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

${\displaystyle {\begin{aligned}{\frac {dN_{1}(t)}{dt}}&=-\lambda _{1}N_{1}(t)\\[3pt]{\frac {dN_{i}(t)}{dt}}&=-\lambda _{i}N_{i}(t)+\lambda _{i-1}N_{i-1}(t)\\[3pt]{\frac {dN_{k}(t)}{dt}}&=\lambda _{k-1}N_{k-1}(t)\end{aligned}}}$

(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.

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

(it can also be expanded with source terms, if more atoms of isotope i are provided externally at a constant rate).[4]

Quantity calculation with the Bateman-Function for 241Pu

While the Bateman formula can be implemented in a computer code, if ${\displaystyle \lambda _{p}\approx \lambda _{j}}$ 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.[5]

For example, for the simple case of a chain of three isotopes the corresponding Bateman equation reduces to

${\displaystyle {\begin{aligned}&A\,{\xrightarrow {\lambda _{A}}}\,B\,{\xrightarrow {\lambda _{B}}}\,C\\[4pt]&N_{B}={\frac {\lambda _{A}}{\lambda _{B}-\lambda _{A}}}N_{A_{0}}\left(e^{-\lambda _{A}t}-e^{-\lambda _{B}t}\right)\end{aligned}}}$

See also

• Harry Bateman
• List of equations in nuclear and particle physics
• Transient equilibrium
• Secular equilibrium
• Pharmacokinetics, loose applicability

References

1. Rutherford, E. (1905). Radio-activity. University Press. p. 331
2. Bateman, H. (1910, June). The solution of a system of differential equations occurring in the theory of radioactive transformations. In Proc. Cambridge Philos. Soc (Vol. 15, No. pt V, pp. 423–427) https://archive.org/details/cbarchive_122715_solutionofasystemofdifferentia1843
3. "Archived copy" (PDF). Archived from the original (PDF) on 2013-09-27. Retrieved 2013-09-22.CS1 maint: archived copy as title (link)
4. http://www.nucleonica.com/wiki/index.php?title=Help%3ADecay_Engine%2B%2B
5. 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