Decay Chain Calculator

The standard Bateman formula divides by (λ2 − λ1), so when the two decay constants are almost identical that denominator approaches zero and the formula becomes numerically unstable. This calculator detects when the difference is smaller than 1e-20 and switches to the secular-equilibrium-buildup limit, A2(t) = A1(0)·λ1·t·e^(−λ1·t), which is the correct limiting form of the same physics rather than an approximation of a different scenario.

This calculator classifies the pair by comparing half-lives directly: secular equilibrium is flagged when the parent's half-life is at least 100 times the daughter's, meaning the daughter eventually decays at essentially the same rate the parent feeds it. Transient equilibrium applies when the parent half-life is longer but by less than that 100x margin — the daughter still tracks the parent but at a shifting ratio. No equilibrium is reported whenever the daughter's half-life is longer than the parent's, since the daughter then simply outlives its source and decays on its own timescale.

At time zero the daughter is assumed to have no activity, so as the parent decays it keeps feeding new daughter atoms faster than the existing daughter atoms decay away, and total daughter activity climbs. Once the parent supply has thinned enough that daughter production falls below the daughter's own decay rate, the daughter activity turns over and falls — the calculator finds that turning point explicitly by solving for tMax = ln(λ2/λ1)/(λ2−λ1) and reports both that time and the peak activity there.

No — it implements only the two-member Bateman equation and assumes the daughter's activity is exactly zero at time zero, which is the standard simplifying case (a freshly separated parent source, for example). If you're analyzing a sample that already contains some daughter activity when your clock starts, or a three-step decay chain (parent → daughter → granddaughter), this calculator's results won't be accurate for that scenario.

No. All of the half-life and elapsed-time inputs are converted by the same multiplier before they're combined into ratios like λ·t, so the unit cancels out of every output — it only sets what unit your Parent Half-Life, Daughter Half-Life, and Elapsed Time numbers are interpreted in, not the resulting activity values.