$β$-Decay Half-Lives Serve as Novel Evidence for the New Magic Number \(N=32\)

Abstract

Conventional signatures of nuclear magic number, including low-lying quadrupole collectivity and mass systematics, face significant challenges when probing emergent shell closures near the drip line. However, $β$-decay half-lives are among the first experimental observables measurable following the discovery of neutron-rich isotopes. This letter demonstrates that $β$-decay half-lives provide evidence for the emergent magic number $N=32$. The observed half-life pattern around the $N=32$ can be attributed to the occupation probabilities of orbitals above this shell gap, which directly reflect the gap's magnitude. Our results reveal a pronounced $N=32$ shell gap in Ca isotopes and a weaker yet apparent gap in K isotopes, consistent with mass and electromagnetic transition data. Furthermore, the analysis indicates no prominent closed-shell signature at $N=32$ in Ar and Cl isotopes.

Figures (4)

• Figure 1: (a)(b) Calculated $\beta$-decay half-lives of Ca and K isotopes using the PKA1 and DD-ME2 Lalazissis2005 interactions. For the PKA1 interaction, the isoscalar pairing strength is fixed at $V_{0} = 200$ MeV for Ca isotopes and $V_{0} = 240$ MeV for K isotopes; for DD-ME2 interaction, it is $V_{0} = 80$ MeV for Ca and $V_{0} = 120$ MeV for K. The isovector pairing strength is set to its standard value $(f_{\text{IV}}= 1.0)$. For comparison, experimental half‑lives Kondev2021 and FRDM Moeller2019 results are shown. (c)(d) The two-neutron gap, defined as the difference of two-neutron separation energies, $\Delta_{2n}=S_{2n}(N)-S_{2n}(N+2)$, for Ca and K isotopes calculated with the PKA1 and DD-ME2 interactions. Experimental values from the AME2020 evaluation Wang2021, and FRDM predictions Moeller2019 are presented for comparison.
• Figure 2: (a)(b)(c) The single-particle energy of $^{50,52,54}$Ca calculated using PKA1 interaction. Dominant GT transitions, critical for $\beta$-decay half-lives, are shown with their strengths [$B_m$(GT)] and $\beta$-decay transition energies ($E_m$) [cf. Eq. \ref{['Eq3']}]. The green arrow indicates the dominant single-particle GT transition in the $\beta$-decay half-life, namely $\nu 1f_{5/2} \rightarrow \pi 1f_{7/2}$. Additionally, the occupation probabilities of the $\nu 2p_{1/2}$ and $\nu 1f_{5/2}$ orbitals are displayed. For more details, see the text.
• Figure 3: (a)(b) The $\beta$-decay half-lives of Ca and K isotopes calculated using PKA1 interaction with the different strength of isovector pairing force. The isoscalar pairing force strength is fixed at $V_{0} = 200$ MeV for Ca isotopes and $V_{0} = 240$ MeV for K isotopes. Experimental data Kondev2021 are included for comparison. (c) The shell gaps at $N = 32$ and $N = 34$ for Ca isotopes, defined as the difference between the corresponding single-particle energies, are presented for different strength of isovector pairing force using the PKA1 interaction. (d) The occupation probabilities for the $\nu 1f_{5/2}$ and $\nu 2p_{1/2}$ orbitals in Ca isotopes obtained by using the PKA1 interaction.
• Figure 4: (a)(b) $\beta$-decay half-lives of Ar and Cl isotopes calculated using the PKA1 and DD-ME2 interactions with different isovector pairing strengths (values indicated in brackets). For the PKA1 interaction, the isoscalar pairing strength is fixed at $V_{0} = 255$ MeV for Ar isotopes and $V_{0} = 270$ MeV for Cl isotopes; for DD-ME2 interaction, it is $V_{0} = 175$ MeV for Ar and $V_{0} = 220$ MeV for Cl. Experimental data QBZeng2025 are provided for comparison. (c)(d) Occupation probabilities for the $\nu 1f_{5/2}$ and $\nu 2p_{1/2}$ orbitals in Ar and Cl isotopes obtained with the PKA1 interaction using different isovector pairing strengths.