DECAY CHAIN
At the daughter’s peak, inflow === outflow to the bit — and the ledger closes at every instant.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Parent decays to daughter decays to stable: A to B to C, with the tempo set by rho, the ratio of the two decay rates. Every natural decay series — uranium’s fourteen steps to lead — is a stack of exactly this link.
The Bateman solution
The daughter population has a closed form — Bateman’s 1910 formula, two exponentials and a divisor, no integrator anywhere. And the three populations sum to N₀ at every instant: nuclei are relabeled, never lost. The node law again, three nodes deep.
The peak
B rises while its parent feeds it faster than it dies, and peaks at t-star equals log-two rho over rho minus one. At that instant, production exactly equals decay — lambda-A N-A equals lambda-B N-B — an identity you can check by substitution, no calculus machinery required. Past the peak the ratio locks into transient equilibrium.
The handoff
The sweep runs ten half-lives of A: the parent drains, the daughter swells to its peak and surrenders, the stable child collects everything. The flat dashed line is the ledger — it never moves. This is the molybdenum-to-technetium generator on every hospital’s weekly schedule: milk the daughter at the peak.
Audit
Audited: the peak identity — inflow equals outflow at t-star — holds bitwise at the defaults and within one part in ten to the fourteen across every rho; conservation emerges rather than being imposed, the independent closed form for C matching the ledger to two parts in ten to the thirteen over the whole plane; and the equilibrium ratio lands on one over rho minus one to two parts in ten to the sixteen. The degenerate rho equals one case gets its own exact formula — special-cased honestly.