs. 1, 6, 5, 10, 11, 12, and 13, and the same shading is used in these figures.

Figure 17

${(0f_{7/2},1p_{3/2})}^8$. The distribution of the overlap $x$ defined in Eq. (17). The results for all $J_{\mathrm{gs}}=0,T_{\mathrm{gs}}=0$ states are unshaded; the fraction of such realizations is 56.3%. Collective realization that in addition to the $0_{\mathrm{gs}}$ state have the $T=0$, $2_1$ first excited state and $b>0.7$ are shaded (their fraction is 8.8%). Solid line shows the Porter-Thomas distribution, which is expected for the overlap between uncorrelated states.