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.

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

${\frac {dN_{1}(t)}{dt}}=-\lambda _{1}N_{1}(t)$

${\frac {dN_{i}(t)}{dt}}=-\lambda _{i}N_{i}(t)+\lambda _{i-1}N_{i-1}(t)$

${\frac {dN_{k}(t)}{dt}}=\lambda _{k-1}N_{k-1}(t)$

(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.

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

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

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

While the Bateman formula can be implemented easily in computer code, if $\lambda _{p}\approx \lambda _{j}$ 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.

See also

• Harry Bateman
• List of equations in nuclear and particle physics
• Transient equilibrium
• Secular equilibrium