On the integrated decay and ingrowth equations used in the measurement of radioactive decay families: the general solution

In the measurement of radioactivity, a finite measurement time is employed to collect data. Usually, this time is small with respect to the half-life of the nuclide being measured and the ‘usual’ decay equations can be used to decay measured activities to a given reference time. In some applications, such as neutron activation, an integrated form of the decay equation needs to be employed as the measurement time is comparable to the half-life and using the non-integrated form introduces a significant error. This correction is well known and is used widely. For radionuclide families, such as the natural decay series of uranium and thorium or simple parent–daughter systems, no such integrated form of the decay and ingrowth series appears to have been published in the open literature. This paper sets out the general solution for integrated decay and ingrowth of sequential decay and illustrates the validity of this theoretical solution by applying it to real examples.