In a recent Letter [Phys. Rev. Lett. 130, 052501 (2023)], we suggested the quasi-SU(3) coupling across the $N=50$ and 82 shell gaps as the main source driving shape evolution and phase transitions. The present article extends the discussion in the Letter. We demonstrate that by varying the coupling strengths of the quasi-SU(3) partner orbits, $(1g_{9/2},$ $2d_{5/2})$ for protons and $(1h_{11/2}$, $2f_{7/2})$ for neutrons, we can well reproduce the features of the IBM U(5), O(6), SU(3), and $\overline{\mathrm{SU}\left(3\right)}$ symmetries and the Iachello E(5), X(5), and $\overline{\mathrm{X}\left(5\right)}$ with critical behavior. Therefore, all known dynamical and critical-point symmetries are realized in our shell-model parameter space. Moreover, in the phase diagram of the parameter space, we find that the line connecting U(5), E(5), and O(6) is characterized by the vanishing spectroscopic quadrupole moment of the first $2^{+}$ state [i.e., $Q_s\left(2_1^{+}\right)=0]$. It gives a new physical meaning to the central axis of the extended Casten triangle and implies that any shape evolution passing through this line can be naturally considered as a prolate-oblate shape phase transition. Thus, we have found microscopic origins for all symmetries originally suggested by algebraic or geometric models. By visualizing the shape using the constrained Hartree-Fock-Bogolyubov (HFB) method based on our shell-model results, we conclude that ${}^{134}\mathrm{Ba}$ is likely the softest nucleus in the nuclear chart, having no definite shape at all in its ground state.

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• Nuclear shapes and moments
• Nuclear structure & decays
• 90 ≤ A ≤ 149
• Shell model