Fission dynamics at low excitation energy

The mass asymmetry in the fission of ${}^{236}\mathrm{U}$ at low excitation energy is clarified by the analysis of the trajectories obtained by solving the Langevin equations for the shape degrees of freedom. It is demonstrated that the position of the peaks in the mass distribution of fission fragments is determined mainly by the saddle point configuration originating from the shell correction energy. The width of the peaks, on the other hand, results from the shape fluctuations close to the scission point caused by the random force in the Langevin equation. We have found out that the fluctuations between elongated and compact shapes are essential for the fission process. According to our results the fission does not occur with continuous stretching in the prolate direction, similarly to that observed in starch syrup, but is accompanied by the fluctuations between elongated and compact shapes. This picture presents a new viewpoint of fission dynamics and the splitting mechanism.

Figure 1

Deformation dependence of the potential energy of ${}^{236}\mathrm{U}$ at $T=0$ calculated by twoctr with fixed $\delta =0,\alpha =0$, and $\epsilon =0.35$. The $V_{\mathrm{LD}}$ and $V_{\mathrm{LD}}+V_{\mathrm{SH}}$ are denoted by the dashed and solid lines, respectively. The short-dashed line shows the $V_{\mathrm{LD}}+\Delta E_{\mathrm{shell}}$.

Figure 2

The comparison of the mass distribution of fission fragments calculated with (dashed line) and without (solid line) account of the $\overline{a}{T}^2$ term in (3).

Figure 3

The examples of trajectories of the fission process of ${}^{236}\mathrm{U}$ at $E^{*}=20\mathrm{MeV}$ projected onto the ${\overline{z}}_0-\alpha$ plane of $V_{\mathrm{LD}}$ $\delta =0.24$ (top) and on the ${\overline{z}}_0-\delta$ plane of $V_{\mathrm{LD}}$ at $\alpha =0$ (bottom). The trajectories calculated without random force start at the saddle point.

Figure 4

Sample trajectory projected onto the ${\overline{z}}_0-\alpha$ plane at $\delta =0.2$ (top) and the ${\overline{z}}_0-\delta$ plane $\alpha =0.0$ (bottom) of $V_{\mathrm{LD}}+E_{\mathrm{shell}}^0$ with $\epsilon =0.35$ for ${}^{236}\mathrm{U}$. The trajectory starts at the ground state $\{{\overline{z}}_0,\delta ,\alpha \}=\{0.0,0.2,0.0\}$ at $E^{*}=20$ MeV. The fission saddle points are indicated by the symbol $\times$. The scission lines are denoted by the white lines.

Figure 5

Nuclear shapes around the scission point of ${}^{236}\mathrm{U}$. The dot, solid, and dashed line corresponds to the nuclear shape at $\{{\overline{z}}_0,\delta ,\alpha \}=\{2.5,-0.2,0.2\},\{2.5,0,0.2\},\text{and}\{2.5,0.2,0.2\}$.

Figure 6

The calculated distribution of fission events in the deformation parameter $\delta$ at the scission point for the fission of ${}^{236}\mathrm{U}$ at $E^{*}=20$ MeV.

Figure 7

The TKE distribution of the fission events of ${}^{236}\mathrm{U}$ at $E^{*}=20$ MeV.

Figure 8

The distribution of fission events of ${}^{236}\mathrm{U}$ at $E^{*}=20$ MeV in the TKE and the parameter $\delta$.

Figure 9

The $z$ dependence of the potential $V(\rho ,z)$ (9) (top) and the example of the equipotential surface of potential $V(\rho ,z)$ (bottom). The neck parameter $\epsilon$ is defined as the ratio of the smoothed potential height $E$ at $z=0$ to the original one $E_0$.

Figure 10

The examples of shapes (13) for a few values of $z_0$ at fixed $\alpha =0,{\delta }_1={\delta }_2=0,\epsilon =0.35$.