From asymmetric to symmetric fission in the fermium isotopes within the time-dependent generator-coordinate-method formalism

Background: Predicting the properties of neutron-rich nuclei far from the valley of stability is one of the major challenges of modern nuclear theory. In heavy and superheavy nuclei, a difference of only a few neutrons is sufficient to change the dominant fission mode. A theoretical approach capable of predicting such rapid transitions for neutron-rich systems would be a valuable tool to better understand $r$-process nucleosynthesis or the decay of superheavy elements.

Purpose: In this work, we investigate for the first time the transition from asymmetric to symmetric fission through the calculation of primary fission yields with the time-dependent generator coordinate method (TDGCM). We choose here the transition in neutron-rich fermium isotopes, which was the first to be observed experimentally in the late 1970s and is often used as a benchmark for theoretical studies.

Methods: We compute the primary fission fragment mass and charge yields for ${}^{254}\mathrm{Fm}, {}^{256}\mathrm{Fm}$, and ${}^{258}\mathrm{Fm}$ from the TDGCM under the Gaussian overlap approximation. The static part of the calculation (generation of a potential energy surface) consists of a series of constrained Hartree-Fock-Bogoliubov calculations based on the D1S, D1M, or D1N parametrization of the Gogny effective interaction in a two-center harmonic oscillator basis. The two-dimensional dynamics in the collective space spanned by the quadrupole and octupole moments $({\widehat{Q}}_{20},{\widehat{Q}}_{30})$ is then computed with the finite element solver Felix-2.0.

Results: The available experimental data and the TDGCM post-dictions are consistent and agree especially on the position in the fermium isotopic chain at which the transition occurs. In addition, the TDGCM predicts two distinct asymmetric modes for the fission of ${}^{254}\mathrm{Fm}$.

Conclusions: Thanks to its intrinsic accounting of shell effects and to its ability to describe the dynamics of the system up to configurations close to scission, the TDGCM is able to describe qualitatively the fission yield transition in the neutron-rich fermium isotopes. This makes it a promising tool to study the evolution of the fission yields far from the valley of stability. The main limitation of the method lies in the presence of discontinuities in the two-dimensional manifold of generator states.

Figure 1

Potential energy surfaces of the 254, 256, and 258 fermium isotopes determined from the D1S Gogny energy density functional. The potential corresponds to the HFB energy corrected from the GCM zero-point energy. The color scale is shifted by 10 MeV between consecutive plots. The red continuous line represents the isoline $Q_N=7.5$ of the neck operator.

Figure 2

Slice of the D1S potential energy surfaces for the three fermium isotopes at the elongations $Q_{20}=140,180,225$ b. To emphasize the difference of topology between nuclei, all the curves are shifted so that $V(Q_{30}=0)=0$.

Figure 3

Primary fragment mass yields obtained with the Gogny D1S effective interaction and compared with various experimental data sets taken from Refs. [2, 3, 6, 10, 12, 46]. All the yields are normalized to 200%. The experimental data points all represent post-neutron evaporation mass yields. The open symbols stand for experimental data associated with spontaneous fission whereas full symbols are related to thermal neutron-induced fission.

Figure 4

Total energy as a function of the heavy fragment charge along the isoline $Q_N=7.5$.

Figure 5

Primary fragment charge yields (normalized to 200%) obtained with the Gogny D1S effective interaction. The black dotted line represents raw results directly obtained from the flux through the frontier, whereas the red full line accounts for the convolution of the raw results with a Gaussian of width $\sigma =4.0\times Z/A$.

Figure 6

Comparison of the primary fragment mass yields obtained with the D1S, D1N, and D1M parametrizations of the Gogny force. All the yields are normalized to 200%.

Figure 7

Evolution of the primary fragment mass yields as a function of the initial energy for ${}^{254}\mathrm{Fm}$ and ${}^{258}\mathrm{Fm}$. The energy is given in MeV relative to the energy of the saddle point of the first fission barrier.

Figure 8

Primary fragment mass yields computed with different Gaussian width ${\sigma }_i$ to build the initial wave packet. The width is given in MeV, and all yields are normalized to 200%.

Figure 9

Comparison of the primary mass yields obtained with the GCM (full red line) and the ATDHF (dashed black line) collective motion approaches. For completeness, we also give the results obtained with the GCM approach without the zero point energy correction on the potential landscape (dotted blue line). All the yields are normalized to 200%.

Figure 10

(a) Isolines $Q_N=7.0,7.5,8.0,8.5$ of the Gaussian neck operator used as frontiers to compute the fission yields of ${}^{254}\mathrm{Fm}$. (b) Variation of the primary fragment mass yields of ${}^{254}\mathrm{Fm}$ with the neck operator isoline used as frontier.

Figure 11

Discontinuity indicator plotted on top of the ${}^{256}\mathrm{Fm}$ PES. The red color scale represents the value of the discontinuity indicator $I\left(k\right)$. Values below 5.5 are not plotted. The background color map represents the potential energy surface. Finally, the black dashed line is the frontier corresponding to $Q_N=7.5$. For the sake of legibility, we removed the points belonging to the non-converged island of high energies in the fusion valley (see Fig. 1 for comparison).