Elimination of degeneracy in the $\gamma$-unstable Bohr Hamiltonian in the presence of an extended sextic potential

In this paper, the $\gamma$-unstable Bohr Hamiltonian is studied in the presence of an extended sextic potential. It is expected that in this situation, we reach some degenerate levels in the energy spectrum, but we are interested in a case in which there is no degeneracy between those levels. This goal can be achieved by including the effects of an interaction made by the Casimir operator of SO(3). When this term is involved in the $\gamma$-unstable Bohr Hamiltonian, the degeneracy will be removed. According to the new scheme of energy levels, after the elimination of degeneracy, some low-lying excited levels are introduced. Then we examine the results by reproducing the experimental data of some isotopes for energy levels and $E2$ electromagnetic transitions.

Figure 1

The upper plot of this figure shows Eq. (22) when $\kappa =1$ and $\eta =0.0$. In this case, there is degeneracy between some levels: those have the same seniority number. The lower plot shows Eq. (22) when $\kappa =1$ and $\eta =0.5$. In this case, the degeneracy has been removed. There are three boxes in the lower plot. These boxes show how we should recognize the ground, $\gamma$ and $\beta$ bands in the considered model. The values are normalized to the first excited level of the ground band.

Figure 2

Behavior of $\frac{4_g}{2_g}$ as a function of the free parameters $\kappa$ for different values of $\eta$.

Figure 3

Energy scheme of ${}^{118}\mathrm{Xe}$. In the fitting process, we used the levels which are connected to their corresponding experimental values. Experimental data for the energy levels were taken from Ref. [21]. The values have been normalized to the first excited levels of the ground band.

Figure 4

Energy scheme of ${}^{120}\mathrm{Xe}$. In the fitting process, we used the levels which are connected to their corresponding experimental values. Experimental data for the energy levels were taken from Ref. [21]. The values have been normalized to the first excited levels of the ground band.

Figure 5

Energy scheme of ${}^{122}\mathrm{Xe}$. In the fitting process, we used the levels which are connected to their corresponding experimental values. Experimental data for the energy levels were taken from Ref. [21]. The values have been normalized to the first excited levels of the ground band.

Figure 6

Energy scheme of ${}^{124}\mathrm{Xe}$. In the fitting process, we used the levels which are connected to their corresponding experimental values. Experimental data for the energy levels were taken from Ref. [21]. The values have been normalized to the first excited levels of the ground band.

Figure 7

Energy scheme of ${}^{126}\mathrm{Xe}$. In the fitting process, we used the levels which are connected to their corresponding experimental values. Experimental data for the energy levels were taken from Ref. [21]. The values have been normalized to the first excited levels of the ground band.

Figure 8

Energy scheme of ${}^{128}\mathrm{Xe}$. In the fitting process, we used the levels which are connected to their corresponding experimental values. Experimental data for the energy levels were taken from Ref. [21]. The values have been normalized to the first excited levels of the ground band.

Figure 9

$S\left(4\right)$ as a function of the free parameters $\kappa$ and $\eta$.

Figure 10

Comparison between the theoretical prediction and experimental staggering for ${}^{118}\mathrm{Xe}$.

Figure 11

Comparison between the theoretical prediction and experimental staggering for ${}^{120}\mathrm{Xe}$.

Figure 12

Comparison between the theoretical prediction and experimental staggering for ${}^{122}\mathrm{Xe}$.

Figure 13

Comparison between the theoretical prediction and experimental staggering for ${}^{124}\mathrm{Xe}$.

Figure 14

Comparison between the theoretical prediction and experimental staggering for ${}^{126}\mathrm{Xe}$.

Figure 15

Comparison between the theoretical prediction and experimental staggering for ${}^{128}\mathrm{Xe}$.

Figure 16

$E2$ electromagnetic transition rate scheme of ${}^{118}\mathrm{Xe}$, shown by purple arrows which are normalized to $B(E2;4_g\to 2_g)$.

Figure 17

$E2$ electromagnetic transition rate scheme of ${}^{120}\mathrm{Xe}$, shown by purple arrows which are normalized to $B(E2;4_g\to 2_g)$.

Figure 18

$E2$ electromagnetic transition rate scheme of ${}^{122}\mathrm{Xe}$, shown by purple arrows which are normalized to $B(E2;4_g\to 2_g)$.

Figure 19

$E2$ electromagnetic transition rate scheme of ${}^{124}\mathrm{Xe}$, shown by purple arrows which are normalized to $B(E2;4_g\to 2_g)$.

Figure 20

$E2$ electromagnetic transition rate scheme of ${}^{128}\mathrm{Xe}$, shown by purple arrows which are normalized to $B(E2;4_g\to 2_g)$.