Bateman equation

In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances.

The model was formulated by Ernest Rutherford in 1905[1] and the analytical solution was provided by Harry Bateman in 1910.

While this can be solved explicitly for i = 2, the formulas quickly become cumbersome for longer chains.

Bateman found a general explicit formula for the amounts by taking the Laplace transform of the variables.

(it can also be expanded with source terms, if more atoms of isotope i are provided externally at a constant rate).

[4] While the Bateman formula can be implemented in a computer code, if

for some isotope pair, catastrophic cancellation can lead to computational errors.