Bateman equation

{{Short description|Mathematical model in nuclear physics}}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 1905Rutherford, E. (1905). Radio-activity. University Press. p. 331 and the analytical solution was provided by Harry Bateman in 1910.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

If, at time t, there are $N\_i(t)$ atoms of isotope $i$ that decays into isotope $i+1$ at the rate $\backslash lambda\_i$, the amounts of isotopes in the k-step decay chain evolves as:

: $$

\begin{align}

\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{align}

(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.{{cite web|url=http://chemistry.sfu.ca/assets/uploads/file/Course%20Materials%2012-1/NUSC%20342/L9.pdf |title=Archived copy |accessdate=2013-09-22 |url-status=dead |archiveurl=https://web.archive.org/web/20130927064244/http://chemistry.sfu.ca/assets/uploads/file/Course%20Materials%2012-1/NUSC%20342/L9.pdf |archivedate=2013-09-27 }} 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.

::$N\_n(t)=N\_1(0)\backslash times\backslash left(\backslash prod\_\{i=1\}^\{n-1\}\backslash lambda\_i\backslash right)\backslash times\backslash sum\_\{i=1\}^n\backslash frac\{e^\{-\backslash lambda\_i\; t\}\}\{\backslash prod\backslash limits\_\{j=1,j\backslash neq\; i\}^\{n\}\backslash left(\backslash lambda\_j-\backslash lambda\_i\backslash right)\}$

(it can also be expanded with source terms, if more atoms of isotope i are provided externally at a constant rate).{{Cite web|url=http://www.nucleonica.com/wiki/index.php?title=Help%3ADecay_Engine%2B%2B|title=Nucleonica}}

File:DecayChain241Pu-eng.svg]]

While the Bateman formula can be implemented in a computer code, if $\backslash lambda\_j\; \backslash approx\; \backslash lambda\_i$ 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.{{Cite journal |last=Harr |first=Logan |date=2007-03-15 |title=Precise Calculation of Complex Radioactive Decay Chains |url=https://scholar.afit.edu/etd/2924 |format=PDF |journal=Theses and Dissertations |publication-date=2007}}

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

: $$

\begin{align}

& 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{align}

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

: $$

\begin{align}

A_B= \frac{\lambda_B}{\lambda_B-\lambda_A}A_{A_0} \left( e^{-\lambda_A t} - e^{-\lambda_B t} \right)

\end{align}