# Bateman equation
The **Bateman equation** is the analytical solution, first published by [Harry Bateman](https://www.edgepedia.org/harry-bateman) in 1910, to the coupled first-order differential equations that govern the amounts of each nuclide in a serial radioactive decay chain. In practice the term covers both the equations themselves and their solution: as an IAEA handbook puts it, the Bateman equation is not a single equation but a method of setting up differential equations describing the decay in the chain of interest.[1](https://www-pub.iaea.org/MTCD/Publications/PDF/Pub1579_web.pdf) • [2](https://www.sciencedirect.com/science/article/pii/S0306454906000284)
| Key fact | Detail |
|---|---|
| Origin | Analytical solution for a linear decay chain given by Bateman (1910)[2](https://www.sciencedirect.com/science/article/pii/S0306454906000284) |
| Governing equations | \( \mathrm{d}N_1/\mathrm{d}t = -\lambda_1 N_1 \) and \( \mathrm{d}N_i/\mathrm{d}t = \lambda_{i-1}N_{i-1} - \lambda_i N_i \) for \( i = 2, \ldots, n \), where \( \lambda_i \) is the decay constant of the \( i \)-th nuclide[2](https://www.sciencedirect.com/science/article/pii/S0306454906000284) |
| Form of solution | Each nuclide's activity is a sum of exponentials \( e^{-\lambda_j t} \), one per chain member[3](https://jtmorrell.github.io/curie/methods.html) |
| Two-member case | \( N_2 = N_0 \frac{\lambda_1}{\lambda_2 - \lambda_1}\left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right) \)[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf) |
| Equilibrium limits | Secular equilibrium for \( \lambda_1 \ll \lambda_2 \); transient equilibrium for a long-lived but measurably declining parent[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf) |
| Known failure mode | The formula is undefined when two chain members have equal decay constants, since the denominators \( \lambda_k - \lambda_j \) vanish, though the physical solution has a finite limiting form[5](https://www-nds.iaea.org/publications/iaea-nds/iaea-nds-0239-1.pdf) • [3](https://jtmorrell.github.io/curie/methods.html) |
| Modern use | Embedded in depletion and decay codes such as ORIGEN, CINDER, and BISON, and in public tools covering the 1252 radionuclides of ICRP Publication 107[2](https://www.sciencedirect.com/science/article/pii/S0306454906000284) • [6](https://epa-dccs.ornl.gov/cgi-bin/chain.pl) |
## The equations
For a linear chain \( 1 \to 2 \to \cdots \to n \) in which each species decays with rate constant \( \lambda_i \) and feeds only the next species, the atom numbers obey[2](https://www.sciencedirect.com/science/article/pii/S0306454906000284)
\[
\frac{\mathrm{d}N_1}{\mathrm{d}t} = -\lambda_1 N_1, \qquad
\frac{\mathrm{d}N_i}{\mathrm{d}t} = \lambda_{i-1}N_{i-1} - \lambda_i N_i \quad (i = 2, \ldots, n).
\]
Each member loses atoms through its own decay and gains atoms through the decay of its parent. The parent decays exponentially, \( N_1(t) = N_0 e^{-\lambda_1 t} \), which is the familiar single-nuclide law; the Bateman equations describe what happens downstream of it.[1](https://www-pub.iaea.org/MTCD/Publications/PDF/Pub1579_web.pdf) The system is closed, with no external production and no loss channels other than decay, and it conserves particles: the sum of all \( \dot{N}_i \) is zero.[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf)
## The solution
Starting from a pure parent, \( N_1(0) = N_1^0 \) and \( N_i(0) = 0 \) for \( i > 1 \), the population of the \( n \)-th member is a weighted sum of exponentials:[7](https://www.physics2u.com/vault/radioactive-decay-bateman-chains.html)
\[
N_n(t) = N_1^0 \left( \prod_{j=1}^{n-1} \lambda_j \right) \sum_{i=1}^{n} \frac{e^{-\lambda_i t}}{\displaystyle\prod_{\substack{k=1 \\ k \neq i}}^{n} (\lambda_k - \lambda_i)}.
