# Bateman equation

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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[2].

If, at time t, there are ${\displaystyle N_{i}(t)}$ atoms of isotope ${\displaystyle i}$ that decays into isotope ${\displaystyle i+1}$ at the rate ${\displaystyle \lambda _{i}}$, the amounts of isotopes in the k-step decay chain evolves as:

{\displaystyle {\begin{aligned}{\frac {dN_{1}(t)}{dt}}&=-\lambda _{1}N_{1}(t)\\[3pt]{\frac {dN_{i}(t)}{dt}}&=-\lambda _{i}N_{i}(t)+\lambda _{i-1}N_{i-1}(t)\\[3pt]{\frac {dN_{k}(t)}{dt}}&=\lambda _{k-1}N_{k-1}(t)\end{aligned}}}

(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]

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

${\displaystyle N_{n}(t)=\sum _{i=1}^{n}\left[N_{i}(0)\times \left(\prod _{j=i}^{n-1}\lambda _{j}\right)\times \left(\sum _{j=i}^{n}\left({\frac {e^{-\lambda _{j}t}}{\prod _{p=i,p\neq j}^{n}(\lambda _{p}-\lambda _{j})}}\right)\right)\right]}$

(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 easily in computer code, if ${\displaystyle \lambda _{p}\approx \lambda _{j}}$ for some isotope pair, cancellation can lead to computational errors. Therefore, other methods such as numerical integration or the matrix exponential method are also in use.[5]

e.g. for the simple case of a chain of three isotopes, the corresponding Bateman equation reduces to:

{\displaystyle {\begin{aligned}&A\,{\xrightarrow {\lambda _{A}}}\,B\,{\xrightarrow {\lambda _{B}}}\,C\\[4pt]&N_{B}={\frac {\lambda _{A}}{\lambda _{B}-\lambda _{A}}}N_{A_{0}}\left(e^{-\lambda _{A}t}-e^{-\lambda _{B}t}\right)\end{aligned}}}