The phase transitions and spectral statistical properties in Nd, Sm, Gd, and Dy isotopes are investigated by spherical mean-field plus standard pairing model. The results of the model calculations successfully reproduce the critical phenomena observed experimentally in the two-neutron separation energy, odd–even mass differences, \(\alpha \)-decay, double \(\beta ^{-}\)-decay energy and the first pairing excitation states of these isotopes with the critical point at the neutron number \(N\sim 90\). As the only parameter in the model, the pairing interaction strength G is determined by fitting the binding energies, the odd–even mass differences and the energies of the first and second pairing excitation states for the \(^{144-157}\)Nd, \(^{146-159}\)Sm, \(^{148-161}\)Gd, and \(^{150-163}\)Dy isotopes. The spectral statistical properties of the excited levels of Sm isotopes show that the quantum chaos exists in \(^{151-153}\)Sm which corresponds to the critical point at \(N\sim 90\). It is inferred that the transitional region is the most sensitive region to perturbation, leading generically to the typical signature of quantum chaos. Moreover, the results indicate that this critical behavior is related not only to the ground-state but also to the excited-state under the present model. It may provide us a microscopic picture that the ground-state phase transition and the quantum chaos behaviors may drive by the competition between the spherical mean-field and the pairing interaction based on the present model for Nd, Sm, Gd, and Dy isotopes.

中文翻译：

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$$ N \ sim 90 $$ N〜90附近原子核的量子相变和光谱统计性质

通过球形平均场加标准配对模型研究了Nd，Sm，Gd和Dy同位素的相变和光谱统计性质。模型计算的结果成功地再现了实验中在两个中子分离能中观察到的临界现象，即奇偶质量差，（\（\ alpha \）-衰变，双\（\ beta ^ {-} \）-衰变能量这些同位素的第一对激发态与临界点在中子数\（N \ sim 90 \）处。作为模型中的唯一参数，配对相互作用强度G通过拟合结合能，奇偶质量差以及第一和第二配对激发态的能量来确定。\（^ {144-157} \） Nd，\（^ {146-159} \） Sm，\（^ {148-161} \） Gd和\（^ {150-163} \） Dy同位素。Sm同位素激发能级的光谱统计性质表明，\（^ {151-153} \） Sm中存在量子混沌，它对应于\（N \ sim 90 \）的临界点。可以推断，过渡区域是对扰动最敏感的区域，通常导致量子混沌的典型特征。而且，结果表明，在本模型下，该临界行为不仅与基态有关，而且与激发态有关。它可以为我们提供一个微观图像，即基于本模型的Nd，Sm，Gd和Dy同位素，球形平均场和配对相互作用之间的竞争可能会驱动基态相变和量子混沌行为。