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.

中文翻译：

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富中子 Cd 同位素低位态的四极声子激发和跃迁概率

在本文中，我们研究了富含中子的低洼状态 ${}^{122,124,126,128}\mathrm{光盘}$ 和 ${}^{130,132,134,136,138}\mathrm{光盘}$在壳模型的核子对近似内。我们采用现象学哈密顿量${}^{122,124,126,128}\mathrm{光盘}$, 和壳模型有效交互 $jj46$ 为了 ${}^{130,132,134,136,138}\mathrm{光盘}$. 我们的计算很好地再现了可用的实验激发能量和四极跃迁概率，我们还对非常富含中子的 Cd 核进行了预测。根据我们的计算，$乙(乙2;0_{\mathrm{G}.\mathrm{秒}.}^{+}\to 2_1^{+})$ 值表现出相对于 $N=82$壳闭合，主要来自质子跃迁矩阵元素的贡献。我们还研究了八个开壳 Cd 核，是否可以通过四极-声子激发连接两个自旋相差 2 的低位 yrast 状态，这里我们将质子和中子四极算子乘以$1/r_0^2$作为我们的四极声子算子。我们根据自旋较低的低层 yrast 态计算了低层 yrast 态和构建的四极声子态之间的显式重叠，结果表明，对于所有八个开壳 Cd 核，$|2_1^{+}〉$ 和 $|4_1^{+}〉$ 可以很好地描述为通过将质子或中子声子耦合到 $|0_{\mathrm{G}.\mathrm{秒}.}^{+}〉$ 和 $|2_1^{+}〉$， 分别。非常有趣的是，对于${}^{126,124,122}\mathrm{光盘}$ 和 ${}^{136,138}\mathrm{光盘}$， 这 $2_1^{+}$状态可以很好地描述为质子声子状态和中子声子状态，这表明这两种声子状态的非正交特征。我们进一步提出了这两种声子态之间重叠的解析关系，这意味着使用四极算子构建的质子和中子声子态和$0_{\mathrm{G}.\mathrm{秒}.}^{+}$ 开壳核中的状态几乎不可能正交。