Innuclear physics, theBateman equation is amathematical model describing abundances and activities in adecay chain as a function of time, based on thedecay rates and initial abundances. The model was formulated byErnest Rutherford in 1905^[1] and the analytical solution was provided byHarry Bateman in 1910.^[2]

If, at timet, 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 thek-step decay chain evolves as:

(this can be adapted to handle decay branches). While this can be solved explicitly fori = 2, the formulas quickly become cumbersome for longer chains.^[3] The Bateman equation is a classicalmaster 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 theLaplace transform of the variables.

${\displaystyle N_n(t)=N_1(0)\times \left(\prod _{i=1}^{n-1}{\lambda }_i\right)\times \sum _{i=1}^n\frac{e^{-{\lambda }_{i}t}}{\prod _{j=1,j\ne i}^n\left({\lambda }_j-{\lambda }_i\right)}}$

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

While the Bateman formula can be implemented in a computer code, if${\displaystyle {\lambda }_j\approx {\lambda }_i}$ for some isotope pair,catastrophic cancellation can lead to computational errors. Therefore, other methods such asnumerical integration or thematrix exponential method are also in use.^[5][6]

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

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

${\displaystyle \begin{array}{r}{A}_B=\frac{{\lambda }_B}{{\lambda }_B-{\lambda }_A}{A}_{A_0}\left(e^{-{\lambda }_{A}t}-e^{-{\lambda }_{B}t}\right)\end{array}}$

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

• Nuclear history of the United Kingdom
• Ordinary differential equations
• Radioactivity

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