Time-dependent generator-coordinate-method study of mass-asymmetric fission of actinides

Low-energy positive and negative parity collective states in the equilibrium minimum, and the dynamics of induced fission of actinide nuclei are investigated in a unified theoretical framework based on the generator coordinate method (GCM) with the Gaussian overlap approximation (GOA). The collective potential and inertia tensor, both at zero and finite temperature, are computed using the self-consistent multidimensionally constrained relativistic mean field model, based on the energy density functional DD-PC1. Pairing correlations are treated in the Bardeen-Cooper-Schrieffer approximation with a separable pairing force of finite range. A collective quadrupole-octupole Hamiltonian characterized by zero-temperature axially symmetric deformation energy surface and perturbative cranking inertia tensor, is used to model the low-lying excitation spectrum. The fission fragment charge distributions are obtained by propagating the initial collective states in time with the time-dependent GCM+GOA that uses the same quadrupole-octupole Hamiltonian, but with the collective potential and inertia tensor computed at finite temperature. The illustrative charge yields of ${}^{228}\mathrm{Th}$, ${}^{234}\mathrm{U}$, ${}^{240}\mathrm{Pu}$, ${}^{244}\mathrm{Cm}$, and ${}^{250}\mathrm{Cf}$ are in very good agreement with experiment, and the predicted mass asymmetry is consistent with the result of a recent microscopic study that has attributed the distribution (peak) of the heavier-fragment nuclei to shell-stabilized octupole deformations.

Figure 1

Axially symmetric quadrupole-octupole energy surface in the ${\beta }_{20}\text{−}{\beta }_{30}$ plane for ${}^{228}\mathrm{Th}$. The contours join points on the surface with the same energy, and the separation between neighboring contours is 0.2 MeV.

Figure 2

Experimental [59, 60] and calculated yrast states of positive and negative parity in ${}^{228}\mathrm{Th}$. The in-band $B\left(E2\right)$ values (dotted) and the $B(E3;3_1^{-}\to 0_1^{+})$ (solid) (both in Weisskopf units) are also shown.

Figure 3

Deformation energy curves (in MeV) along the least-energy fission path as functions of the quadrupole deformation parameter ${\beta }_{20}$, for ${}^{228}\mathrm{Th}$, ${}^{234}\mathrm{U}$, ${}^{240}\mathrm{Pu}$, ${}^{244}\mathrm{Cm}$, and ${}^{250}\mathrm{Cf}$. All curves are normalized to their values at equilibrium minimum.

Figure 4

Axially symmetric quadrupole-octupole deformation energy surfaces in the (${\beta }_{20},{\beta }_{30}$) plane for ${}^{234}\mathrm{U}$, ${}^{240}\mathrm{Pu}$, ${}^{244}\mathrm{Cm}$, and ${}^{250}\mathrm{Cf}$. In each panel the energies are normalized with respect to the corresponding value at the equilibrium minimum. The contours join points on the surface with the same energy and the separation between neighboring contours is 2.0 MeV.

Figure 5

Deformation (free) energy (in MeV) of ${}^{228}\mathrm{Th}$ in the (${\beta }_{20},{\beta }_{30}$) plane at zero temperature and at $T=0.85$ MeV. In both panels the energies are normalized with respect to the corresponding value at the equilibrium minimum, and the contours join points on the surface with the same energy. The contour interval is 1.0 MeV.

Figure 6

Free energy (in MeV) of ${}^{228}\mathrm{Th}$ along the least-energy fission path as function of the quadrupole deformation (a). The values of the deformation parameter ${\beta }_{20}$ (b), the free energy (c), and the heavy-fragment charge number, along the frontier of the domain defined by $Q_N>3.0$ at zero temperature and at $T=0.85$ MeV. The position on the scission contour is labeled by the corresponding ${\beta }_{30}$ value.

Figure 7

Charge yields for photoinduced fission of ${}^{228}\mathrm{Th}$. The collective potentials and perturbative cranking inertia tensors for zero and finite temperature are used in the TDGCM+GOA calculation. $T=0.85$ MeV corresponds to the intrinsic excitation energy of $E_{\mathrm{int}}^{*}\approx 11$ MeV, equivalent to the peak value of the photon energy distribution.

Figure 8

Same as in Fig. 7, but for ${}^{234}\mathrm{U}$, ${}^{240}\mathrm{Pu}$, ${}^{244}\mathrm{Cm}$, and ${}^{250}\mathrm{Cf}$. See text for the description.

Figure 9

Evolution of the height of the second barrier along the reflection-asymmetric (RA) (least-energy cf. Fig. 3) and reflection-symmetric (RS) fission paths in the deformation (free) energy of ${}^{228}\mathrm{Th}$, as functions of temperature. The empty (filled) symbols denote the barrier heights obtained without (AS), and with (TA) inclusion of the triaxial quadrupole collective degree of freedom, respectively.