Triaxial rotor in the SU(3) limit of the interacting boson model

A mapping from a triaxial rotor Hamiltonian to that of the SU(3) limit description in the interacting boson model (IBM) is established, which is achieved by the SU(3) realization of the triaxial rotor. A detailed comparison between the triaxial dynamics generated from the quadrupole-deformed rotor and those from the IBM image is made. The results indicate that the mapping can be well realized. A preliminary test for ${}^{128}\mathrm{Ba}$ further confirms the finite-$N$ effect of the mapping. It thus provides an alternative way to understand the triaxiality in the finite-$N$ system and additional insight into understanding the SU(3) IBM theory from microscopic point of view via the SU(3) shell model.

Figure 1

The level patterns of the ground-state and $\gamma$ bands obtained from the quadrupole-deformed rotors and the corresponding IBM images, where all the energy values are normalized to that of the first $2^{+}$ state in each case. (a) The levels in the ground-state band for $\gamma =16.{47}^{\circ }$, where $R_a$ and $R_b$ represent the results obtained from the rotor model of the rigid type with $\beta =1.0$ and $0.5$, respectively; $R_c$ represents those from the rotor model of irrotational type; and IBM represents those obtained from the IBM images of the corresponding rotor model. (b) The same as in (a) but for $24.{00}^{\circ }$; (c) the same as in (a) but for the $\gamma$ band; (d) the same as in (c) but for $\gamma =24.{00}^{\circ }$. In the IBM calculations, $t^{\prime }=9.0$ keV and $t=41.7$ keV are used for the case of $\gamma =16.{47}^{\circ }$ corresponding to the SU(3) irrep (16,6) with $N=14$, while $t^{\prime }=9.0$ keV and $t=33.4$ keV are used for the case of $\gamma =24.{00}^{\circ }$ corresponding to the SU(3) irrep (12,8).

Figure 2

The reduced intraband $E2$ transitional rate $B(E2;L\to L-2)$ in the ground-state band obtained from each case given in Fig. 1 as a function of the angular momentum $L$, where IBM represents the results obtained from the IBM Hamiltonian (23) with a given $\gamma$ value since the IBM images of the quadrupole-deformed rotor of different types at the same $\gamma$ deformation result in the same eigenstate (see the text). All the results have been normalized to $B(E2;2_1\to 0_1)=100$.

Figure 3

The same as Fig. 2 but for the transitions $B(E2;L\to L-2)$ in the $\gamma$ band.

Figure 4

The same as in Fig. 2 but for the transitions $B(E2;L\to L-1)$ in the $\gamma$ bands.

Figure 5

Two typical interband $B\left(E2\right)$ ratios.

Figure 6

(a) The reduced $E2$ transitional rate $B(E2;L\to L-2)$ in the ground-state band obtained from each case changing as a function of $L$, where $R_a$ and $R_b$ represent the results as those shown in Fig. 2 in the case of $\gamma ={24}^{\circ }$, $\mathrm{IBM}\left(N=14\right)$ represents the results obtained from the correspond IBM image with the SU(3) irrep (16,6) and $N=14$, $\mathrm{IBM}\left(N=21\right)$ represents the IBM results with the SU(3) irrep (18,12) and $N=21$, and $\mathrm{IBM}\left(N=38\right)$ represents the IBM results with the SU(3) irrep (32,22) and $N=38$. (b) The same as in (a) but for the $\gamma$ band; (c) the same as (b) but for $L=\mathrm{odd}$.

Figure 7

A typical level pattern of the triaxial rotor described by (1) and that of the corresponding IBM image described by (23) for $N=14$. In the calculation, the inertia parameters in the triaxial rotor model are taken as $A_1=10.0$ keV, $A_2=8.0$ keV, and $A_3=14.0$ keV, and the parameters in the IBM are taken as $t^{\prime }=15.0$ keV and $t=55.7$ keV.

Figure 8

Some low-lying levels of ${}^{128}\mathrm{Ba}$ involving those in the ground-state and $\gamma$ band, where the results calculated from the irrotational-type rotor at $\gamma ={22}^{\circ }$ and $\beta =0.236$ are labeled Rotor, and those calculated from the IBM within the irrep (12,2) and (8,4). The parameters in the IBM Hamiltonian corresponding to (12,2) are given as $t=50$ keV, $t^{\prime }=7.5$ keV, $a=108.7$ keV, $b=10.97$ keV, and $c=-0.326$ keV, and those corresponding to (8,4) are given as $t=50$ keV, $t^{\prime }=10.8$ keV, $a=90.1$ keV, $b=11.9$ keV, and $c=0.355$ keV. All the level energies have been normalized to that of the $2_1$ state for each case and the experimental data are taken from Ref. [32].

Figure 9

(a) The $B\left(E2\right)$ values (in W.u.) in the ground-state band corresponding to the cases shown in Fig. 8. (b) The same as in (a) but for the $\gamma$ band. The experimental data are taken from Ref. [32], and the theoretical results are normalized to the experimental value of $B(E2;2_1\to 0_1)$.