In this paper we study the low-lying states of neutron-rich ${}^{122,124,126,128}\mathrm{Cd}$ and ${}^{130,132,134,136,138}\mathrm{Cd}$ within the nucleon-pair approximation of the shell model. We adopt the phenomenological Hamiltonian for ${}^{122,124,126,128}\mathrm{Cd}$, and the shell-model effective interaction $jj46$ for ${}^{130,132,134,136,138}\mathrm{Cd}$. The available experimental excitation energies and quadrupole transition probabilities are well reproduced by our calculation, and we also make predictions for very neutron-rich Cd nuclei. Based on our calculation, the $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ values exhibit an asymmetric feature with respect to the $N=82$ shell closure, which mainly comes from the contributions of the proton transition matrix elements. We also investigate for the eight open-shell Cd nuclei, whether two low-lying yrast states with spins differing by 2 can be connected by a quadrupole-phonon excitation, and here we take the proton and neutron quadrupole operators multiplied by $1/r_0^2$ as our quadrupole phonon operators. We calculate explicit overlaps between the low-lying yrast states and the constructed quadrupole-phonon states based on the low-lying yrast states with lower spins, and the results indicate that, for all eight open-shell Cd nuclei, $|2_1^{+}\rangle$ and $|4_1^{+}\rangle$ can be well described to be the states constructed by coupling the proton or neutron phonon to $|0_{\mathrm{g}.\mathrm{s}.}^{+}\rangle$ and $|2_1^{+}\rangle$, respectively. Very interestingly, for ${}^{126,124,122}\mathrm{Cd}$ and ${}^{136,138}\mathrm{Cd}$, the $2_1^{+}$ state can be well described to be both the proton phonon state and the neutron phonon state, which indicates a nonorthogonal feature of these two phonon states. We further present an analytic relation for the overlap between these two phonon states, which implies that the proton and neutron phonon states constructed using the quadrupole operators and the $0_{\mathrm{g}.\mathrm{s}.}^{+}$ state in an open-shell nucleus are almost impossibly orthogonal.

• Received 5 March 2021
• Revised 1 May 2021
• Accepted 11 August 2021

DOI:https://doi.org/10.1103/PhysRevC.104.034312