\]
For the first daughter this reduces to[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf)
\[
N_2 = N_0 \frac{\lambda_1}{\lambda_2 - \lambda_1} \left( e^{-\lambda_1 t} - e^{-\lambda_2 t} \right),
\]
and the daughter activity relative to the parent is
\[
A_2 = A_1 \frac{\lambda_2}{\lambda_2 - \lambda_1} \left( 1 - e^{-(\lambda_2 - \lambda_1)t} \right).
\]
The stable end of the chain is found by subtraction, \( N_n = N_0 - \sum_{i=2}^{n-1} N_i \), reflecting conservation of the total number of atoms.[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf)
## Equilibrium regimes
The two-member solution has limits with practical names.[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf)
**Secular equilibrium.** When the parent is very long-lived compared with the daughter (\( \lambda_1 \ll \lambda_2 \)), the parent's activity is effectively constant on the timescale of measurement, and the daughter activity rises as \( A_2 = A_1(0)(1 - e^{-\lambda_2 t}) \) toward equality with the parent. This is the working principle of the [natural decay series](https://www.edgepedia.org/natural-decay-series): in ancient uranium ore, every daughter down to the stable lead end has an activity equal to that of 238U, so counting one member measures the whole chain.[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf) • [8](https://notes.amittai.studio/nuclear-physics/radioactive-decay/decay-kinetics-equilibrium)
**Transient equilibrium.** When the parent is long-lived but its decline is measurable, the daughter activity rises quickly to match the parent and then follows it closely, remaining slightly greater than the parent's with a temporal offset of about one daughter mean-life \( \tau_2 \).[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf)
**Equal decay constants.** If \( \lambda_1 = \lambda_2 \), the general formula's denominators vanish. The limit can be taken by series expansion, giving \( A_2(t) = A_1(t)\,\lambda t \); the case is described as more of a mathematical curiosity than a physical regime.[4](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf)
## Extensions
**Branching.** Real decay schemes often branch, for example the decay of 212Bi to either 212Po or 208Tl. The simple product-of-rates prefactor is then replaced by a sum over paths, each weighted by its product of branching ratios; a parent with partial decay constants for two channels decays with their sum, and a channel's branching ratio is its partial constant divided by the total.[7](https://www.physics2u.com/vault/radioactive-decay-bateman-chains.html) • [8](https://notes.amittai.studio/nuclear-physics/radioactive-decay/decay-kinetics-equilibrium) The Curie software library handles this by enumerating every distinct path from the parent to the isotope of interest and summing the contributions.[3](https://jtmorrell.github.io/curie/methods.html)
**External production.** With a continuous production rate \( R(t) \), as in reactor irradiation or cosmogenic production, each equation gains a source term and the homogeneous Bateman solution must be convolved with \( R \). For constant production into a single isotope the result is the saturation curve \( A(t) = R(1 - e^{-\lambda t}) \), in which activity builds from zero toward the value \( R \) where production and decay balance.[7](https://www.physics2u.com/vault/radioactive-decay-bateman-chains.html) • [3](https://jtmorrell.github.io/curie/methods.html)
## Numerical use and limitations
The Bateman solution underlies most decay and depletion calculations. In the linear chain method, a branched transmutation network is broken into a set of linear chains and the 1910 solution is applied to each; codes using this approach include BISON (Furuta et al., 1987) and CINDER (Wilson et al., 1998). Matrix-based methods, of which ORIGEN (Croff, 1980) is the most used, treat the same system with matrix calculus. Both families generally assume that all decay constants in a chain are different, an assumption that branching and chain loops in real transmutation problems can violate, causing loss of numerical convergence.[2](https://www.sciencedirect.com/science/article/pii/S0306454906000284)
