Shell-model description of the electron spectra for the second-forbidden nonunique $β^-$ decays of the $fp$-shell nuclei $^{46}$Sc and $^{59,60}$Fe

Anil Kumar, Praveen C. Srivastava, Jouni Suhonen

Published 2021-01-08Version 1

In the present work, we have computed the shape factors and electron spectra for the second-forbidden nonunique $\beta^-$-decay transitions of $^{46}$Sc, and $^{59,60}$Fe in the framework of the nuclear shell model. We have performed the shell-model calculations of all the involved wave functions in the $\beta$-decay rate by using the KB3G and GXPF1A interactions in the full $fp$ model space. When compared with the available data, these effective interactions predict the low-energy spectra and electromagnetic properties of the involved nuclei quite successfully. This success paves the way for a reliable computation of the $\beta$-decay properties, and comparison with data. We use the spectrum-shape method (SSM), including the next-to-leading-order corrections in the shape factor, in order to compute the electron spectral shapes as functions of the weak axial coupling constant $g_{\rm A}$. We have also constrained the value of the relativistic vector matrix element, $^V\mathcal{M}^{(0)}_{KK-11}$, using the conserved vector-current hypothesis (CVC) and found that this procedure influences the electron spectral shapes. Based on the "CVC-inspired" SSM calculations we find that the spectral shapes of $^{59,60}$Fe depend strongly on the value of $g_{\rm A}$, thus making these nuclei as perfect test cases of the revised SSM for future $\beta$-decay experiments that are able to resolve electron spectral shapes.