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Shapes and shape evolution in the mass-130 region, including the Te, Xe, and Ba isotopes, have long been a focus of discussion in nuclear physics. This mass region consists of complex many-body systems that can behave in astonishingly simple and regular ways, as classified in the Casten symmetry triangle. By applying the shell model Hamiltonian proposed recently, we carry out calculations using the Hartree-Fock-Bogolyubov plus generator coordinate method, in the large model space containing the (1g_{9/2},1g_{7/2},2d_{5/2},2d_{3/2},3s_{1/2},1h_{11/2},2f_{7/2}) orbits. Based on good reproduction of the experimentally known energy levels, spectroscopic quadrupole moments, and E2 transition probabilities, we identify the quasi-SU(3) couplings across the N=50 and 82 shell gaps, which play a role in driving shape evolution and phase transition discussed in the extended Casten triangle. Specifically, we demonstrate that the quasi-SU(3) coupling mechanism in the proton partner orbits (1g_{9/2}, 2d_{5/2}) tends to drive the system to be more γ soft, and that in the neutron partner orbits (1h_{11/2}, 2f_{7/2}) are responsible for the oblate-to-prolate shape phase transition. With an emphasis on discussing spectroscopic quadrupole moments, our Letter uncovers hidden symmetries from the vast shell-model configurations and adds microscopical insights into the empirical symmetry triangle.

β decay of proton-rich nuclei plays an important role in exploring isospin mixing. The β decay of ^{26}P at the proton drip line is studied using double-sided silicon strip detectors operating in conjunction with high-purity germanium detectors. The T=2 isobaric analog state (IAS) at 13 055 keV and two new high-lying states at 13 380 and 11 912 keV in ^{26}Si are unambiguously identified through β-delayed two-proton emission (β2p). Angular correlations of two protons emitted from ^{26}Si excited states populated by ^{26}P β decay are measured, which suggests that the two protons are emitted mainly sequentially. We report the first observation of a strongly isospin-mixed doublet that deexcites mainly via two-proton decay. The isospin mixing matrix element between the ^{26}Si IAS and the nearby 13 380-keV state is determined to be 130(21) keV, and this result represents the strongest mixing, highest excitation energy, and largest level spacing of a doublet ever observed in β-decay experiments.

β-delayed one-proton emissions of ^{22}Si, the lightest nucleus with an isospin projection T_{z}=-3, are studied with a silicon array surrounded by high-purity germanium detectors. Properties of β-decay branches and the reduced transition probabilities for the transitions to the low-lying states of ^{22}Al are determined. Compared to the mirror β decay of ^{22}O, the largest value of mirror asymmetry in low-lying states by far, with δ=209(96), is found in the transition to the first 1^{+} excited state. Shell-model calculation with isospin-nonconserving forces, including the T=1, J=2, 3 interaction related to the s_{1/2} orbit that introduces explicitly the isospin-symmetry breaking force and describes the loosely bound nature of the wave functions of the s_{1/2} orbit, can reproduce the observed data well and consistently explain the observation that a large δ value occurs for the first but not for the second 1^{+} excited state of ^{22}Al. Our results, while supporting the proton-halo structure in ^{22}Al, might provide another means to identify halo nuclei.

For mirror nuclei with masses A=42-95, the effects of isospin-nonconserving nuclear forces are studied with the nuclear shell model using the Coulomb displacement energy and triplet displacement energy as probes. It is shown that the characteristic behavior of the displacement energies can be well reproduced if the isovector and isotensor nuclear interactions with J=0 and T=1 are introduced into the f(7/2) shell. These forces, with their strengths being found consistent with the nucleon-nucleon scattering data, tend to modify nuclear binding energies near the N=Z line. At present, no evidence is found that these forces are needed for the upper fp shell. Theoretical one- and two-proton separation energies are predicted accordingly, and locations of the proton drip line are thereby suggested.

The anomaly in Coulomb energy differences (CEDs) between the isospin T=1 states in the odd-odd N=Z nucleus 70Br and the analogue states in its even-even partner 70Se has remained a puzzle. This is a direct manifestation of isospin-symmetry breaking in effective nuclear interactions. Here, we perform large-scale shell-model calculations for nuclei with A=66 to 78 using the new filter diagonalization method based on the Sakurai-Sugiura algorithm. The calculations reproduce well the experimental CED. The observed negative CED for A=70 are accounted for by the cross-shell neutron excitations from the fp shell to the g(9/2) intruder orbit with the enhanced electromagnetic spin-orbit contribution at this special nucleon number.