Cassini-oval description of the multidimensional potential energy surface for ${}^{236}\mathrm{U}$: Role of octupole deformation and calculation of the most probable fission path

Multidimensional potential energy surfaces (PES) are calculated using the microscopic-macroscopic approach. The nuclear shapes are described by Cassinian ovals generalized by the inclusion of ${\alpha }_1$, ${\alpha }_3$, and ${\alpha }_4$ shape parameters in addition to the main fission coordinate $\alpha$. The influence of the octupole deformation $\left({\alpha }_3\right)$ on the PES is studied in the case of the nuclear fission of ${}^{236}\mathrm{U}$. It is found that ${\alpha }_3$ plays an important role in the last stage of the fission process; for instance, it lowers the third minimum and the third barrier. Two methods to calculate the static fission path are investigated. They are found to be consistent in the sense that they lead to the same fission barrier. In certain subspaces, the least energy paths from the ground state to scission present discontinuities around one of the saddles. They are caused by sharp changes in the nuclear shapes involved occurring without a change in energy. Such transitions are smoothed out by the principle of stationary action, which transforms a discontinuous path into a continuous one. Finally, various macroscopic models have been employed in order to study their influence on the energies and positions of the saddles and the minima.

Figure 1

Pure Cassini ovals that conserve the volume at five values of their deformation $\varepsilon$.

Figure 2

Coordinate transformation used in the von Ritz method.

Figure 3

Total deformation energy as a function of overall elongation ($\alpha$) and mass asymmetry (${\alpha }_1$) for ${}^{236}\mathrm{U}$ between a sphere and the scission line. On it the fission path calculated in three different ways is drawn. The red circles represent the minimum energy, the blue squares follow the steepest descent, and the solid curve represents the path of minimum action (von Ritz). The fission barriers along each path are plotted in the upper part.

Figure 4

The same as in Fig. 3 but with inclusion of (${\alpha }_4$) and minimization with respect to it.

Figure 5

Total deformation energy as a function of overall elongation ($\alpha$) and hexadecapole (${\alpha }_4$) for ${}^{236}\mathrm{U}$ between the ground state and the second minimum. The points of minimum deformation energy (red circles), of steepest descent (blue squares), and the path of minimum action (solid black) are also shown. The fission barriers along each path are plotted in the upper part.

Figure 6

Total deformation energy as a function of overall elongation ($\alpha$) and hexadecapole (${\alpha }_4$) for ${}^{236}\mathrm{U}$ between the ground state and the second minimum. The lines corresponding to $\frac{\partial E_{\mathrm{def}}}{\partial a_4}$=0 (red) and $\frac{\partial E_{\mathrm{def}}}{\partial a}$=0 (blue) are superposed. Their intersection precisely defines the extreme points on the PES, i.e., the GS, the 1stB, and the 2ndM.

Figure 7

Total deformation energy as a function of overall elongation ($\alpha$) and fragment mass ($A_F$) for ${}^{236}\mathrm{U}$ between the second minimum and the scission line. ${\alpha }_4=0$. The points of minimum deformation energy (red) and the path of minimum action (black) are also shown.

Figure 8

Total deformation energy as a function of overall elongation ($\alpha$) and fragment mass ($A_F$) for ${}^{236}\mathrm{U}$ between second minimum and scission. At each point, the energy is minimized as a function of the octupole deformation ${\alpha }_3$. The points of minimum deformation energy (red) and the path of minimum action (black) are also shown.

Figure 9

The same as in Fig. 8 but with additional minimization on ${\alpha }_4$.

Figure 10

Dependence of the minimum action integral $S$ on the number of terms $N$ in Eq. (9).

Figure 11

Influence of the number of terms $N$ in Eq. (9) on the minimum path. $N=7$ (red), $N=8$ (blue), $N=9$ (green). The PES is minimized with respect to ${\alpha }_3$.

Figure 12

Influence of the initial trial path (parabola or points of minimum deformation energy) on the minimum path. $N=5$. The two paths are identical. The PES is minimized with respect to ${\alpha }_3$.

Figure 13

The same as in Fig. 12 but for $N=7$, 8, and 9.

Figure 14

The liquid-drop fission barrier for three sets of parameters ($a_s$, $a_c$). The arrows indicate the barrier tops.