Structure of ${}^{46,47}\mathrm{Ca}$ from the ${\beta }^{-}$ decay of ${}^{46,47}\mathrm{K}$ in the framework of the nuclear shell model

In the present work, we report a comprehensive theoretical study of the $logft$ values for the allowed and forbidden ${\beta }^{-}$ decay transitions of ${}^{46}\mathrm{K}$ and ${}^{47}\mathrm{K}$ corresponding to recently available experimental data from the GRIFFIN spectrometer at TRIUMF-ISAC [Phys. Rev. C 100, 054327 (2019); Phys. Rev. C 102, 054314 (2020)]. We perform the nuclear shell-model calculation in $sdpf$-valence space, with the SDPF-MU interaction, to calculate low-lying energy spectra of ${}^{46,47}\mathrm{K}$ and ${}^{46,47}\mathrm{Ca}$. For further investigation, we also calculate spectroscopic properties of ${}^{46,47}\mathrm{K}$ and ${}^{46,47}\mathrm{Ca}$ and compare the results with the experimental data. We suggest the spin-parity of several levels, which was previously tentative, in both the calcium isotopes. Based on the energy and $logft$ values, we conclude that the state at 3.984 MeV in ${}^{46}\mathrm{Ca}$ could be either $3^{-}$ or $2^{+}$ and a state at 4.432 MeV might be $3^{-}$. The spin-parity of the states at 2.875, 3.951, and 4.453 MeV are predicted to be $1/2^{-}, 3/2^{-}$, and $1/2^{-}$ or $1/2^{+}$, respectively, in ${}^{47}\mathrm{Ca}$. We also give the confirmation to the spin-parity of 2.850, 3.889, and 4.606 MeV states for ${}^{47}\mathrm{Ca}$. Our results of level schemes of ${}^{46,47}\mathrm{K}$ are in reasonable agreement with the experimental data, while the agreement of calculated $logft$ values for the ${\beta }^{-}$ decay with the experimental data is limited.

Figure 1

Comparison of calculated and experimental average shape factors for Gamow-Teller and first-forbidden transitions to obtain the quenching factor. For extraction of the quenching factor, we have used only confirmed transitions of ${}^{46,47}\mathrm{Ca}$.

Figure 2

Comparison of theoretical and experimental energy spectra of ${}^{46}\mathrm{K}$ and ${}^{46}\mathrm{Ca}$. Experimental data are taken from Refs. [31, 64].

Figure 3

Comparison of theoretical and experimental energy spectra of ${}^{46}\mathrm{K}$ and ${}^{46}\mathrm{Ca}$. Experimental data are taken from Refs. [31, 64].

Figure 4

Comparison of excitation energies of low-lying excited states in ${}^{46}\mathrm{Ca}$ obtained from SDPF-MU interaction with previously obtained results in $s{d}_{3/2}+f_{7/2}{p}_{3/2}, pf$, and $pf{\mathrm{g}}_{9/2}$ valence spaces using $NN+3N$ forces [30, 45]. Experimental data are also shown in first column for comparison.

Figure 5

Comparison of excitation energies of low-lying excited states in ${}^{47}\mathrm{Ca}$ obtained from SDPF-MU interaction with previously obtained results in $pf$ and $pf{\mathrm{g}}_{9/2}$ valence spaces using $NN+3N$ forces [45]. Experimental data are also shown in first column for comparison.