${\beta }^{-}$-delayed fission in the coupled quasiparticle random-phase approximation plus Hauser-Feshbach approach

Beta-delayed neutron emission and $\beta$-delayed fission ($\beta \mathrm{df}$) probabilities were calculated for heavy, neutron-rich nuclei using the Los Alamos coupled quasiparticle random-phase approximation plus Hauser-Feshbach ($\mathrm{QRPA}+\mathrm{HF}$) approach. In this model, the compound nucleus is initially populated by $\beta$ decay and is followed through subsequent statistical decays taking into account competition among neutrons, $\gamma$ rays, and fission. The primary output of these calculations includes branching ratios along with neutron and $\gamma$-ray spectra. We find a relatively large region of heavy nuclides where the probability of $\beta \mathrm{df}$ is near 100%. For a subset of nuclei near the neutron drip line, delayed neutron emission and the probability of fission are both large, which leads to the possibility of multichance $\beta \mathrm{df}$ (mc-$\beta \mathrm{df}$). We comment on prospective neutron-rich nuclei that could be probed by future experimental campaigns and provide a full table of branching ratios in ASCII format in the Supplemental Material for use in various applications.

Figure 1

Schematic of the combined $\mathrm{QRPA}+\mathrm{HF}$ approach when applied to $\beta \mathrm{df}$. Initial population of the daughter nucleus ($Z+1,A$) is determined by the $\beta$-strength function ($S_{\beta }$) using QRPA. The competition among neutron emission, $\gamma$ deexcitation, and fission is then handled in the HF framework, for which the transmission coefficients ($T_{\text{n}}$, $T_{\gamma }$, $T_{\text{f}}$) are calculated respectively at each stage of the statistical decay. In some cases, the fission barrier heights ($E_A$, $E_B$) may contain only a single hump ($E_B=0$), and could possibly be much larger than denoted by this schematic. Multichance $\beta \mathrm{df}$ occurs for the daughter nucleus ($Z+1,A-1$) and beyond. The trailing dots denotes that the statistical decay continues until the total available excitation energy ($Q_{\beta }$) is exhausted.

Figure 2

Competition among the neutron, $\gamma$, and fission channels in the case of $\beta$-delayed fission of precursor nucleus $Z=97$, $A=282$, shown as a ratio of the respective transmission coefficient to the total sum. The total transmission coefficient sum (dashed light line), which spans many orders of magnitude, can be read from the right $Y$ axis. The energies in the middle and upper panel are shifted relative to the ground state of ${}^{282}\mathrm{Cf}(Z=98$).

Figure 3

Individual probabilities for delayed neutron and delayed fission as a function of $j$, neutrons emitted from the daughter nucleus. ${}^{295}\mathrm{Fm}$ is dominated by fission while ${}^{290}\mathrm{Am}$ shows more competition between neutron emission and fission.

Figure 4

(a) The cumulative probabilities for emitting neutrons (blue) or fission (red) after $\beta$ decay for neutron-rich plutonium ($Z=94$) isotopes. The summation of these two terms yields 100%. The average neutron multiplicity after $\beta$ decay, $\langle n\rangle$, is also displayed by a dashed line and read from the right $Y$ axis. (b) The same quantities shown for the berkelium ($Z=97$) isotopic chain. The solid grey vertical line indicates the one-neutron drip line in FRDM2012.

Figure 5

The one-neutron separation energies and maximum fission barriers along the same isotopic chains as in Fig. 4. The interplay between these two quantities strongly impacts $P_{j\text{n}}$ and $P_{j\text{f}}$ predictions.

Figure 6

(a) The average neutron multiplicity, $\langle n\rangle$, after $\beta$ decay for heavy neutron-rich nuclei computed with the $\mathrm{QRPA}+\mathrm{HF}$ framework. (b) Average $\beta$-delayed neutron emission energy in units of MeV. (c) The cumulative probability of $\beta \mathrm{df}$ occurring in the region. Nuclei with large cumulative probability and neutron multiplicity are classified as multichance $\beta \mathrm{df}$, denoted by solid black border. The black dots denote the $r$-process highway, roughly $S_{\text{n}}\approx 1$ to $2\mathrm{MeV}$, using FRDM2012 masses. Nuclei that have 100% probability for $\beta \mathrm{df}$ are filled with the darkest shade of purple; this color does not appear on the colorbar associated with panel (c).

Figure 7

The difference in cumulative probabilities of emitting neutrons between the results of Ref. [9] and this work. These differences highlight the primary impact of the first-forbidden (FF) contributions.

Figure 8

The percentage of models that predict $\beta \mathrm{df}$ with $P_{\text{f}}>1\%$ just outside the range of current measurements (see text for details). The grey region indicates the extent of the NuBase (2020) evaluation and the black line is five neutrons from it. Shaded nuclei in region between these two bounds represent the first potential candidates for future exploration; these nuclei are summarized in Table 1.

Figure 9

Using model variations, the number of nuclei with at least 1% chance of $\beta \mathrm{df}$ increases quadratically (dotted gray) with neutrons beyond NuBase (2020) (green). See text for details.