Energy and pairing dependence of dissipation in real-time fission dynamics

This work presents a microscopic study of dissipation in fission dynamical evolutions with the time-dependent Hartree-$\mathrm{Fock}+\mathrm{BCS}$ method in terms of energy dependence and pairing dependence. The friction coefficients and dissipation energies are extracted by mapping the symmetric fission process of ${}^{258}\mathrm{Fm}$ into a classical equation of motion. Density-constrained calculations are used to obtain the dynamical potential. The obtained friction coefficients have a strong dependence of deformations, and averagely match the coefficients adopted in statistical models. The dissipation indeed increase with increasing initial excitation energies or with decreasing pairing strengths. It is also shown that dissipations play a significant role in the final stage of scission. The demonstrated characteristics of dissipations will be valuable for calibrations of various fission models.

Figure 1

In the symmetric fission process of ${}^{258}\mathrm{Fm}$, the nuclear potentials as a function of quadrupole potential $Q_{20}$ are obtained by static $\mathrm{HF}+\mathrm{BCS}$ calculations (called adiabatic potential $V_{ad}$) and density-constrained HF-BCS calculations with densities from TD-BCS evolutions (called dynamical potential $V_{dyn}$). The potentials at different temperatures (or excitation energies) are shown: $T=0$ (a), $T=0.5$ MeV (b), $T=0.75$ MeV (c), and $T=1.0$ MeV (d).

Figure 2

Panel (a) shows the dissipated energies obtained via Eq. (3) as a function of $Q_{20}$ at different temperatures. Panel (b) shows the extracted friction coefficients $\gamma \left(Q\right)$ in Eq. (2). The vertical lines denote the scission deformations at different temperatures.

Figure 3

Panel (a) shows dynamical nuclear potentials as a function of $Q_{20}$ from density-constrained calculations with varying pairing strengths. Panel (b) shows the number of particles in neck in the evolution, which is defined as the number of particles in a neck of 2 fm length at the density minimum. Panel (c) shows the deformation dependent pairing energies with varying pairing strengths.

Figure 4

Similar to Fig. 2, panel (a) shows the dissipated energy with varying pairing strengths from $80\%$ to $110\%$. Panel (b) shows the obtained friction coefficients correspondingly. The scission deformations are denoted by the vertical lines.