Thermal fission rates with temperature dependent fission barriers

Background: The fission processes of thermal excited nuclei are conventionally studied by statistical models which rely on inputs of phenomenological level densities and potential barriers. Therefore the microscopic descriptions of spontaneous fission and induced fission are very desirable for a unified understanding of various fission processes.

Purpose: We propose to study the fission rates, at both low and high temperatures, with microscopically calculated temperature-dependent fission barriers and collective mass parameters.

Methods: The fission barriers are calculated by the finite-temperature Skyrme-Hartree-Fock+BCS method. The mass parameters are calculated by the temperature-dependent cranking approximation. The thermal fission rates can be obtained by the imaginary free energy approach at all temperatures, in which fission barriers are naturally temperature dependent. The fission at low temperatures can be described mainly as a barrier-tunneling process. While the fission at high temperatures has to incorporate the reflection above barriers.

Results: Our results of spontaneous fission rates reasonably agree with other studies and experiments. The temperature dependencies of fission barrier heights and curvatures have been discussed. The temperature dependent behaviors of mass parameters have also been discussed. The thermal fission rates from low to high temperatures with a smooth connection have been given by different approaches.

Conclusions: Since the temperature dependencies of fission barrier heights and curvatures, and the mass parameters can vary rapidly for different nuclei, the microscopic descriptions of thermal fission rates are very valuable. Our studies without free parameters provide a consistent picture to study various fissions such as that in fast-neutron reactors, astrophysical environments, and fusion reactions for superheavy nuclei.

Figure 1

The fission barrier of ${}^{260}\text{Fm}$ is shown to illustrate calculations of fission processes with a decay energy $E_0$. The potential frequencies (or curvatures) around the equilibrium point $s_0$ and the saddle point $s_b$ are labeled as ${\omega }_0$ and ${\omega }_b$, respectively.

Figure 2

(a) The calculated spontaneous fission barriers of ${}^{260}\text{Fm}$ with different Skyrme forces: ${\mathrm{SkM}}^{*}$, SLy6, and UNEDF1, respectively. (b) The calculated collective mass parameters of ${}^{260}\text{Fm}$ with the three Skyrme forces.

Figure 3

The calculated temperature dependent fission barriers as a function of quadrupole deformation ${\beta }_2$, (a) for ${}^{260}\text{Fm}$ and (b) for ${}^{292}114$. In ${}^{292}114$, the reflection asymmetric deformation has been taken into account.

Figure 4

The calculated potential curvatures (or frequencies) around the equilibrium point (${\omega }_0$) and the barrier saddle point (${\omega }_b$) as a function of temperature, (a) for ${}^{260}\text{Fm}$ and (b) for ${}^{292}\text{Fl}$.

Figure 5

(a) The calculated temperature-dependent mass parameters as a function of deformations, (a) for ${}^{260}\text{Fm}$ and (b) for ${}^{292}114$.

Figure 6

The calculated thermal fission lifetimes of ${}^{260}\text{Fm}$ as a function of excitation energies by different formulas, in which ‘$\mathrm{Im}F$-L’ denotes the low-temperature $\mathrm{Im}F$ formula Eq. (10), ‘$\mathrm{Im}F$-H’ denotes the high-temperature $\mathrm{Im}F$ formula Eq. (16), ‘Boltzmann-L’ denotes the low-temperature Boltzmann thermal fission formula Eq. (9), ‘Kramers’ denotes the Kramers fission formula Eq. (17).