Nuclear double-$\beta$ decay with two neutrinos is a rare and important process for natural radioactivity of unstable nuclei. The experimental data of nuclear double-${\beta }^{-}$ decay with two neutrinos are analyzed and a systematic law to calculate the half-lives of this rare process is proposed. It is the first analytical and simple formula for double-$\beta$-decay half-lives where the leading effects from both the Coulomb potential and the nuclear structure are included. The systematic law shows that the logarithms of the half-lives are inversely proportional to the decay energies for the ground-state transitions between parent nuclei and daughter nuclei. The calculated half-lives are in agreement with the experimental data of ground-state transitions of all known 11 nuclei with an average factor of 3.06. The half-lives of other possible double-$\beta$-decay candidates with two neutrinos are predicted and these can be useful for future experiments. The law, without introducing any extra adjustment, is also generalized to the calculations of double-$\beta$-decay half-lives from the ground states of parent nuclei to the first $0^{+}$ excited states of daughter nuclei, and the calculated half-lives agree very well with the available data. The calculated half-lives from the ground states of parent nuclei ${}^{48}$Ca and ${}^{150}$Nd to the first $0^{+}$ excited states of daughter nuclei are the first theoretical results as far as we know. The similarity and difference between the law of $\alpha$ decay and that of double-${\beta }^{-}$ decay are also analyzed and discussed.

• Received 5 March 2014
• Revised 14 April 2014

DOI:https://doi.org/10.1103/PhysRevC.89.064603