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Nuclear fission is a decay process by which a heavy nucleus splits into two or more lighter nuclei. It plays a crucial role in the synthesis of superheavy elements, the rapid neutron-capture process, nuclear energy application and so on. The fission barrier is an important property of heavy nuclei, because its height and width directly relate with the lifetime of heavy nuclei, and affect charge yield, mass yield, and kinetic energy of fission fragments. In our study, the potential energy curves of actinide nuclei are obtained from the relativistic density functional theory in 3D lattice when the axial symmetry, reflection symmetry and $V_4$ symmetry are broken in turn. The effects of all the quadrupole and octupole deformation degrees of freedom on the inner barrier, outer barrier, and the fission isomeric state are investigated. It is found that breaking the reflection symmetry can lower the outer fission barriers significantly, breaking the axial symmetry can lower both the inner and outer barriers, breaking the $V_4$ symmetry has little effect on the inner and outer barriers, and the fission isomeric state is almost unaffected by symmetry breaking. Based on the relativistic density functional PC-PK1 and monopole pairing interaction, our results well reproduce the empirical values of the inner and outer barriers extracted from experiments, and the energies of the fission isomeric states are slightly underestimated. All the data presented in this paper is openly available at
https://www.doi.org/10.57760/sciencedb.j00213.00229 .-
Keywords:
- relativistic density functional theory /
- 3D lattice space /
- fission barrier /
- $V_{4}$ symmetry
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图 1 $ Z=94 $的偶偶核中子对隙(蓝色空心方框)和实验上提取的中子经验对隙(黑色实心方框)随质量数的变化. $ N=146 $的偶偶核的质子对隙(红色空心圆)和实验上提取的质子经验对隙(黑色实心圆)随质量数的变化
Figure 1. The neutron pairing gap calculated by 3DRDFT (blue open square) and extracted from experiments (black solid square) versus mass number for the even-even nuclei with $ Z = 94 $. The proton pairing gap calculated by 3DRDFT (red open circle) and extracted from experiments (black solid circle) versus mass number for the even-even nuclei with $ N = 146 $.
图 2 $ ^{240}{\rm{Pu}} $分别在保持轴对称和反射对称性(紫色点划线), 轴对称性(蓝色短划线), $ V_{4} $对称性(红色实线)时的势能曲线以及$ V_{4} $对称性破缺后在裂变内垒, 外垒和同核异能态附近的势能(绿色空心圆). 经验裂变内垒$ B_f^i $, 外垒$ B_f^o $, 和同核异能态$ E_{{\rm{iso}}} $的能量由黑色横线标记. 取$ ^{240}{\rm{Pu}} $的基态能量为$ 0 $
Figure 2. The potential energy curve of $ ^{240}{\rm{Pu}} $ with axial symmetry and reflection symmetry (purple dot-dashed line), axial symmetry (blue dashed line), $ V_{4} $ symmetry (red solid line) and potential energy of $ ^{240}{\rm{Pu}} $ nearby inner barrier, outer barrier, isomeric state with $ V_{4} $ symmetry breaking (green open circle). The empirical inner barrier $ B_{f}^{i} $, outer barrier $ B_{f}^{o} $ and isomeric state $ E_{iso} $ is denoted by the black dash line. The energy is normalized with respect to the energy of the ground state.
表 1 在$ V_{4} $对称性破缺时, 7 个锕系原子核的裂变内垒, 外垒和同核异能态的能量经验值[75,76]$ \Delta E_{{\rm{Exp}}} $和基于3DRDFT计算的能量$ \Delta E_{{\rm{Theo}}} $, 四极形变$ \beta_{20} $, $ \beta_{22} $和八级形变$ \beta_{30} $, $ \beta_{31} $, $ \beta_{32} $, $ \beta_{33} $
Table 1. The energies extracted from experiments $ \Delta E_{{\rm{Exp}}} $ and the energies $ \Delta E_{{\rm{Theo}}} $, quadrupole deformations $ \beta_{20} $, $ \beta_{22} $ and octupole deformations $ \beta_{30} $, $ \beta_{31} $, $ \beta_{32} $, $ \beta_{33} $ of fission inner barrier, outer barrier, and isomeric states of 7 actinide nuclei obtained by 3DRDFT with $ V_{4} $ symmetry breaking.
核素 $ ^{232}{\rm{U}} $ $ ^{234}{\rm{U}} $ $ ^{236}{\rm{U}} $ $ ^{238}{\rm{U}} $ $ ^{236}{\rm{Pu}} $ $ ^{238}{\rm{Pu}} $ $ ^{240}{\rm{Pu}} $ 内垒 $ \Delta E_{{\rm{Exp}}}\ [{\rm{MeV]}} $ 4.90 4.80 5.00 6.30 - 5.60 6.05 $ \Delta E_{{\rm{Theo}}}\ [{\rm{MeV]}} $ 4.59 5.24 5.13 5.70 5.94 5.75 6.12 $ \beta_{20} $ 0.60 0.60 0.60 0.65 0.60 0.60 0.65 $ \beta_{22} $ 0.04 0.06 0.06 0.06 0.06 0.06 0.06 $ \beta_{30} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{31} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{32} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{33} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 外垒 $ \Delta E_{{\rm{Exp}}}\ [{\rm{MeV]}} $ 5.40 5.50 5.67 5.50 - 5.10 5.15 $ \Delta E_{{\rm{Theo}}}\ [{\rm{MeV]}} $ 5.45 6.06 5.58 5.78 5.15 5.02 5.12 $ \beta_{20} $ 1.20 1.20 1.35 1.35 1.20 1.25 1.40 $ \beta_{22} $ 0.03 0.03 0.03 0.03 0.03 0.03 0.02 $ \beta_{30} $ 0.41 0.37 0.39 0.50 0.35 0.30 0.51 $ \beta_{31} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{32} $ 0.02 0.01 0.00 0.00 0.02 0.01 0.01 $ \beta_{33} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 同核异能态 $ \Delta E_{{\rm{Exp}}}\ [{\rm{MeV]}} $ - - 2.3 2.6 - 2.4 2.25 $ \Delta E_{{\rm{Theo}}}\ [{\rm{MeV]}} $ 2.1 1.2 1.2 1.2 1.4 1.4 1.4 $ \beta_{20} $ 0.85 0.90 0.90 1.00 0.90 0.90 0.95 $ \beta_{22} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{30} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{31} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{32} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 $ \beta_{33} $ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
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