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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.<ref>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 </ref>
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| If at time t there is <math>N_i(t)</math> atoms of isotope <math>i</math> that decays into isotope <math>i+1</math> at rate <math>\lambda_i</math> the amounts of isotopes in the k-step decay chain evolves as
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| :<math>\frac{dN_1(t)}{dt}=-\lambda_1 N_1(t)</math> | |
| :<math>\frac{dN_i(t)}{dt}=-\lambda_i N_i(t) + \lambda_{i-1}N_{i-1}(t)</math>
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| :<math>\frac{dN_k(t)}{dt}= \lambda_{k-1}N_{k-1}(t)</math>
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| (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.<ref>http://chemistry.sfu.ca/assets/uploads/file/Course%20Materials%2012-1/NUSC%20342/L9.pdf</ref>
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| Bateman found a general explicit formula for the amounts by taking the [[Laplace transform]] of the variables.
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| ::<math>N_n(t) = \prod_{j=1}^{n-1} \lambda_j \sum_{i=1}^n \sum_{j=i}^n
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| \left ( \frac{N_i(0)e^{-\lambda_j t}}{\prod_{p=i, p\neq j}^n (\lambda_p-\lambda_j)} \right )</math>
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| (it can also be expanded with source terms, if more atoms of isotope i are provided externally at a constant rate).<ref>http://www.nucleonica.com/wiki/index.php?title=Help:Decay_Engine</ref>
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| While the Bateman formula can be implemented easily in computer code, if <math>\lambda_p \approx \lambda_j</math> for some isotope pair [[Loss of significance|cancellation]] can lead to computational errors. Other methods, for example using [[Numerical methods for ordinary differential equations|numerical integration]] or the [[matrix exponential]], are hence also in use.<ref>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</ref>
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| ==See also==
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| * [[Harry Bateman]]
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| * [[List of equations in nuclear and particle physics]]
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| * [[Transient equilibrium]]
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| * [[Secular equilibrium]]
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| ==References==
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| {{reflist}}
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| [[Category:Radioactivity]]
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30 year old Community Arts Worker Ahuna from Gander, likes to spend some time caving, home improvement and base jumping. Has enrolled in a world contiki tour. Is very thrilled specifically about touring Etruscan Necropolises of Cerveteri and Tarquinia.
My blog post: Home Improvement Loans Usaa