The contribution of the first forbidden transitions to the nuclear β~--decay half-life

β-decay half-life is a key quantity for nuclear structure and nucleosynthesis studies.There exist large uncertainties in the contributions of allowed and forbidden transitions to the total β-decay life,which limits the resolution of the predicted β-decay half-life.We systematically study the contribution of the first forbidden(FF) transitions to the β~--decay half-life,and quantify it with a formula based on simple physics considerations.We also propose a new formula for calculation of the β~--decay half-life that includes the FF contribution.It is shown that the inclusion of the contribution of FF transitions significantly improves the precision of calculations of the β~--decay half-life.By fitting of the RQRPA results for neutron-rich Z=47,57 isotopes and N=80,94 isotones,the formula for the contribution of the FF transitions gives similar results as the RQRPA calculations.However,because of limited experimental data for the branching ratios of unstable nuclei,the fit parameters are not fully constrained.Therefore,the proposed formula for the β~--decay half-life is more suitable for calculations of half-lives than of the FF contributions.The formula could be used to predict the β~--decay half-life in nuclear structure studies as well as nucleosynthesis calculations in stars.