Shape coexistence and shell effect of medium mass nuclei

LIU Dong; GUO Jianyou

doi:10.7498/aps.74.20250095

Abstract
The atomic nucleus is an extremely complex quantum many- body system composed of nucleons, and its shape is determined by the number of nucleons and their interactions. The study of atomic nuclear shapes is one of the most fascinating topics in nuclear physics, providing rich insights into the microscopic details of nuclear structure. Physicists have observed significant shape coexistence phenomena and stable triaxial deformation in isotopes of Zn, Ge, Se, and Kr. This paper aims to delve deeper into the influences of shape coexistence and triaxiality on the ground-state properties of atomic nuclei, as well as to verify new magic numbers. We employ the density-dependent meson-exchange model within the framework of the relativistic Hartree-Bogoliubov (RHB) theory to systematically study the ground-state properties of even-even Zn, Ge, Se, and Kr isotopes with neutron numbers N = 32–42. The calculated potential energy surfaces clearly demonstrate the presence of shape coexistence and triaxial characteristics in theseisotopes. By analyzing the ground-state energy, deformation parameters, two-neutron separation energy, neutron radius, proton radius, and charge radius of the atomic nucleus, we discuss the closure of nuclear shells. Our results reveal that at N = 32, there is anotable abrupt change in the two-neutron separation energy values of ⁶²Zn and ⁶⁴Ge. At N = 34, a significant decrease in the two-neutron separation energy values of ⁶⁸Se and ⁷⁰Kr is observed, accompanied by an abrupt change in their charge radii. Meanwhile, at N = 40, clear signs of shell closure are observed. The maximum specific binding energy may be correlated with the emergence of spherical nuclear structures. The shell closure not only enhances nucleon binding energy but also suppresses nuclear deformation through symmetry constraints. Our findings support N = 40 as a new magic number, and some results also suggest that N = 32 and N = 34 can be new magic numbers. Notably, triaxial deformation plays a crucial role here. Furthermore, we explore the potential correlation between triaxiality and shape coexistence in the ground-state properties of atomic nuclei and analyze the physical mechanisms behind these changes.The discrepancies between current theoretical predictions and experimental data reflect the limitations of modeling higher-order many-body correlations (e.g. three-nucleon forces) and highlight challenges in experimental measurements for extreme nuclear regions(including neutron-rich and near-proton-drip-line regions). Future studies will combine tensor force corrections, large-scale shell model calculations, and high-precision data from next-generation radioactive beam facilities (e.g. FRIB and HIAF) to clarify the interplay among nuclear force parameterization, proton-neutron balance, and emergent symmetry, thereby providing a more comprehensive theoretical framework for studying the nuclear structures under extreme conditions.

Keywords:
shape coexistence /

shell effect /

new magic numbers
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图 1 采用DD-ME2有效相互作用, 通过约束四极变形的三轴RHB计算生成中子数N = 32, N = 34, N = 40的Zn, Ge, Se, Kr同位素的势能曲面. 所有势能面经过归一化处理, 最小值代表基态能量. 等高线由能量相同点连成, 相邻等高线之间的能差为0.6 MeV

Figure 1. The potential energy surfaces of Zn, Ge, Se, Kr isotopes, with neutron numbers N = 32, N = 34 and N = 40. These surfaces are generated through triaxial RHB calculations with constrained quadrupole deformation, employing the DD-ME2 effective interactions. The scale of the potential energy is consistent across all surfaces, and the lowest minimum represents the ground state. The contours join points on the surface with the same energy, and the separation between neighboring contours is 0.6 MeV.

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图 2 采用DD-ME2有效相互作用, 通过约束四极变形的三轴RHB计算生成中子数N = 32到42的Zn, Ge, Se, Kr偶偶核同位素的结合能与比结合能, 并与参考文献[38,39]中的实验值进行对比

Figure 2. The binding energy per nucleon and the total binding energy of even-even Zn, Ge, Se, Kr isotopes in the triaxial RHB calculations using both the DD-ME2 interactions, with comparisons made to experimental data from Ref. [38,39].

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图 3 采用DD-ME2有效相互作用, 通过约束四极变形的三轴RHB计算生成中子数N = 32—42的Zn, Ge, Se, Kr偶偶核同位素的双中子分离能, 并与参考文献[38,39]中的实验值进行对比

Figure 3. The two-neutron separation energy for even-even Zn, Ge, Se, Kr isotopes, obtained through triaxial RHB calculations using the DD-ME2 interactions, with comparisons made to experimental data from Ref. [38,39].

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图 4 采用DD-ME2有效相互作用, 通过约束四极变形的三轴RHB计算生成中子数N = 32—42的Zn, Ge, Se, Kr偶偶核同位素的中子、质子半径与电荷半径, 并与参考文献[38,39]中电荷半径的实验值进行对比

Figure 4. The neutron, proton, and charge radii for even-even Zn, Ge, Se, Kr isotopes in the triaxial RHB calculations using the DD-ME2 interactions, with comparisons made to charge radii experimental data from Ref. [38,39].

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表 1 势能曲面极小值点, 坐标(β, γ)为极小值点, 第一极小值能量最低, 对应基态位置

Table 1. The minima of the potential energy surfaces. The coordinates (β, γ) specify the locations of these minima. The primary minimum, which is the deepest, corresponds to the ground state of the nucleus.

原子核	第一极小值	第二极小值	实验四极形变值β
⁶²Zn	(0.25, 20)	无	0.216
⁶⁴Zn	(0.23, 0)	(0.24, 60)	0.236
⁷⁰Zn	(0.00, 0)	无	0.216
⁶⁴Ge	(0.26, 27)	无	0.259
⁶⁶Ge	(0.25, 60)	(0.24, 0)	0.172
⁷²Ge	(0.00, 0)	(0.21, 60)	0.240
⁶⁶Se	(0.26, 60)	(0.25, 0)
⁶⁸Se	(0.27, 60)	(0.25, 0)	0.242
⁷⁴Se	(0.22, 60)	(0.00, 0)	0.302
⁶⁸Kr	(0.27, 38)	无
⁷⁰Kr	(0.30, 60)	(0.25, 0)
⁷⁶Kr	(0.00, 0)	(0.19, 60)	0.290

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