Shape phase transition at $N=88\text{–}90$ in ${}^{144,146}\mathrm{Ba}$

Background: The neutron-rich Ba isotopes with only six valence protons represent the beginning of the collective rotation-vibration band structure. The sharp shape phase transition at $N=88\text{–}90$ observed in Ce-Gd isotones is not exhibited in ${}^{144\text{–}146}\mathrm{Ba}$, which renders their analysis interesting. Also there are ambiguities in the spin and band assignments.

Purpose: To study their spectra empirically and to compare with predictions from an interacting boson model and microscopic dynamic pairing plus quadrupole model to explain the smooth shape transition at $N=88\text{–}90$.

Method: We compare the results of the calculation in the interacting boson models (IBM-1) and IBM-2 and the dynamic pairing plus quadrupole model with experiment and illustrate the variation in level structure of the Ba isotopes with N. The absence of sharp phase transition at $N=88\text{–}90$ is examined.

Results: The ambiguous spin and parity of levels of the vibrational bands are assigned on the basis of calculated K components and the decay characteristics.

Conclusion: The second $I^{\pi }=2^{+}$ states in ${}^{144,146}\mathrm{Ba}$ have $K=0$ predominance, and $I^{\pi }=2_3^{+}$ states are $K=2$. The smooth transition at $N=88\text{–}90$ is explained.

Figure 1

Level energy $E\left(2_1^{+}\right)$ [4] in the ground band of Ba-Er for $N=88$ and 90 isotones. Maximum rise and fall are at $Z=64$ Gd. At $Z=56$, the change is minimum.

Figure 2

The energy ratio [4] $R_{I/2}\left(=E_I/E_2\right)$ versus spin I in the ground-state bands of ${}^{144,146}\mathrm{Ba}$. The linear curve [Eq. (2)] is for ${}^{146}\mathrm{Ba}$.

Figure 3

The partial energy-level spectra [4] of ${}^{144,146}\mathrm{Ba}$.

Figure 4

The partial energy-level spectra of $N=88$ isotones of Ba and Ce.

Figure 5

Energy levels in ${}^{140\text{–}148}\mathrm{Ba}$ (Ref. [4]).

Figure 6

Energy-level spectra of ${}^{144}\mathrm{Ba}$ compared with IBM-1 (set 1), the DPPQ model, and IBM-2.

Figure 7

Energy-level spectra of ${}^{146}\mathrm{Ba}$ compared with IBM-1 (set 2), the DPPQM, and IBM-2.

Figure 8

The static potential-energy curve $V(\beta ,\gamma =0^{\circ })$ for ${}^{146}\mathrm{Ba}$ from the DPPQ model. The horizontal line labeled ‘ZPE’ denotes zero point energy.