Microscopic self-consistent description of induced fission: Dynamical pairing degree of freedom

The role of dynamical pairing in induced fission dynamics is investigated using the time-dependent generator coordinate method in the Gaussian overlap approximation, based on the microscopic framework of nuclear energy density functionals. A calculation of fragment charge yields for induced fission of ${}^{228}\mathrm{Th}$ is performed in a three-dimensional space of collective coordinates that, in addition to the axial quadrupole and octupole intrinsic deformations of the nuclear density, also includes an isoscalar pairing degree of freedom. It is shown that the inclusion of dynamical pairing has a pronounced effect on the collective inertia, the collective flux through the scission hypersurface, and the resulting fission yields, reducing the asymmetric peaks and enhancing the contribution of symmetric fission, in better agreement with the empirical trend.

Figure 1

Two-dimensional projections of the deformation-energy manifold of ${}^{228}\mathrm{Th}$ on the quadrupole-octupole axially symmetric plane, calculated with the RMF $+$ BCS model based on the functional DD-PC1 for selected values of the pairing coordinate ${\lambda }_2$. Contours join points on the surface with the same energy, and the separation between neighboring contours is 1 MeV. The red curves denote static fission paths of minimum energy for each value of ${\lambda }_2$.

Figure 2

Pairing gaps for neutrons ${\mathrm{\Delta }}_n$ (upper panel) and protons ${\mathrm{\Delta }}_p$ (lower panel) along the static fission path as functions of the axial quadrupole deformation for selected values of the isoscalar pairing collective coordinate ${\lambda }_2$.

Figure 3

Perturbative cranking masses ${\mathcal{M}}_{11}^{Cp},{\mathcal{M}}_{22}^{Cp}$, and the nonperturbative cranking mass ${\mathcal{M}}_{33}^C$ (in ${\hslash }^2{\mathrm{MeV}}^{-1}$) (logarithmic scale) along the static fission path for several values of ${\lambda }_2$.

Figure 4

The scission contour of ${}^{228}\mathrm{Th}$ on the (${\beta }_2,{\beta }_3$) deformation plane for several values of the collective pairing coordinate ${\lambda }_2$.

Figure 5

Charge yields for induced fission of ${}^{228}\mathrm{Th}$, calculated in the 3D collective space built from the deformation ${\beta }_2,{\beta }_3$ and dynamical pairing ${\lambda }_2$ coordinates (solid red curve). The yields are shown in comparison to the results obtained in the 2D space of shape degrees of freedom ${\beta }_2$ and ${\beta }_3$ with static pairing correlations adjusted to empirical ground-state pairing gaps (100% pairing strength) and enhanced by 10% (110% pairing strength). The data for photoinduced fission correspond to photon energies in the interval 8–14 MeV and peak value of $E_{\gamma }=11\mathrm{MeV}$ [45].

Figure 6

Two-dimensional deformation-energy surface of ${}^{228}\mathrm{Th}$ on the ${\beta }_2\text{−}{\beta }_3$ axially symmetric plane, calculated with the RMF $+$ BCS model based on the functional DD-PC1 and with normal (100%) and enhanced (110%) static pairing strengths as determined by the empirical pairing gaps. Contours join points on the surface with the same energy, and the separation between neighboring contours is 1 MeV.

Figure 7

Time-integrated collective flux $B\left({\lambda }_2\right)$ Eq. (10) through the scission contour as a function of the pairing collective coordinate ${\lambda }_2$.