Bateman equation

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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 atoms of isotope that decays into isotope at the rate , the amounts of isotopes in the k-step decay chain evolves as:

(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]

Bateman found a general explicit formula for the amounts by taking the Laplace transform of the variables.

(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 easily in computer code, if 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]

e.g. for the simple case of a chain of three isotopes, the corresponding Bateman equation reduces to:

See also[edit]


  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)
  3. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2013-09-27. Retrieved 2013-09-22. 
  4. ^
  5. ^ Logan J. Harr. Precise Calculation of Complex Radioactive Decay Chains. M.Sc thesis Air Force Institute of Technology. 2007.