\begin{document}$ \beta $\end{document} decays with two neutrinos (\begin{document}$ 2\nu\beta\beta $\end{document}) is performed based on the analytical formula proposed by Primakoff and Rosen. We improve the formula by considering the shell effects and refining the energy dependence of the phase-space factor. This improved formula can closely describe all available experimental half-lives of \begin{document}$ 2\nu\beta^{-}\beta^{-} $\end{document} decays, both for ground-state transitions and transitions from ground states of parent nuclei to the first \begin{document}$ 0^{+} $\end{document} excited states of daughter nuclei. The calculated half-lives agree with the experimental data of ground-state transitions of all known eleven nuclei with an average factor of 2.3. Further predictions are provided for \begin{document}$ 2\nu\beta\beta $\end{document}-decay candidates with decay energies above 0.5 MeV. We compare different theoretical predictions and emphasize the importance of experimental measurements on the half-lives of double-\begin{document}$ \beta $\end{document} transitions between the ground state of \begin{document}$ ^{48} $\end{document}Ca, \begin{document}$ ^{76} $\end{document}Ge, and \begin{document}$ ^{136} $\end{document}Xe and the first \begin{document}$ 0^{+} $\end{document} excited states of their corresponding daughter nuclei, which will be very useful for understanding the underlying mechanisms of double-\begin{document}$ \beta $\end{document} decays and for further studying the shell effects on nuclear transition matrix elements."> Calculation of double-<i>β</i> decay half-lives using an improved Primakoff-Rosen formula -
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