Jacobi-type transitions in the interacting boson model

We examine, within the framework of the interacting boson model, the structural evolution along the excitation energy across the saddle points of potential wells in a finite nuclear system, and its connection with excited-state quantum phase transitions (ESQPTs). We find that transitions between $\gamma$-rigid and $\gamma$-soft shapes occur as a function of angular momentum, $L$, analogous to the Jacobi transitions occurring in rotating stars. Empirical evidence of these transitions is found in the rotational spectra of ${}^{156–162}\mathrm{Gd}$.

Figure 1

The phase diagram of the IBM.

Figure 2

Potential energy surfaces ($N=15$), according to Eq. (2), for the selected two cases.

Figure 3

(a) Behavior of the effective order parameter $\rho$ as a function of the excitation energy for $L^{\pi }=0^{+}$ in the case with $(\eta ,\chi )=(0.65,-1.3228)$. (b) The same as in panel (a) but for $L^{\pi }=4^{+}$. (c) Behavior of $\rho$ for $L^{\pi }=0^{+}$ in the case with $(\eta ,\chi )=(1.0,-0.5)$. (d) The same as in panel (c) but for $L^{\pi }=4^{+}$. In addition, the dashed red lines are included to guide the eye.

Figure 4

(a) Behavior of the quantity $R/\varepsilon$ as a function of the rotational energy for the ground band (up to $L=20$) and excited bands (up to $L=16$) in the case corresponding to $(\eta ,\chi )=(0.65,-1.3228)$. The curves denoted by $\gamma$ rigid and $\gamma$ soft are those given by $R\propto (4-\frac{2}{L})$ and $R\propto (1+\frac{2}{L})$, respectively. (b) The same as in panel (a) but for $(\eta ,\chi )=(1.0,-0.5)$.

Figure 5

Comparison between the experimental values of $R$ for the ground bands (up to $L=16$) in ${}^{152–162}\mathrm{Gd}$ [34] and IBM calculations. The scale factor $\varepsilon$ is fixed by fitting the experimental values of $E\left(2_1^{+}\right)$ in each case, and values of ($\eta ,\chi$) shown in each panel are taken from Ref. [35] with slightly changes in those for panels (c) and (d).