Dynamical simulation of the fission process and anisotropy of the fission fragment angular distributions of excited nuclei produced in fusion reactions

Abstract. A stochastic approach based on four-dimensional Langevin equations was applied to calculate the anisotropy of fission fragment angular distributions, average prescission neutron multiplicity, and the fission probability in a wide range of fissile parameters for the compound nuclei ${}^{197}\mathrm{Tl},{}^{225}\mathrm{Pa},{}^{248}\mathrm{Cf}$, and ${}^{264}\mathrm{Rf}$ produced in fusion reactions. Three collective shape coordinates plus the projection of total spin of the compound nucleus to the symmetry axis $K$ were considered in the four-dimensional dynamical model. In the dynamical calculations, nuclear dissipation was generated through the chaos-weighted wall and window friction formula. Furthermore, in the dynamical calculations the dissipation coefficient of $K,{\gamma }_k$ was considered as a free parameter, and its magnitude inferred by fitting measured data on the anisotropy of fission fragment angular distributions for the compound nuclei ${}^{197}\mathrm{Tl},{}^{225}\mathrm{Pa},{}^{248}\mathrm{Cf}$, and ${}^{264}\mathrm{Rf}$. Comparison of the calculated results for the anisotropy of fission fragment angular distributions with the experimental data showed that the results of the calculations are in good agreement with the experimental data by using values of the dissipation coefficient of $K$ equal to (0.185–0.205), (0.175–0.192), (0.077–0.090), and (0.075–0.085) ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ for the compound nuclei ${}^{197}\mathrm{Tl},{}^{225}\mathrm{Pa},{}^{248}\mathrm{Cf}$, and ${}^{264}\mathrm{Rf}$, respectively. It was also shown that the influence of the dissipation coefficient of $K$ on the results of the calculations of the prescission neutron multiplicity and fission probability is small.

Figure 1

The cross sections for the reactions ${}^{16}\mathrm{O}+{}^{181}\mathrm{Ta}\to {}^{197}\mathrm{Tl}$ and ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}\to {}^{248}\mathrm{Cf}$ as a function of spin and at projectile energies of 100 and 130 MeV.

Figure 2

The Helmholtz free energies for the compound nuclei ${}^{197}\mathrm{Tl}$ and ${}^{248}\mathrm{Cf}$ as a function of the collective coordinates $q_1$ and $K$ at $T=2\mathrm{MeV}$ and $I=40\hslash$. The dashed line curves show the dependence of saddle-point deformations on $K$. The numbers on the contour lines represent the potential-energy surface values in MeV.

Figure 3

The anisotropy of fission fragment angular distributions for the compound nuclei ${}^{197}\mathrm{Tl},{}^{225}\mathrm{Pa},{}^{248}\mathrm{Cf}$, and ${}^{264}\mathrm{Rf}$ as a function of projectile energy in the laboratory system. The filled circles are the experimental data [49, 50, 51, 52, 53]. The dashed and dotted curves correspond to fitted values calculated with the 4D dynamical model and by using different values of ${\gamma }_K$. The dashed-dotted and dashed-double-dotted curves correspond to fitted values calculated by the SPTS and SCTS models, respectively.

Figure 4

The anisotropy of fission fragment angular distributions for the compound nuclei ${}^{225}\mathrm{Pa}$ and ${}^{248}\mathrm{Cf}$ as a function of the scattering angle and at different values of projectile energy in the laboratory system. The curves correspond to fitted values calculated with the 4D dynamical model and by using different values of ${\gamma }_K$. The experimental data (filled circles) are taken from Refs. [53, 54].

Figure 5

The average prescission neutron multiplicity for the compound nuclei ${}^{197}\mathrm{Tl}$ and ${}^{248}\mathrm{Cf}$ as a function of excitation energy and different values of ${\gamma }_K$. The experimental data (filled circles) are taken from Refs. [55, 56].

Figure 6

Results of the calculations for fission probabilities as a function of the parameter ${\mathrm{Z}}^2/A$. The open squares and open circles are the calculated results calculated with the 4D dynamical model and by using different values of ${\gamma }_K$ which were extracted in the present paper for the compound nuclei ${}^{197}\mathrm{Tl},{}^{225}\mathrm{Pa},{}^{248}\mathrm{Cf}$, and ${}^{264}\mathrm{Rf}$. The closed symbols are the experimental data [57, 58, 59].

Figure 7

The extracted values of the dissipation coefficient of $K$ (solid lines) for the compound nuclei ${}^{197}\mathrm{Tl},{}^{225}\mathrm{Pa},{}^{248}\mathrm{Cf}$, and ${}^{264}\mathrm{Rf}$. The shaded area represents the variation of the best-fit values of the dissipation coefficient of $K$ with the compound nucleus mass number.