Theoretical study of the almost sequential mechanism of true ternary fission

We consider the collinear ternary fission which is a sequential ternary decay with a very short time between the ruptures of two necks connecting the middle cluster of the ternary nuclear system and outer fragments. In particular, we consider the case where the Coulomb field of the first massive fragment separated during the first step of the fission produces a lower pre-scission barrier in the second step of the residual part of the ternary system. In this case, we obtain a probability of about ${10}^{-3}$ per binary fission for the yield of massive clusters such as ${}^{70}\mathrm{Ni},{}^{80-82}\mathrm{Ge},{}^{86}\mathrm{Se}$, and ${}^{94}\mathrm{Kr}$ in the ternary fission of ${}^{252}\mathrm{Cf}$. These products appear together with the clusters having mass numbers of $A=132–140$. The results show that the yield of a heavy cluster such as ${}^{68-70}\mathrm{Ni}$ would be followed by a product of $A=138–148$ with a large probability as observed in the experimental data obtained with the FOBOS spectrometer at the Joint Institute for Nuclear Research. The third product is not observed. The landscape of the potential-energy surface shows that the configuration of the $\mathrm{Ni}+\mathrm{Ca}+\mathrm{Sn}$ decay channel is lower by about 12 MeV than that of the $\mathrm{Ca}+\mathrm{Ni}+\mathrm{Sn}$ channel. This leads to the fact that the yield of $\mathrm{Ni}$ and $\mathrm{Sn}$ is large. The analysis on the dependence of the velocity of the middle fragment on mass numbers of the outer products leads to the conclusion that, in the collinear tripartition channel of ${}^{252}\mathrm{Cf}$, the middle cluster has a very small velocity, which does not allow it to be found in experiments.

Figure 1

The CCT fission mechanism of a heavy nucleus in a sequential decay [11]. $A_1,A_2$, and $A_3$ are mass numbers of the fragments formed in the trinuclear system.

Figure 2

The variables of the trinuclear system used in the analysis of the interaction energy between its fragments. Here, $Z_i$ is the charge number of fragment $i(i=1,2,3)$ and $R_{ij}$ is the distance between the mass centers of fragments $i$ and $j$.

Figure 3

The potential-energy surface of the ${}^{252}\mathrm{Cf}(\mathrm{sf},\mathrm{fff})$ reaction. The wide solid rectangle “CCT” shows the area of the mass numbers $Z_1\left(A_1\right)$ and $Z_3\left(A_3\right)$ which corresponds to the CCT products. The red dashed rectangle “FFF” shows the area of formation of three fragments with similar mass numbers. The solid (yellow) and dashed (pink) lines show the TNS configurations having ${}^{132}\mathrm{Sn}$ and ${}^{134}\mathrm{Te}$, respectively, as the outer nucleus $Z_2$.

Figure 4

The pre-scission barriers $B_{\mathrm{Ni}\mathrm{Ca}}$ and $B_{\mathrm{Ca}\mathrm{Sn}}$ keeping TNS fragments together.

Figure 5

The total nucleus-nucleus interaction potential $V_{\mathrm{int}}$ as a function of intercenter distances $R_{13}$ and $R_{32}$ between fragments of the TNS with collinear geometry.

Figure 6

The dependence of the change of the pre-scission barrier $\Delta B_{13}$ for the decay of the binary system $\mathrm{Ni}+\mathrm{Ca}$ on the distance $R_{32}$ due to the Coulomb interaction of the massive third fragment $\mathrm{Sn}$ in the collinear geometry.

Figure 7

Theoretical results for the yield of the outer fragments ${}^{{\mathrm{A}}_1}{\mathrm{Z}}_1$ and ${}^{{\mathrm{A}}_2}{\mathrm{Z}}_2$ of the TNS formed at the spontaneous fission of ${}^{252}\mathrm{Cf}$. The yield is high at $Z_2\sim 50$.

Figure 8

Theoretical results for the yield of the outer ${}^{{\mathrm{A}}_1}{\mathrm{Z}}_1$ and middle ${}^{{\mathrm{A}}_3}{\mathrm{Z}}_3$ fragments of the TNS formed in the spontaneous fission of ${}^{252}\mathrm{Cf}$.

Figure 9

The contour map of the calculated velocity $v_3^{}$ (in cm/ns) of the middle ${}^{{\mathrm{A}}_3}{\mathrm{Z}}_3$ fragment of the TNS formed at the spontaneous fission of ${}^{252}\mathrm{Cf}$ as a function of the charge and mass numbers of the outer fragments ${}^{{\mathrm{A}}_1}{\mathrm{Z}}_1$ and ${}^{{\mathrm{A}}_2}{\mathrm{Z}}_2$. The negative values of $v_3^{}$ mean that its direction is opposite the direction of $v_2$.