Calculators / Decay
Decay chain calculator: the Bateman equations
When a daughter is itself radioactive, its activity grows in, peaks or settles, and has to be solved together with its parent. The app solves whole branching chains; the examples below check it against the closed-form answer.
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The formula (parent and one daughter)
A₂(t) = A₁(0) · λ₂/(λ₂ − λ₁) · (e^(−λ₁t) − e^(−λ₂t)) secular (T½₁ ≫ T½₂): A₂ → A₁ transient (T½₁ > T½₂): A₂ → A₁ · λ₂/(λ₂ − λ₁)
Secular: Rn-222 growing into radium
| TIME | A(Rn-222)/A(Ra-226), APP | CLOSED FORM |
|---|---|---|
| 1 d | 0.1658 | 0.1658 |
| 3.8235 d | 0.5 | 0.5 |
| 7 d | 0.7189 | 0.7189 |
| 14 d | 0.921 | 0.921 |
| 30 d | 0.9957 | 0.9957 |
Transient: La-140 from Ba-140
| TIME | A(La-140)/A(Ba-140), APP | CLOSED FORM |
|---|---|---|
| 1 d | 0.347 | 0.347 |
| 3 d | 0.7588 | 0.7588 |
| 7 d | 1.058 | 1.058 |
| 14 d | 1.144 | 1.144 |
| 30 d | 1.152 | 1.152 |
La-140 overtakes its parent and settles at 1.152× its activity. Treating this as secular equilibrium understates the La-140 by 13 %, and La-140 carries most of the gamma dose.
Assumptions
- Pure parent at t = 0. Every branch is followed with its IAEA branching ratio; branch intensities NUBASE2020 leaves unevaluated are resolved so atoms are conserved, and shown as published.
Sources
- Bateman, H. (1910), Proc. Camb. Phil. Soc. 15, 423.
- Half-lives and branching: IAEA NUBASE2020. Lamarsh & Baratta §2.
Numbers on this page computed by the physics code of Nuclear Toolkit 1.4.1, the same code as the app. How it is checked.
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