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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. 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 $N_{i}(t)$ 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:

{\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.

N_{n}(t)=N_{1}(0)\times \left(\prod _{i=1}^{n-1}\lambda _{i}\right)\times \sum _{i=1}^{n}{\frac {e^{-\lambda _{i}t}}{\prod \limits _{j=1,j\neq i}^{n}\left(\lambda _{j}-\lambda _{i}\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 plutonium-241

While the Bateman formula can be implemented in a computer code, if $\lambda _{j}\approx \lambda _{i}$ for some isotope pair, catastrophic 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

{\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}}

Which gives the following formula for activity of isotope $B$ (by substituting $A=\lambda N$ )

{\begin{aligned}A_{B}={\frac {\lambda _{B}}{\lambda _{B}-\lambda _{A}}}A_{A_{0}}\left(e^{-\lambda _{A}t}-e^{-\lambda _{B}t}\right)\end{aligned}}

References

^ Rutherford, E. (1905). Radio-activity. University Press. p. 331
^ 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
^ "Archived copy" (PDF). Archived from the original (PDF) on 2013-09-27. Retrieved 2013-09-22.{{cite web}}: CS1 maint: archived copy as title (link)
^ "Nucleonica".
^ Harr, Logan (2007-03-15). "Precise Calculation of Complex Radioactive Decay Chains" (PDF). Theses and Dissertations (published 2007).

[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.{{cite web}}: CS1 maint: archived copy as title (link)

[4] "Nucleonica".

[5] Harr, Logan (2007-03-15). "Precise Calculation of Complex Radioactive Decay Chains" (PDF). Theses and Dissertations (published 2007).

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