Research on convergence of the nuclear matrix elements for 2νββ decays

In this work, the characteristics of \begin{document}$2\nu\beta\beta$\end{document} decays for six nuclei (\begin{document}$^{36}$\end{document}Ar, \begin{document}$^{46}$\end{document}Ca, \begin{document}$^{48}$\end{document}Ca, \begin{document}$^{50}$\end{document}Cr, \begin{document}$^{70}$\end{document}Zn, and \begin{document}$^{136}$\end{document}Xe) in a mass range from \begin{document}$A = 36$\end{document} to \begin{document}$A = 136$\end{document} are studied within the nuclear shell model (NSM) framework. Calculations are presented for the half-lives, nuclear matrix elements (NMEs), phase space factors (\begin{document}$G_{2\nu}$\end{document}), and convergence of the NMEs. The theoretical results agree well with the experimental data. In addition, we predict the half-lives of \begin{document}$2\nu\beta\beta$\end{document} decays for four nuclei. We focus on the convergence of the NMEs by analyzing the number of contributing intermediate \begin{document}$1^{+}$\end{document} states (\begin{document}$N_{\rm{C}}$\end{document}) for the nuclei of interest. We assume that \begin{document}$N_{\rm{C}}$\end{document} is safely determined when the accumulated NMEs saturate 99.7% of the final calculated magnitude. From the calculations of the involved nuclei, we discover a connection between \begin{document}$N_{\rm{C}}$\end{document} and the total number of intermediate \begin{document}$1^{+}$\end{document} states (\begin{document}$N_{\rm{T}}$\end{document}). According to the least squares fit, we conclude that the correlation is \begin{document}$N_{\rm{C}}=\left( 10.8\pm 1.2\right) \times N_{\rm{T}}^{\left( 0.29\pm 0.02\right)}$\end{document}.