Fundamental uncertainty equations for nuclear dating applied to the ¹⁴⁰Ba-¹⁴⁰La and ²²⁷Th-²²³Ra chronometers

Basic equations for age dating through activity ratio measurements are presented and applied to nuclear chronometers based on parent-daughter decay. Uncertainty propagation formulae are derived which relate the relative uncertainty on the half-lives and measured activity ratios with the relative uncertainty on the calculated time of a nuclear event. Particular attention is paid to the case of relatively short-lived radionuclides for which the change in decay rate during the measurement is non-negligible. Mathematical solutions are presented to correct the perceived activity ratio and adapt the uncertainty propagation formulae to complete the uncertainty budget. The formulae have been applied to ¹⁴⁰Ba-¹⁴⁰La chronometry, which is particularly useful for dating a nuclear explosion through measurement of the produced activity ratio of ¹⁴⁰La and ¹⁴⁰Ba in a finite time interval. They were also applied to the ²²⁷Th-²²³Ra parent-daughter pair produced for therapeutic use. The impact of inaccuracies in the nuclear decay data on the performance of these nuclear chronometers is shown and discussed.