If we are going to use analytic solutions such as the Bateman solution
we have to be careful to specify what portion of a decay chain we are
discussing, and what the starting conditions are. Then we need to
determine if the analytic solution pertains.
If we have a steady influx of radon from uranium in the soil, then we
have an established secular equilibrium clear back to uranium and
extending at least to 210-Pb. In this case everything "decays" at the
uranium rate, which for us mortals is essentially no decay. Bateman is
not needed.
If we stop the influx of radon, but still have the initial radon
present, then we have a secular equilibrium from 222-Rn to 210-Pb and
everything decays with the 3.8-day half-life of 222-Rn. Bateman is not
needed.
If we take a wipe from a computer monitor and collect mostly 214-Pb,
214-Bi and a bit of 218-Po and even smaller 214-Po, then we are closer
to the Bateman equations, but not quite because we are starting with
equilibrium values of all these whereas the Bateman equations as usually
presented assume we start with only the nuclide at the top of the chain
(essentially 214-Pb in this case).
BTW, even in 1969 when my copy of Harvey: Intro to Nuclear Physics and
Chemistry was published, it states that decay chain calculations are
usually done numerically by computer rather than analytically by
Bateman-type equations.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu