$g_{\mathrm{A}}$-driven shapes of electron spectra of forbidden $\beta$ decays in the nuclear shell model

The evolution of the shape of the electron spectra of 16 forbidden ${\beta }^{-}$ decays as a function of $g_{\mathrm{A}}$ was studied using the nuclear shell model in appropriate single-particle model spaces with established, well-tested nuclear Hamiltonians. The $\beta$ spectra of ${}^{94}\mathrm{Nb}\left(6^{+}\right)\to {}^{94}\mathrm{Mo}\left(4^{+}\right)$ and ${}^{98}\mathrm{Tc}\left(6^{+}\right)\to {}^{98}\mathrm{Ru}\left(4^{+}\right)$ were found to depend strongly on $g_{\mathrm{A}}$, which makes them excellent candidates for the determination of the effective value of $g_{\mathrm{A}}$ with the spectrum-shape method (SSM). A strong $g_{\mathrm{A}}$ dependence is also seen in the spectrum of ${}^{96}\mathrm{Zr}\left(0^{+}\right)\to {}^{96}\mathrm{Nb}\left(6^{+}\right)$. This decay could be used for determining the quenching of $g_{\mathrm{A}}$ in sixth-forbidden decays in the future, when the measurement of the spectrum becomes experimentally feasible. The calculated shell-model electron spectra of the ground-state-to-ground-state decays of ${}^{87}\mathrm{Rb}$, ${}^{99}\mathrm{Tc}$, and ${}^{137}\mathrm{Cs}$ and the decay of ${}^{137}\mathrm{Cs}$ to the isomeric $11/2^{-}$ state in ${}^{137}\mathrm{Ba}$ were found to be in excellent agreement with the spectra previously calculated using the microscopic quasiparticle-phonon model. This is further evidence of the robust nature of the SSM observed in the previous studies.

Figure 1

Normalized electron spectra for the second-forbidden nonunique ${\beta }^{-}$ decays of ${}^{36}\mathrm{Cl}$ [panel (a)] and ${}^{60}\mathrm{Fe}$ [panel (c)], and the fourth-forbidden nonunique decay of ${}^{50}\mathrm{V}$ [panel (b)]. The value $g_{\mathrm{V}}=1.00$ was adopted and the color coding represents the value of $g_{\mathrm{A}}$.

Figure 2

Same as Fig. 1 but for the second-forbidden nonunique decays of ${}^{94}\mathrm{Nb}$ [panel (a)], ${}^{98}\mathrm{Tc}$ [panel (b)], and ${}^{126}\mathrm{Sn}$ [panel (c)].

Figure 3

Same as Fig. 1 but for the decays ${}^{48}\mathrm{Ca}\left(0^{+}\right)\to {}^{48}\mathrm{Sc}(4^{+},5^{+},6^{+})$ [panels (a), (b), and (c)].

Figure 4

Same as Fig. 1 but for the decays ${}^{96}\mathrm{Zr}\left(0^{+}\right)\to {}^{96}\mathrm{Nb}(4^{+},5^{+},6^{+})$ [panels (a), (b), and (c)].

Figure 5

Presently calculated shell-model spectra and the MQPM spectra, first published in Ref. [30], for the ground-state-to-ground-state decays of ${}^{87}\mathrm{Rb}$ [panels (a) and (b)], ${}^{99}\mathrm{Tc}$ [panels (c) and (d)], and ${}^{137}\mathrm{Cs}$ [panel (f)], and the decay of ${}^{137}\mathrm{Cs}$ to the isomeric $11/2^{-}$ state in ${}^{137}\mathrm{Ba}$ [panel (e)]. The color coding in the spectra of ${}^{87}\mathrm{Rb}$ and ${}^{99}\mathrm{Tc}$ refers to the value of $g_{\mathrm{A}}$. The shape of the electron spectra of the two ${}^{137}\mathrm{Cs}$ decays is $g_{\mathrm{A}}$ independent, and the color coding in these decays refers to the adopted nuclear model. The free-nucleon value $g_{\mathrm{V}}=1.00$ was used in all calculations.