$\beta$-decay half-lives of the $r$-process waiting-point isotones of $N=81$ and 82 nuclei

In this paper we perform systematic calculations of $\beta$-decay half-lives for the $r$-process waiting-point isotones of $N=81$ and 82, in the framework of nucleon-pair approximation. A phenomenological form of the Hamiltonian, which includes single-particle energies, pairing plus quadrupole interactions, and Gamow-Teller interactions is assumed, and the theoretical $\beta$-decay energies are taken based on the Bayesian machine learning. A good agreement with experimental values of half-lives is achieved by using the nucleon-pair truncated model space in which the $S$-pair condensation dominates. Our calculated half-lives are found to be very sensitive to $\beta$-decay energies on the one hand, and are quite robust with respect to extension of the model space and variation of interaction parameters on the other hand.

Figure 1

Energy levels of three nuclei, (a) ${}^{130}\mathrm{Sn}$, (b) ${}^{130}\mathrm{Cd}$, and (c) ${}^{130}\mathrm{In}$, which are used to fix parameters in our calculations. Experimental data (denoted by “EXP”) are taken from Ref. [64]. The results denote by “NSM” are the nuclear shell-model results calculated by using the parameters in Table 1.

Figure 2

$Q_{\beta }$ values for $N=81$ and 82 isotones versus proton number $Z$. Experimental data and those predicted in the AME2020 database [70] are plotted by using black balls in solid and hollow black balls. The theoretical values extracted from the FRDM [72], the systematics of proton-neutron interactions [73] and the WS4 model [74] are presented by using blue squares, red triangles, and green diamonds, respectively. The regions in dark grey are plotted by using the confidence intervals of $Q_{\beta }$ predicted by the Bayesian machine learning (BML) mass model [71].

Figure 3

Calculated energy spectra of parent nuclei. (a) corresponds to $N=82$ isotones with odd numbers of protons; (b) corresponds to $N=81$ isotones with even numbers of protons, and (c) corresponds to $N=81$ isotones with odd numbers of protons. In (c) we use different colors for configurations of different orbits in which the unpaired nucleons fill. See text for detailed description.

Figure 4

Occupation probability of the proton orbits in the ground states of $N=82$ isotones by our optimized $S$-pair truncation. Results of the generalized seniority scheme are plotted for comparison.

Figure 5

$\beta$-decay half-lives for $N=82$ and 81 isotones versus proton number $Z$. (a) corresponds to $N=82$, and (b) corresponds to $N=81$. Experimental data, and theoretical results from the NSM [26], the $\mathrm{QRPA}+\mathrm{HFB}21$ [21], and the $\mathrm{RQRPA}+\mathrm{RHB}$ [23] methods, are also presented in black, blue, red, and orange, respectively. The range in dark grey represents our predicted half-lives with uncertainties originated from the nuclear masses.

Figure 6

Gamow-Teller decay rates of the parent ground state (${\lambda }_{ij}^{\mathrm{GT}}$ with $j=0$) versus excitation energy of populated states for daughter nuclei. (a)–(l) correspond to $N=82$ isotones, and (m)–(x) correspond to $N=81$ isotones. Proton number $38⩽Z⩽49$.

Figure 7

Gamow-Teller decay rates from the ground state of ${}^{128}\mathrm{Pd}$, ${}^{126}\mathrm{Ru}$, and ${}^{124}\mathrm{Mo}$ in the $SD$-pair (triangles in blue) and $S$-pair (solid squares in black) truncated spaces, versus excitation energies of daughter nuclei, populated by the GT transitions.

Figure 8

Half-life of ${}^{126}\mathrm{Ru}$ versus multipliers on the interaction parameters and the $\beta$-decay energy. $x_P$, $x_{QQ}$, and $x_{\mathrm{GT}}$ are multipliers of the $H_P$, $H_{QQ}$, and $H_{\mathrm{GT}}$ terms, respectively, in Eq. (4). The range of $Q_{\beta }$ is adopted to be the theoretical uncertainty in the BML mass model.