Estimation of radiative half-life of $^{229m}$Th by half-life measurement of other nuclear excited states in $^{229}$Th

We perform coincidence measurements between $alpha$ particles and $gamma$ rays from a $^{233}$U source to determine the half-lives of the excited state in a $^{229}$Th nucleus. We first prove that the half-lives of 42.43- and 164.53-keV states are consistent with literature values, whereas that of the 97.14-keV state (93(7) ps) deviates from a previously measured value (147(12) ps). The half-lives of 71.83- and 163.25-keV states are determined for the first time. Based on the obtained half-lives and the Alaga rule, we estimate the radiative half-life of the low-energy isomeric state ($^{229m}$Th) to be $5.0(11)times10^{3}$ s, which is one of the key parameters for the frequency standard based on $^{229}$Th.