U-238 decays through 14 radioactive daughters before reaching stable Pb-206. Each step obeys dNi/dt = λi-1Ni-1 − λiNi, with λ = ln2/T½. This coupled linear system has an exact closed-form solution, the Bateman equation:
N_k(t) = N0 · Σ(i=1..k) [ (Πλ_1..λ_k-1, skip i) · e^(−λ_i t) / Π(λ_j − λ_i) ]
Every curve on the chart is computed from this exact formula, not a stochastic approximation. Short-lived daughters (Po-214, T½=164µs; Pa-234m, T½=70s) rise almost instantly to a quasi-steady population where their decay rate equals their production rate from the parent — secular equilibrium: Niλi ≈ Nparentλparent. Zoom the time window to see it happen at very different timescales for each link in the chain.
- Chain diagram — node brightness/size = current population share (log-scaled). α steps shrink the mass number by 4, β⁻ steps leave it unchanged.
- Population chart — log-scale fraction N/N0 vs. elapsed time in the selected window; the highlighted isotope is drawn bold.