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

TL;DR

This paper introduces a simple, analytical systematic law for calculating double beta-decay half-lives with two neutrinos, incorporating Coulomb and nuclear effects, and validates it against experimental data.

Contribution

It presents the first analytical formula for double beta-decay half-lives that includes Coulomb and nuclear structure effects, accurately matching experimental data.

Findings

The law accurately predicts half-lives of 11 known nuclei with an average factor of 3.06.

It generalizes to excited state transitions with good agreement.

Predictions for unknown candidates can guide future experiments.

Abstract

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.