Multimodal fission from self-consistent calculations

Background: When multiple fission modes coexist in a given nucleus, distinct fragment yield distributions appear. Multimodal fission has been observed in a number of fissioning nuclei spanning the nuclear chart, and this phenomenon is expected to affect the nuclear abundances synthesized during the rapid neutron-capture process ($r$-process).

Purpose: In this study, we generalize the previously proposed hybrid model for fission-fragment yield distributions to predict competing fission modes and estimate the resulting yield distributions. Our framework allows for a comprehensive large-scale calculation of fission-fragment yields suited for $r$-process nuclear network studies.

Methods: Nuclear density functional theory is employed to obtain the potential energy and collective inertia tensor on a multidimensional collective space defined by mass multipole moments. Fission pathways and their relative probabilities are determined using the nudged elastic band method. Based on this information, mass and charge fission yields are predicted using the recently developed hybrid model.

Results: Fission properties of fermium isotopes are calculated in the axial quadrupole-octupole collective space for three energy density functionals (EDFs). Disagreement between the EDFs appears when multiple fission modes are present. Within our framework, the ${\mathrm{UNEDF}1}_{\text{HFB}}$ EDF agrees best with experimental data. Calculations in the axial quadrupole-octupole-hexadecapole collective space improve the agreement with the experiment for ${\mathrm{SkM}}^{*}$. We also discuss the sensitivity of fission predictions on the choice of EDF for several superheavy nuclei.

Conclusions: Fission-fragment yield predictions for nuclei with multiple fission modes are sensitive to the underlying EDF. For large-scale calculations in which a minimal number of collective coordinates is considered, ${\mathrm{UNEDF}1}_{\text{HFB}}$ provides the best description of experimental data, though the sensitivity motivates robust quantification of the uncertainties of the theoretical model.

Figure 1

(a) Mass and (b) charge yield distributions of ${}^{284}\mathrm{Cn}$. The magenta patterned regions are calculated with the ${\mathrm{UNEDF}1}_{\text{HFB}}$ functional (our model). Also shown are the results from the BSM [19] (black solid lines), SPM [14] (blue dashed lines), and GEF [20] (red dash-dotted lines) models.

Figure 2

Potential energy surfaces of ${}^{254,256,258,260}\mathrm{Fm}$ (in MeV) calculated using ${\mathrm{UNEDF}1}_{\text{HFB}}$ (left column), ${\mathrm{SkM}}^{*}$ (center column), and D1S (right column) EDFs. The symmetric (dashed lines) and asymmetric (solid and dotted lines) least action paths are drawn from the ground state (solid circle) to the fission isomer (asterisk) to the exit point (open square). The white contour denotes the outer turning line $V_{\text{eff}}=\mathrm{\Delta }E=0$. Gray contours are marked at 1-MeV intervals for $0<V_{\text{eff}}<5$ MeV.

Figure 3

Fission-fragment mass (left) and charge (right) yields for ${}^{254,256,258,260}\mathrm{Fm}$ calculated with ${\mathrm{UNEDF}1}_{\text{HFB}}$ (magenta vertical patterns), ${\mathrm{SkM}}^{*}$ (black horizontal patterns), and D1S (green $\times$ patterns). Experimental yields (circles) [75, 76, 77, 78] are shown where available.

Figure 4

Panels (a)–(c): Similar to Fig. 2 but for ${}^{258}\mathrm{Rf}, {}^{262}\mathrm{Sg}$, and ${}^{262}\mathrm{Hs}$ using ${\mathrm{UNEDF}1}_{\text{HFB}}$. Panels (d)–(f) and (g)–(i): Mass and charge yields, respectively, for ${\mathrm{UNEDF}1}_{\text{HFB}}$ (magenta vertical patterns) and ${\mathrm{SkM}}^{*}$ (black horizontal patterns).

Figure 5

The fragment mass (a) and charge (b) yields for ${}^{254}\mathrm{Fm}$ calculated with ${\mathrm{SkM}}^{*}$ in the 2D (green solid lines) and 3D (blue dashed lines) collective spaces. Experimental yields from Refs. [75, 76] are shown as black and red circles, respectively.

Figure 6

The least action paths in 3D for ${}^{258}\mathrm{Fm}$ using ${\mathrm{SkM}}^{*}$ (a) and ${\mathrm{UNEDF}1}_{\text{HFB}}$ (b). The outer turning surface is shown. A 2D PES is shown for constant $Q_{40}=16{\mathrm{b}}^2$ ($Q_{40}=32 {\mathrm{b}}^2$) for ${\mathrm{SkM}}^{*}$ (${\mathrm{UNEDF}1}_{\text{HFB}}$). Neutron localizations for the identified precission configurations are shown in the insets.

Figure 7

The fragment mass (left) and charge (right) yields for ${}^{258}\mathrm{Fm}$ using ${\mathrm{SkM}}^{*}$ (top) and ${\mathrm{UNEDF}1}_{\text{HFB}}$ (bottom) calculated in the 2D (green solid lines) and 3D (blue dashed lines) collective spaces. Experimental data from Ref. [78] are shown as red solid circles.

Figure 8

Similar to Fig. 6 but for ${}^{306}122$. The 2D PESs are at constant $Q_{40}=0$.

Figure 9

The fragment mass (a) and charge (b) yields for ${}^{306}122$ calculated with ${\mathrm{UNEDF}1}_{\text{HFB}}$ (black solid lines) and ${\mathrm{SkM}}^{*}$ (magenta dashed lines).