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.^[1]

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.^[2]

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).^[3]

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.^[4]

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

$A{\xrightarrow {\lambda _{A}}}B{\xrightarrow {\lambda _{B}}}C$

N_{B}={\frac {\lambda _{A}}{\lambda _{B}-\lambda _{A}}}N_{A_{0}}(e^{-\lambda _{A}t}-e^{-\lambda _{B}t})

References

↑ H. Bateman. "Solution of a System of Differential Equations Occurring in the Theory of Radio-active Transformations," Proc. Cambridge Phil. Soc. IS, 423 (1910) https://archive.org/details/cbarchive_122715_solutionofasystemofdifferentia1843
↑ "Archived copy" (PDF). Archived from the original (PDF) on 2013-09-27. Retrieved 2013-09-22.
↑ http://www.nucleonica.com/wiki/index.php?title=Help%3ADecay_Engine%2B%2B
↑ Logan J. Harr. Precise Calculation of Complex Radioactive Decay Chains. M.Sc thesis Air Force Institute of Technology. 2007. http://www.dtic.mil/dtic/tr/fulltext/u2/a469273.pdf

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Bateman Equation

See also

References