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 atoms of isotope that decays into isotope at the rate , 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caecf8e62ad5ccd9e315dfe79d94084d0bb6b69f)
(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]
While the Bateman formula can be implemented in a computer code, if 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]
For example, for the simple case of a chain of three isotopes the corresponding Bateman equation reduces to
- https://en.wikipedia.org/wiki/Bateman_equation
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b549bceca2d0a608e8ae6b15d819c257f39658b1)
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