Systematic Law for Half-lives of Double $\beta$-decay with Two Neutrinos

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 effect from both the Coulomb potential and nuclear structure is 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 all known eleven 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. Some calculated half-lives are the first theoretical results of double $\beta$-decay half-lives from the ground states of parent nuclei to the first $0^+$ excited states of daughter nuclei. The similarity and difference between the law of $\alpha$-decay and that of double $\beta^{-}$-decay are also analyzed and discussed.