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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.

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

$\frac{d{N}_1(t)}{dt}=-{\lambda }_1{N}_1(t)$

$\frac{d{N}_i(t)}{dt}=-{\lambda }_i{N}_i(t)+{\lambda }_{i-1}{N}_{i-1}(t)$

$\frac{d{N}_k(t)}{dt}={\lambda }_{k-1}{N}_{k-1}(t)$

(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.

$N_n(t)=\prod _{j=1}^{n-1}{\lambda }_j\sum _{i=1}^n\sum _{j=i}^n\left(\frac{N_i(0)e^{-{\lambda }_{j}t}}{\prod _{p=i,p\ne j}^n({\lambda }_p-{\lambda }_j)}\right)$

(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. Other methods, for example using numerical integration or the matrix exponential, are hence also in use.

See also

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

References

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