Effects of microscopic transport coefficients on fission observables calculated by the Langevin equation

Nuclear fission is treated by using the Langevin dynamical description with macroscopic and microscopic transport coefficients (mass and friction tensors), and it is elucidated how the microscopic (shell and pairing) effects in the transport coefficients, especially their dependence on temperature, affects various fission observables. We found that the microscopic transport coefficients, calculated by linear response theory, change drastically as a function of temperature: in general, the friction increases with growing temperature while the mass tensor decreases. This temperature dependence brings a noticeable change in the mass distribution and kinetic energies of fission fragments from nuclei around ${}^{236}\mathrm{U}$ at an excitation energy of 20 MeV. The prescission kinetic energy decreases from 25 MeV at low temperature to about 2.5 MeV at high temperature. In contrast, the Coulomb kinetic energy increases as the temperature increases. Interpolating the microscopic transport coefficients among the various temperatures enabled our Langevin equation to use the microscopic transport coefficients at a deformation-dependent local temperature of the dynamical evolution. This allowed us to compare directly the fission observables of both macroscopic and microscopic calculations, and we found almost identical results under the conditions considered in this work.

Figure 1

The two-center shell model potential (1) for $\rho =0$.

Figure 2

The shape of a nuclear surface calculated with the two-center shell model parametrization ($\delta =0.2, \epsilon =0.35$, solid lines) and the modification (6) (dashed lines).

Figure 3

The deformation energy of ${}^{236}\mathrm{U}$ as function of elongation (the distance $R_{12}$ between the centers of mass of left and right parts of the nucleus) and the mass asymmetry $\alpha$.

Figure 4

Comparison of the $z_0{z}_0$ component of the microscopic mass tensor (19) calculated at several values of $T_{\mathrm{tr}}$ with the Werner-Wheeler approximation (9) (top) and a similar comparison of the $z_0{z}_0$ component of the microscopic friction tensor (18) for several values of $T_{\mathrm{tr}}$ with the macroscopic approximation (17) (bottom).

Figure 5

The components of the microscopic mass tensor (19) calculated for $\delta =0.2$ and $T=1\mathrm{MeV}$.

Figure 6

The fission product yield (FPY) of ${}^{236}\mathrm{U}$ at the excitation energy $E_x=20\mathrm{MeV}$. The circles denote the experimental data from the library JENDL/FPY-2011 [46], while the present results obtained by using macroscopic transport coefficients are shown as a histogram. The error bars mark the statistical uncertainty of present results. The yield is normalized to 2.

Figure 7

The comparison of fission product yield of the multichance fission calculation (blue histogram), the first-chance fission calculation (black histogram), and JENDL data (red circles). The calculations were performed with the macroscopic transport coefficients.

Figure 8

The fission product yield for the ${}^{236}\mathrm{U}$ compound nucleus at an excitation energy of 20 MeV calculated with the microscopic transport coefficients, which depend on deformation-dependent local temperature.

Figure 9

The fission product yield for ${}^{234}\mathrm{U}$ (top) and ${}^{240}\mathrm{Pu}$ (bottom) compound nuclei at an excitation energy of 20 MeV calculated with the temperature-independent macroscopic (broken blue histogram) and microscopic (black histogram) transport coefficients at deformation-dependent local temperature compared with data given in JENDL-4 (red circles).

Figure 10

The fission product yield for ${}^{236}\mathrm{U}$ at an excitation energy of 20 MeV. The top figure compares the Langevin calculation with different choices of pairing correction energy (or factor) to the potential energy with macroscopic transport coefficients, while the bottom figure compares those calculated with microscopic transport coefficients, which depends on deformation-dependent local temperature.

Figure 11

The prescission kinetic energy of fission fragments as a function of temperature $T_{\mathrm{tr}}$ at which the transport coefficients were calculated for ${}^{236}\mathrm{U}$ at $E_x=20\mathrm{MeV}$. The horizontal red and blue lines denote the values calculated with the macroscopic transport coefficients and the local-temperature-dependent microscopic transport coefficients, respectively.

Figure 12

The average deformation $〈\delta 〉$ of fission fragments at scission as function of $T_{\mathrm{tr}}$ (top) and the Coulomb interaction energy of fission fragments at scission (bottom) for ${}^{236}\mathrm{U}$ at $E_x=20\mathrm{MeV}$.

Figure 13

Comparison of calculated total kinetic energy of fission fragments for ${}^{236}\mathrm{U}$ with the experimental data of Dyachenkov and Kuzminov [47] (full circles) and Zeynalov et al. [48] (open circles) for ${}^{235}\mathrm{U}+n$.