$\beta$-delayed neutron-emission and fission calculations within relativistic quasiparticle random-phase approximation and a statistical model

Background: $\beta$-delayed neutron emission and fission are essential in $r$-process nucleosynthesis. Although the number of experimental studies covering $r$-process nuclei has recently increased, the uncertainties of $\beta$-delayed neutron emission and fission are still large for $r$-process simulations.

Purpose: Our aim is to introduce a theoretical framework for the description of $\beta$-delayed neutron-emission and fission rates based on relativistic nuclear energy density-functional and statistical models and investigate their properties throughout the nuclide map.

Methods: To obtain $\beta$ strength functions, the relativistic proton-neutron quasiparticle random-phase approximation is employed. Particle evaporations and fission from highly excited nuclear states are estimated by the Hauser-Feshbach statistical model. $\beta$-delayed neutron branching ratios $P_n$ are calculated and compared with experimental data, and the $\beta$-delayed fission branching ratio $P_f$ are also assessed by using different fission barrier data.

Results: Calculated $P_n$ are in a good agreement with the experimental data and the root mean square deviation is comparable to results of preceding works. It is found that energy withdrawal by $\beta$-delayed neutron-emission sensitivity varies $P_n$, especially for nuclei near the neutron drip line. $P_f$ depend sensitively on fission barrier data. It is found that not only the barrier height but also the number of barrier humps is important to evaluate $P_f$.

Conclusions: The framework introduced in this work provides an improved theoretical description of the $\beta$-delayed neutron emission and fission. Since $P_f$ as well as $P_n$ depend strongly on fission barrier information, four kinds of fission barrier data are used in this work to allow further sensitivity studies of the $r$-process nucleosynthesis on the nuclear fission. More studies on fission barrier are highly requested to assess the role of $\beta$-delayed fission in the $r$-process study. A complete set of calculated data for $\beta$-delayed neutron emission and fission are summarized as a table in supplemental material for its use in $r$-process studies as well as to complement a part of nuclear data in which no experimental data are available.

Figure 1

Weight functions of Gaussian type (the dotted line) of Eq. (8) and Lorentzian (the solid line) type of Eq. (9) with $E^{*}=10$ MeV and width $\mathrm{\Gamma }=0.6$ MeV. The inserted panel is depicted in a linear scale

Figure 2

Estimated ${\sigma }_{\mathrm{rms}}^{1n}$ value given in Eq. (15) as a function of the width $\mathrm{\Gamma }$ calculated with the Gaussian-type function (8) and the Lorentzian type function (9).

Figure 3

(a) Ratio of $P_{1n}$ for $pn$-RQRPA $+$ HFM to that for $pn$-RQRPA. Panels (b) and (c) represent ratio of calculated to experimental data of $P_{1n}$ ($\mathrm{C}/\mathrm{E}$) for $pn$-RQRPA $+$ HFM (this work) and $pn$-RQRPA [15], respectively.

Figure 4

Total $\beta$-delayed neutron branching ratios ($P_n={\sum }_x{P}_{xn}$) calculated by the $pn$-RQRPA $+$ HFM with HFB-14 fission barrier data [76]. The black filled squares stand for stable or long-lived nuclei.

Figure 5

$\beta$-delayed neutron branching ratio of $P_{1n}$ to $P_{5n}$ in the $N\text{−}Z$ plane.

Figure 6

Same as Fig. 5, but for $P_{6n}$ to $P_{10n}$.

Figure 7

Average difference $M_x$ for various $x$ defined in Eq. (16).

Figure 8

(a) $\beta$ strength function of ${}^{140}\mathrm{Sn}$. $x$-neutron ($xn$) emission energy windows are separated by the dotted lines. (b) Isotope production ratios, $p_i$ of ${}^{137–140}\mathrm{Sb}$ as a function of excitation energy, assuming $1^{+}$ excited states of ${}^{140}\mathrm{Sb}$ (the daughter nucleus of ${}^{140}\mathrm{Sn}$). The calculation is performed by the HFM.

Figure 9

(a) $\beta$ strength function of ${}^{162}\mathrm{Sn}$. $xn$ emission energy windows are separated by the dotted lines. (b) Isotope production ratios of ${}^{162–156}\mathrm{Sb}$ from $1^{+}$ excited states of ${}^{162}\mathrm{Sb}$ (the daughter nucleus of ${}^{162}\mathrm{Sn}$) as a function of excitation energy. The calculation is performed by the HFM.

Figure 10

$\beta$-delayed neutron branching ratio $P_{1n}$ of Pd, Ag, Os, and Ir isotopes as a function of $A$. The experimental data are taken from Ref. [82].

Figure 11

BDNE spectrum of (a) ${}^{89}\mathrm{Br}$ and (b) ${}^{138}\mathrm{I}$. The calculated result is shown by the solid line (red). Experimental data are taken from Rudstam et al. [91] and Shalev et al. [92].

Figure 12

BDNE branching ratios $P_n$ (left panels) and BDF branching ratios $P_f$ (right panels) in unit of % calculated by the fission barrier data of (a) HFB-14 [76], (b) FRDM [78], (c) ETFSI [77], and (d) SBM [79]. Nuclei where no fission barrier or $\beta$-decay data are provided are given as blank.

Figure 13

Same as Fig. 12, but for $P_{1nf}$.

Figure 14

Same as Fig. 12, but for $P_{2nf}$.

Figure 15

Same as Fig. 12, but for $P_{3nf}$.

Figure 16

Same as Fig. 12, but for $P_{4nf}$.

Figure 17

Same as Fig. 12, but for $P_{5nf}$.

Figure 18

Fission barrier height of Fm ($Z=100$) isotopes for HFB-14 [76], FRDM [78], ETFSI [77], and SBM [79] models. The numbers in parentheses indicate $1:\text{inner}$, $2:\text{middle}$, and $3:\text{outer}$ fission barrier heights.

Figure 19

Delayed alpha branching ratios $P_{\alpha }$ (%) in the case of FRDM fission barrier data. The black filled squares indicate stable or long-lived nuclei.

Figure 20

Mean number of emitted neutrons in the $N\text{−}Z$ plane. The calculation is performed by the HFB-14 fission barrier data [76].

Figure 21

Mean number of emitted neutrons for nickel, tin, and fermium isotopes.

Figure 22

Mean energy of emitted neutrons in the $N\text{−}Z$ plane.

Figure 23

Mean energy of $\beta$-delayed neutron $\langle E_n\rangle$. For fermium isotopes, the results with different fission barrier data are also plotted together.