What it does
Many radioactive nuclides decay into something that is itself radioactive. Molybdenum-99 decays into technetium-99m, radium-226 into radon-222 and then a string of short-lived “progeny”. The amount of each daughter depends on how fast it is made (by its parent) and how fast it decays. The Bateman equations give the answer exactly for any linear chain A → B → C → …, and this page solves them for a parent and up to seven daughters.
You get the activity and number of atoms of every member at any time, a chart of the ingrowth, the time at which each daughter’s activity peaks, and whether the chain settles into equilibrium with its parent.
How to use it
- Pick an example chain or edit the members. Each member has a name, a half-life with its unit, a starting activity (what is there at time zero) and a branching percentage: the share of its decays that produce the next member. The rest is assumed to go somewhere outside the chain.
- Add a daughter or remove the last to change the length (one to eight members).
- Choose the activity unit (Bq to Ci) and the time unit, then the time at which you want the activities.
- Plot up to sets the length of the chart; leave it blank and the page chooses a span that shows the approach to equilibrium. Switch the activity axis between logarithmic and linear.
- Equilibrium within sets how close (in percent) a daughter’s activity ratio must be to its final value before it counts as “in equilibrium”.
What the results mean
- Secular equilibrium: the parent lives far longer than the daughter (here, 100 times or more). After several daughter half-lives, the daughter’s activity equals the parent’s (times the branching).
- Transient equilibrium: the parent lives longer, but not hugely longer. The daughter’s activity ends up a fixed amount above the parent’s, λ₂ / (λ₂ − λ₁) times it, and both then fall together with the parent’s half-life.
- No equilibrium: the daughter lives longer than the parent. The parent dies away and the daughter then decays on its own.
- Peak time: for a pure parent with no daughter at the start, the daughter’s activity is highest at t = ln(λ₂/λ₁) / (λ₂ − λ₁), the moment the two activity curves cross.
Worked example
The default chain is a technetium generator: 1,000 MBq of Mo-99 (half-life rounded to 66 h), freshly separated so there is no Tc-99m (6.0 h), with 88 % of Mo-99 decays feeding Tc-99m. After 24 h the page shows Mo-99 777.2 MBq and Tc-99m 691.8 MBq. Tc-99m peaks at 22.83 h, which is why generators are usually “milked” once a day. The pair is in transient equilibrium: Tc-99m ends up at 0.968 times the Mo-99 activity (0.88 × 66 / (66 − 6)), within 1 % after 43.85 h.
Choose the radium-226 chain and the page shows, at 30 days, Ra-226 almost unchanged at 1,000 MBq and every daughter at 995.6 MBq, within 0.5 % of it: secular equilibrium, with Rn-222 within 1 % of equilibrium after 25.38 days.
Tips
- After a separation, set the daughter’s starting activity to 0. If you start with an old sample, give each member its present activity instead.
- Very short half-lives (microseconds) and very long ones (thousands of years) can sit in the same chain; the maths stays exact.
- The CSV has the activity of every member at 241 times across the chart, for your own plots.
FAQ
Can it handle equal half-lives? Yes. The textbook formula divides by the difference of decay constants, which fails when two are equal; the page uses a form that stays exact.
Does it work out doses? No. It is for study and planning, not a substitute for professional judgement, a calibrated instrument or your local radiation safety rules.
Where do the example half-lives come from? They are rounded general values. Check current evaluated nuclear data before relying on them.
Where the numbers come from
- Bateman, H. (1910), Solution of a system of differential equations occurring in the theory of radioactive transformations, Proceedings of the Cambridge Philosophical Society 15, 423 to 427 (the equations; public domain by age).
- Divided differences of the exponential for close decay constants: the Taylor-series approach of McCurdy, A. C., Ng, K. C. and Parlett, B. N. (1984), Mathematics of Computation 43, 501 to 528 (method only).
- Checked against a 60-digit matrix exponential of each chain's rate matrix (mpmath expm) and SciPy's minimiser for the peak time, written for the tests.
- Half-lives in the example chains are rounded general values, editable on the page; check current nuclear data (for example NNDC or DDEP) for real work.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.