In physics, the Bateman equations are a set of first-order differential equations, which describe the time evolution of nuclide concentrations undergoing serial or linear decay chain. The model was formulated by Ernest Rutherford in 1905 and the analytical solution for the case of radioactive decay in a linear chain was provided by Harry Bateman in 1910. This model can be also used in nuclear depletion codes to solve nuclear transmutation and decay problems.
For example, ORIGEN is a computer code system for calculating the buildup, decay, and processing of radioactive materials. ORIGEN uses a matrix exponential method to solve a large system of coupled, linear, first-order ordinary differential equations (similar to the Bateman equations) with constant coefficients.
The Bateman equations for radioactive decay case of n – nuclide series in linear chain describing nuclide concentrations are as follows:
Bateman Equations for Nuclear Transmutation
As was written, this model can be also used in nuclear depletion codes to solve nuclear transmutation and decay problems as well. In case of transmutation the decay constants that govern Bateman equations for a decay case are substituted by transmutation constants. By the transmutation constant λi,j we understand probability of the ith nuclide production per time unit from the jth nuclide destruction, as a result of nuclear interaction with the whole spectrum of interacting particles or due to the natural nuclear decay.
These equations are usually used for the exact evolution of isotopic changes in the nuclear fuel during fuel depletion. Fuel depletion is usually modelled mathematically as a set of differential equations known as evolution equations.
Special Reference: Jerzy Cetnar, General solution of Bateman equations for nuclear transmutations. Annals of Nuclear Energy 33 (2006). January 2006.