Impact of pairing interactions on fission in the deformed mean-field plus standard pairing model

The role of pairing interactions on fission barriers and static fission paths in Th, U and Pu isotopes is investigated by using the deformed mean-field plus standard pairing model. It is found that pairing interaction plays different roles in different stages of the fission process. The first and second saddle point of the fission process are sensitive to the pairing interaction strength, while the ground-state and prescission points are mainly driven by nuclear deformation. It is also demonstrated that pairing interactions play a crucial role in the region of the inner and outer barriers. The barrier heights obtained by using the present model are found to reproduce the experimental data remarkably well and yield better results than those obtained by using the BCS approximation, thus indicating the importance of exact solutions of pairing interactions. Through numerical analysis, we provide possible microscopic pictures of spontaneous fission in the current pairing model.

Figure 1

The odd-even mass differences (in MeV) for ${}^{220–235}\mathrm{Th}$, ${}^{222–237}\mathrm{U}$ and ${}^{224–239}\mathrm{Pu}$. Experimental values are denoted by “Exp.”, and are taken from [35], and theoretical values calculated in the extended pairing model are denoted by “Th.”.

Figure 2

Contour map of the potential-energy surface of the nucleus ${}^{226}\mathrm{Th}$ (in MeV), minimized over (${\alpha }_4,{\alpha }_5,{\alpha }_6$) with the pairing strengths $G^{\nu }=0.1$ and $G^{\pi }=0.15$ (in MeV). The red trajectory shows the static fission path. Positions of the ground state, the first saddle point, the second saddle point, and the prescission point are indicated by the black balls. Energies are plotted relatively to the ground-state value.

Figure 3

The energy curves constrained by the elongation $\epsilon$ and minimized in the remaining degrees of freedom (${\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6$) for ${}^{226}\mathrm{Th}$ with different pairing strengths $G^{\nu \left(\pi \right)}$ (in MeV). Energies are normalized to zero energy at the ground-state value.

Figure 4

Neutron and proton pairing correction energies $E_{\mathrm{pair}}^{\nu \left(\pi \right)}$ calculated from Eq. (1) in ${}^{226}\mathrm{Th}$ as functions of $\epsilon$ deformation for indicated values of $G^{\nu }\left(G^{\pi }\right)$ (in MeV). The arrows show the gap between the $E_{\mathrm{pair}}^{\nu \left(\pi \right)}$ values with different $G^{\nu }\left(G^{\pi }\right)$ values (in MeV).

Figure 5

The inner and outer barrier heights of ${}^{226}\mathrm{Th}$ as functions of $G^{\nu \left(\pi \right)}$ (in MeV).

Figure 6

The barrier heights as a function of $G^{\nu }$ with $G^{\pi }=0.05$ MeV (left panel) and as a function of $G^{\pi }$ with $G^{\nu }=0.05$ MeV (right panel).

Figure 7

Experimental and calculated heights of the inner and outer fission barriers in actinide nuclei. Experimental data are taken from Ref. [43] (in MeV). A typical uncertainty in the experimental values, as suggested by the differences among various compilations, is of the order of 0.5 MeV [43].

Figure 8

The information entropy $I_H$ for the four characteristic points of ${}^{226}\mathrm{Th}$ as functions of $G^{\nu }\left(G^{\pi }\right)$ (in MeV) in the present model. The blue balls are the realistic values of $G^{\nu }=0.1$ MeV and $G^{\pi }=0.15$ MeV.

Figure 9

The derivative of information entropy $d{I}_H/dG$ for the four characteristic points of ${}^{226}\mathrm{Th}$ as functions of $G^{\nu }\left(G^{\pi }\right)$ (in MeV). The black ball are the realistic values $G^{\nu }=0.1$ MeV and $G^{\pi }=0.15$ MeV. The blue balls are the realistic values of $G^{\nu }=0.1$ MeV and $G^{\pi }=0.15$ MeV. The shadowed area indicating the width of the peak in $d{I}_H/dG$ is provided to guide to the eye.

Figure 10

The derivative of information entropy $d{I}_H/dG$ for the second saddle point as functions of $G^{\nu }\left(G^{\pi }\right)$ (in MeV) with different octupole deformation parameter ${\alpha }_3$ values. The blue balls are the realistic values of $G^{\nu }=0.1$ MeV and $G^{\pi }=0.15$ MeV.