Efficient method for estimation of fission fragment yields of $r$-process nuclei

Background: More than half of all the elements heavier than iron are made by the rapid neutron capture process (or $r$ process). For very-neutron-rich astrophysical conditions, such at those found in the tidal ejecta of neutron stars, nuclear fission determines the $r$-process endpoint, and the fission-fragment yields shape the final abundances of $110\le A\le 170$ nuclei. The knowledge of fission-fragment yields of hundreds of nuclei inhabiting very-neutron-rich regions of the nuclear landscape is thus crucial for the modeling of heavy-element nucleosynthesis.

Purpose: In this study, we propose a model for the fast calculation of fission-fragment yields based on the concept of shell-stabilized prefragments defined with help of the nucleonic localization functions.

Methods: To generate realistic potential-energy surfaces and nucleonic localizations, we apply Skyrme density-functional theory. The distribution of the neck nucleons among the two prefragments is obtained by means of a statistical model.

Results: We benchmark the method by studying the fission yields of ${}^{178}\mathrm{Pt},{}^{240}\mathrm{Pu},{}^{254}\mathrm{Cf}$, and ${}^{254,256,258}\mathrm{Fm}$ and show that it satisfactorily explains the experimental data. We then make predictions for ${}^{254}\mathrm{Pu}$ and ${}^{290}\mathrm{Fm}$ as two representative cases of fissioning nuclei that are expected to significantly contribute during the $r$-process nucleosynthesis occurring in neutron-star mergers.

Conclusions: The proposed framework provides an efficient alternative to microscopic approaches based on the evolution of the system in a space of collective coordinates all the way to scission. It can be used to carry out global calculations of fission-fragment distributions across the $r$-process region.

Figure 1

NLFs of selected nuclei used in the calculation of spontaneous fission-fragment yield distributions. Plots (a), (b), (g)–(n) are calculated with ${\mathrm{UNEDF}1}_{\text{HFB}}$ EDF and plots (c)–(f) are calculated with ${\mathrm{SkM}}^{*}$ EDF. Dashed lines indicate equatorial planes of the prefragments.

Figure 2

Density distributions of prefragments (bottom) and neck-nucleons (top) in the configuration $\mathcal{C}$ of ${}^{254}\mathrm{Fm}$ indicated in Fig. 3, obtained with ${\mathrm{UNEDF}1}_{\text{HFB}}$.

Figure 3

PESs of ${}^{254,256,258}\mathrm{Fm},{}^{254}\mathrm{Cf}$, and ${}^{240}\mathrm{Pu}$ calculated with ${\mathrm{UNEDF}1}_{\text{HFB}}$ (left panels) and ${\mathrm{SkM}}^{*}$ (right panels). For each nucleus, the energy (in MeV) is normalized to the ground-state energy $E_{\text{GS}}$. The contours $E=E_{\text{GS}}$ are indicated by thick solid lines. Most probable fission paths are calculated dynamically for $E^{*}=0$ with constant inertia [dashed lines in panels (a)–(d)] and nonperturbative cranking inertia [dotted line in panel (d)] while static pathways are computed for $E^{*}>0$ [dash-dotted lines in panels (e) and (f)]. The configurations $\mathcal{C}$ used to compute prefragments are marked by squares. For ${}^{258}\mathrm{Fm}$, two configurations are shown: ${\mathcal{C}}_s$ (symmetric fission pathway) and ${\mathcal{C}}_a$ (asymmetric fission pathway).

Figure 4

Prefragments for different fissioning systems calculated with ${\mathrm{SkM}}^{*}$ (squares) and ${\mathrm{UNEDF}1}_{\text{HFB}}$ (stars): (top) proton number, (bottom) neutron number. Heavy and light prefragments are marked with closed and open symbols, respectively. Note that, for ${\mathcal{C}}_s$ configurations of ${}^{258}\mathrm{Fm}$ and ${}^{290}\mathrm{Fm}$, only symmetric fission is expected and hence only one symbol is shown.

Figure 5

Single-particle levels calculated with ${\mathrm{SkM}}^{*}$ along the most probable fission path of ${}^{240}\mathrm{Pu}$ from the outer turning point to scission. Fermi energies are indicated by thick dashed lines.

Figure 6

Calculated yield distributions for SF (shaded for ${\mathrm{SkM}}^{*}$ and vertically patterned for ${\mathrm{UNEDF}1}_{\text{HFB}})$ and thermal fission (horizontally patterned) with two-particle uncertainty in each prefragment. Open circles show experimental data for SF of ${}^{240}\mathrm{Pu}$ [68], ${}^{254}\mathrm{Cf}$ [69], ${}^{254}\mathrm{Fm}$ [70, 71], ${}^{256}\mathrm{Fm}$ [72], and ${}^{258}\mathrm{Fm}$ [73]. Closed circles show experimental data for thermal fission of ${}^{240}\mathrm{Pu}$ [74] and ${}^{256}\mathrm{Fm}$ [75], and heavy-ion induced fission of ${}^{178}\mathrm{Pt}$ [57].

Figure 7

Yield distributions for ${}^{294}\mathrm{Og}$ calculated in ${\mathrm{SkM}}^{*}$ using the statistical model described in this work (shaded region) and the Langevin approach of Ref. [35] (dashed lines). Only heavy-fragment distributions are plotted.

Figure 8

Predicted fission properties of $r$-process nucleus ${}^{254}\mathrm{Pu}$. (left) Potential-energy surfaces calculated with (a) ${\mathrm{UNEDF}1}_{\text{HFB}}$ and (b) ${\mathrm{SkM}}^{*}$ functionals. The configurations $\mathcal{C}$ are marked by square symbols. (right) Predicted (c) mass- and (d) charge-yield distributions (shaded and patterned regions for ${\mathrm{SkM}}^{*}$ and ${\mathrm{UNEDF}1}_{\text{HFB}}$, respectively, given by our method. The GEF [30] and ABLA [31] results are shown by solid and dashed lines, respectively.

Figure 9

Similar as in Fig. 8 but for ${}^{290}\mathrm{Fm}$. The predictions of ${\mathrm{UNEDF}1}_{\text{HFB}}$ and ${\mathrm{SkM}}^{*}$ are practically identical.