Bateman decay-chain framework: N₁ → N₂ → stable with N₂(t) = (λ₁/(λ₂−λ₁))·N₁(0)·(e^(−λ₁t) − e^(−λ₂t))

The Bateman 1910 solution for a two-member radioactive decay chain N_1 → N_2 → stable: N_1(t) = N_1(0)·e^(−λ₁t); N_2(t) = (λ₁/(λ₂−λ₁))·N_1(0)·(e^(−λ₁t) − e^(−λ₂t)). Secular-equilibrium limit (λ₁ ≪ λ₂): after t ≫ 1/λ₂ the daughter activity…