**Equal and nearly equal constants.** When two members of a chain have the same decay constant, the usual Bateman solution for distinct decay constants becomes undefined, although the physical solution has a finite limiting form.[3](https://jtmorrell.github.io/curie/methods.html) One practical workaround is to shift the equal constants artificially, for example by 0.001%, and accept an approximate solution.[5](https://www-nds.iaea.org/publications/iaea-nds/iaea-nds-0239-1.pdf) Even distinct constants can cause trouble: the denominators \( \prod_{k \neq j}(\lambda_k - \lambda_j) \) vanish when database rounding makes two half-lives identical, so the formula becomes undefined even though a finite limiting solution exists.[3](https://jtmorrell.github.io/curie/methods.html)
**Half-life disparity.** Double-precision floating-point arithmetic, with roughly 15 decimal places, loses accuracy when a chain contains nuclides with disparate half-lives. The chain of 254Es is an extreme case, containing 238U with a 4.468 billion year half-life and 214Po with a 164.3 microsecond half-life, a spread of 20 orders of magnitude. The radioactivedecay Python package addresses this by evaluating the matrix-form analytical solution with 320 significant figures in its high-precision mode, which is sufficient for any physically relevant decay calculation.[9](https://arxiv.org/abs/2203.09761)
**Applications.** The Bateman model is used wherever ingrowth and decay of progeny matter: nuclear engineering, medical physics, radiation protection, environmental science, and archaeology, among other fields.[9](https://arxiv.org/abs/2203.09761) The US EPA's Superfund risk assessment decay chain tool uses a dedicated C++ Bateman solver covering all 1252 radionuclides identified in ICRP Publication 107, alongside a Python implementation using the Chebyshev Rational Approximation Method; a user-entered initial parent activity is multiplied by stored unit-activity files to give the state of the chain at any time.[6](https://epa-dccs.ornl.gov/cgi-bin/chain.pl) In reactor physics, decay heats and delayed neutron yields following a fission event are computed as weighted sums over the Bateman solution \( N_i(t) \) using fission yields and average emitted energies from evaluated decay data libraries.[5](https://www-nds.iaea.org/publications/iaea-nds/iaea-nds-0239-1.pdf)
## References
1. [Radiotracer Generators for Industrial Applications, IAEA Radiation Technology Series No. 5](https://www-pub.iaea.org/MTCD/Publications/PDF/Pub1579_web.pdf)
2. [General solution of Bateman equations for nuclear transmutations, Annals of Nuclear Energy (ScienceDirect)](https://www.sciencedirect.com/science/article/pii/S0306454906000284)
3. [Theory & Methodology, Curie 0.3.0 documentation](https://jtmorrell.github.io/curie/methods.html)
4. [Radioactive Decay (Chapter 13), University of Michigan NERS 311 course library](https://websites.umich.edu/~ners311/CourseLibrary/book.pdf)
5. [IAEA-NDS-0239-1, nuclear data and β-deay chain calculations](https://www-nds.iaea.org/publications/iaea-nds/iaea-nds-0239-1.pdf)
6. [Radionuclide Decay Chain, Superfund Risk Assessment, US EPA](https://epa-dccs.ornl.gov/cgi-bin/chain.pl)
7. [Radioactive Decay and Bateman Equations, physics2u](https://www.physics2u.com/vault/radioactive-decay-bateman-chains.html)
8. [Serial Decay, the Bateman Equations, and Radioactive Equilibrium, Amittai study notes](https://notes.amittai.studio/nuclear-physics/radioactive-decay/decay-kinetics-equilibrium)
9. [radioactivedecay: A Python package for radioactive decay calculations (Malins & Lemoine, arXiv 2022)](https://arxiv.org/abs/2203.09761)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Particles and nuclei › Nuclear physics › Radioactivity and nuclear decay › Decay kinetics and decay chains › Bateman equations and chain kinetics*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 11, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license