If, at time t, there are atoms of isotope that decays into isotope at the rate , the amounts of isotopes in the k-step decay chain evolves as:
(this can be adapted to handle decay branches). While this can be solved explicitly for i = 2, the formulas quickly become cumbersome for longer chains.[3] The Bateman equation is a classical master equation where the transition rates are only allowed from one species (i) to the next (i+1) but never in the reverse sense (i+1 to i is forbidden).
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]
Quantity calculation with the Bateman-Function for plutonium-241