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