Activity Correction Method
Isotopes such as I-131 and Te-132 have half-lives of several days. This means that over a few days, these isotopes transform into other, non-radioactive isotopes. Since we want to know how much of these isotopes were present when the sample was collected, the constant decrease in their activity as time goes by must be corrected for in our calculations.
The radiation levels we have reported have now been corrected for this radioactive decay between sample collection and radiation detection, as well as the changing rate of decay during the long sample measurement. Reported numbers now correspond to the radiation levels in the sample at the time of collection. The corrected numbers are slightly higher than previous calculations because there is about one day delay before measurement due to three effects: the long sample collection time, any sample preparation time such as distilling rainwater to a smaller volume, and the long sample counting time.
We assume there is an initial activity A_0 of a given isotope at a time t=0, defined to be the midpoint of the sample collection time period. At a later time t, the remaining activity is A(t) = A_0 * 2^(-t/t_H), where t_H is the half-life of the isotope. Additionally, the initial number of atoms of the isotope is defined as N_0 = A_0 * t_H / ln(2), and the number of atoms at time t is N(t) = A(t) * t_H / ln(2) = N_0 * 2^(-t/t_H).
The beginning of the measurement is defined as t_1, and the end of measurement is t_2. The number of decays occuring during the measurement time is DN = N(t_1) - N(t_2) = N_0 * [2^(-t_1/t_H) - 2^(-t_2/t_H)]. Expressed in terms of the initial activity, DN = [A_0 * t_H / ln(2)]*[2^(-t_1/t_H) - 2^(-t_2/t_H)]. We also define DT = t_2 - t_1 as the measurement time. Then the activity measured from our detector, after correction for the finite detection efficiency, is simply A_measured = DN / DT = [A_0 * t_H / (ln(2) * DT)] * [2^(-t_1/t_H) - 2^(-t_2/t_H)].
Solving for A_0 in terms of A_measured, we get A_0 = [A_measured * DT * ln(2) / t_H] / [2^(-t_1/t_H) - 2^(-t_2/t_H)].
One exception is I-132. I-132 has a short half life (2.3 hours) and is produced by the decay of Te-132 (t_H = 3.2 days). I-132 is essentially in secular equilibrium with Te-132, meaning that the amount of I-132 is proportional to the amount of Te-132, and the activities of the two isotopes are approximately equal. Then the initial amount of I-132 does not depend on the half life of I-132, but on the half life of Te-132. We have used the Te-132 half life to approximate the initial quantity of I-132 using the formula above.
The naturally occuring isotopes Ra-226, Pb-212, Tl-208, and K-40 depend on the decay of primordial isotopes U-238, Th-232, and K-40 which have very long half lives (>billions of years). We have ignored the infinitesimal decay of these isotopes during our collection and measurement times.