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We perform coincidence measurements between $\alpha$ particles and $\gamma$rays from a $^{233}$U source to determine the half-lives of the excited statein a $^{229}$Th nucleus. We first prove that the half-lives of 42.43- and164.53-keV states are consistent with literature values, whereas that of the97.14-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 thefirst time. Based on the obtained half-lives and the Alaga rule, we estimatethe 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 frequencystandard based on $^{229}$Th.

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