Ternary-fragmentation-driving potential energies of ${}^{252}\mathrm{Cf}$

Within the framework of a simple macroscopic model, the ternary-fragmentation-driving potential energies of ${}^{252}\mathrm{Cf}$ are studied. In this work, all possible ternary-fragment combinations of ${}^{252}\mathrm{Cf}$ are generated by the use of atomic mass evaluation-2016 (AME2016) data and these combinations are minimized by using a two-dimensional minimization approach. This minimization process can be done in two ways: (i) with respect to proton numbers ($Z_1$, $Z_2$, $Z_3$) and (ii) with respect to neutron numbers ($N_1$, $N_2$, $N_3$) of the ternary fragments. In this paper, the driving potential energies for the ternary breakup of ${}^{252}\mathrm{Cf}$ are presented for both the spherical and deformed as well as the proton-minimized and neutron-minimized ternary fragments. From the proton-minimized spherical ternary fragments, we have obtained different possible ternary configurations with a minimum driving potential, in particular, the experimental expectation of Sn + Ni + Ca ternary fragmentation. However, the neutron-minimized ternary fragments exhibit a driving potential minimum in the true-ternary-fission (TTF) region as well. Further, the $Q$-value energy systematics of the neutron-minimized ternary fragments show larger values for the TTF fragments. From this, we have concluded that the TTF region fragments with the least driving potential and high $Q$ values have a strong possibility in the ternary fragmentation of ${}^{252}\mathrm{Cf}$. Further, the role of ground-state deformations (${\beta }_2$, ${\beta }_3$, ${\beta }_4$, and ${\beta }_6$) in the ternary breakup of ${}^{252}\mathrm{Cf}$ is also studied. The deformed ternary fragmentation, which involves $Z_3=12–19$ fragments, possesses the driving potential minimum due to the larger oblate deformations. We also found that the ground-state deformations, particularly ${\beta }_2$, strongly influence the driving potential energies and play a major role in determining the most probable fragment combinations in the ternary breakup of ${}^{252}\mathrm{Cf}$.

Figure 1

A schematic illustration of the orientation of axial-symmetric deformed ternary fragments in the collinear touching configuration. The touching configuration is defined by the surface separation distance between interacting nuclei and it is equal to zero. Here, the lightest fragment $A_3$ is kept in the middle of two main fragments ($A_1$ and $A_2$). The horizontal solid line represents the axial-symmetric axis.

Figure 2

Ternary-fragmentation-driving potential energies of ${}^{252}\mathrm{Cf}$ for the collinear touching configuration of spherical fragments. Panels (a) and (b) corresponds to the proton-minimized and the neutron-minimized ternary fragments, respectively.

Figure 3

The $Q$-value energy systematics of the neutron-minimized fragment combinations.

Figure 4

Ternary fragmentation driving potential energies of ${}^{252}\mathrm{Cf}$ for the collinear touching configuration of deformed fragments. Panels (a) and (b) correspond to the proton-minimized and the neutron-minimized ternary fragments, respectively.