Microscopic Phase-Space Exploration Modeling of ${}^{258}\mathrm{Fm}$ Spontaneous Fission

We show that the total kinetic energy (TKE) of nuclei after the spontaneous fission of ${}^{258}\mathrm{Fm}$ can be well reproduced using simple assumptions on the quantum collective phase space explored by the nucleus after passing the fission barrier. Assuming energy conservation and phase-space exploration according to the stochastic mean-field approach, a set of initial densities is generated. Each density is then evolved in time using the nuclear time-dependent density-functional theory with pairing. This approach goes beyond the mean-field theory by allowing spontaneous symmetry breaking as well as a wider dynamical phase-space exploration leading to larger fluctuations in collective space. The total kinetic energy and mass distributions are calculated. New information on the fission process: fluctuations in scission time, strong correlation between TKE and collective deformation, as well as prescission particle emission, are obtained. We conclude that fluctuations of the TKE and mass are triggered by quantum fluctuations.

Figure 1

TKE (a) and fragment mass (b) distributions obtained starting from $Q_2^{\mathrm{ini}}=160\mathrm{b}$ (shaded area) and 125 b (dashed line). The solid line is the experimental data [51]. In (a) the arrow indicates the mean TKE obtained in Ref. [28]. In the inset, the correlation between the average TKE and heaviest fragment mass (red squares) is shown. Comparison is made with results of the scission point model [53] in dotted and dot-dashed lines and ${}^{257}\mathrm{Fm}$ data [54] in a solid line.

Figure 2

Distribution of the time elapsed from $Q_2^{\mathrm{ini}}=125\mathrm{b}$ to the scission point including (shaded area) or not including (solid line) dynamical pairing. The time needed to reach scission using TDDFT with dynamical pairing and without initial fluctuation is shown in the inset as a function of initial deformation.

Figure 3

Calculated distributions of quadrupole ${\beta }_2$ (a) and octupole ${\beta }_3$ (b) deformation parameters of fragments at the scission point (shaded area) and at 26 fm (black solid line). The insets (c) [resp. (e)] show the correlation between the final TKE and ${\beta }_2$ (resp. ${\beta }_3$) at scission. The inset (d) [resp. (f)] shows the event by event correlation at scission between the ${\beta }_2$ (resp. ${\beta }_3$) of the heaviest ($\mathit{H}$) nucleus and the ${\beta }_2$ (resp. ${\beta }_3$) of the lightest ($L$) nucleus.

Figure 4

Probability distribution of the number $N_{\mathrm{pre}}$ of neutrons (red shaded area) and protons (blue shaded area) emitted before reaching scission. Neutron (resp. proton) distribution is shifted by a $-0.5$ ($\mathrm{resp}\mathrm{.}+0.5$) unit for display reasons. The equivalent distribution obtained without dynamical pairing is also shown for the neutron (solid line) and proton (dashed line).