Microscopic self-consistent description of induced fission dynamics: Finite-temperature effects

The dynamics of induced fission of ${}^{226}\mathrm{Th}$ is investigated in a theoretical framework based on the finite-temperature time-dependent generator coordinate method (TDGCM) in the Gaussian overlap approximation (GOA). The thermodynamical collective potential and inertia tensor at temperatures in the interval $T=0–1.25$ MeV are calculated using the self-consistent multidimensionally constrained relativistic mean-field (MDC-RMF) model, based on the energy density functional DD-PC1. Pairing correlations are treated in the BCS approximation with a separable pairing force of finite range. Constrained RMF+BCS calculations are carried out in the collective space of axially symmetric quadrupole and octupole deformations for the asymmetric fissioning nucleus ${}^{226}\mathrm{Th}$. The collective Hamiltonian is determined by the temperature-dependent free energy surface and perturbative cranking inertia tensor, and the TDGCM+GOA is used to propagate the initial collective state in time. The resulting charge and mass fragment distributions are analyzed as functions of the internal excitation energy. The model can qualitatively reproduce the empirical triple-humped structure of the fission charge and mass distributions already at $T=0$, but the precise experimental position of the asymmetric peaks and the symmetric-fission yield can only be accurately reproduced when the potential and inertia tensor of the collective Hamiltonian are determined at finite temperature, in this particular case between $T=0.75$ MeV and $T=1$ MeV.

Figure 1

Free energy (in MeV) along the least-energy fission pathway in ${}^{226}\mathrm{Th}$ for finite temperatures $T=0.0,0.5,0.75,1.0$, and 1.25 MeV. All curves are normalized to their values at equilibrium minimum.

Figure 2

Evolution of the first ($B_{\mathrm{I}}$), second ($B_{\mathrm{II}}$), and third ($B_{\text{III}}$) barrier heights in the free energy of ${}^{226}\mathrm{Th}$, as functions of temperature.

Figure 3

Temperature dependence of the pairing energy in the RMF+BCS equilibrium minimum, in the fission isomer, and at the top of the first and second barriers in ${}^{226}\mathrm{Th}$.

Figure 4

Free energy $F$ of ${}^{226}\mathrm{Th}$ in the $({\beta }_{20},{\beta }_{30})$ plane for finite temperatures $T=0.0,0.5,0.75,1.0$, and 1.25 MeV. In each panel energies are normalized with respect to the corresponding value at the equilibrium minimum, and contours join points on the surface with the same energy (in MeV). The energy surfaces are calculated with the relativistic density functionals DD-PC1 [54], and the pairing interaction Eq. (12). The contour interval is 1.0 MeV.

Figure 5

The ${\mathcal{M}}_{11}$ component of the mass tensor of ${}^{226}\mathrm{Th}$ as function of the quadrupole deformation ${\beta }_{20}$ (top panel), and the ${\mathcal{M}}_{22}$ component as function of the octupole deformation ${\beta }_{30}$ (bottom panel) for finite temperatures $T=0.0,0.75,1.0$, and 1.25 MeV.

Figure 6

Charge yields for induced fission of ${}^{226}\mathrm{Th}$. The collective potentials and perturbative cranking inertia tensors for the finite temperatures $T=0.0,0.5,0.75,1.00$, and 1.25 MeV are used in the calculations. The corresponding internal excitation energies are $E_{\mathrm{int}.}^{*}=0.0,2.58,8.71,16.56$, and 27.12 MeV, respectively. The average excitation energy of the initial state ($E_{\mathrm{coll}.}^{*}$) is chosen 1 MeV above the corresponding second fission barrier $B_{\mathrm{II}}$. The experimental charge yields for ${}^{226}\mathrm{Th}(\gamma ,f)$ are from Ref. [2].

Figure 7

Same as Fig. 6, but for preneutron emission mass yields.