A parent decays into a daughter, the daughter into a granddaughter — a cascade of coupled first-order ODEs. Rutherford wrote the system in 1905; in 1910 Bateman solved it in closed form. Watch the daughter build up from zero, peak, and — when the parent is long-lived — settle into secular equilibrium, its activity locked to the parent's. Down the center: the initial atoms go in, the equations integrate, the populations come out. The blue team builds it; the red team tries to break it.
source H. Bateman, “The solution of a system of differential equations occurring in the theory of radioactive transformations,” Proc. Camb. Phil. Soc. 15 (1910) 423–427 — archive.org/details/mathematical-proceedings-cambridge-philosophical-society_1908-1910_15. Rendered, not quoted.
A three-link chain A → B → C (C stable), each parent feeding the next:
dNA/dt = −λANA
dNB/dt = λANA − λBNB (production − decay)
dNC/dt = λBNB
Bateman's closed solution for NA(0)=N0, NB(0)=NC(0)=0:
NB(t) = N0 · λA/(λB−λA) · (e−λAt − e−λBt). The panel integrates the ODEs by RK4 and the witness matches them to this formula to 1e−6.
One link is the-radioactive-decay: N = N0e−λt, a single exponential. Chain two of them and the daughter no longer just decays — it is fed, so it climbs from zero before it falls.
Bateman (1910) solved Rutherford's 1905 system for any length of chain. The reward is secular equilibrium: a long-lived parent pins its short-lived daughter's activity to its own. That single decay is this sphere's premise; this sphere is its multi-step extension.
The blue team's live check: re-integrate the chain and confirm the daughter builds up (peaks above zero), that A+B+C is conserved, and that the analytic and numeric NB agree. If red drops the feed term, this badge catches it.
Feed the chain three numbers: how fast each link decays, and how many parent atoms you start with.
| quantity | symbol | meaning |
|---|---|---|
| parent rate | λA | ln2 / T½(A) |
| daughter rate | λB | ln2 / T½(B) |
| start | N0 | NA(0), with B=C=0 |
Set the two half-lives below; the ratio T½(A)/T½(B) decides whether the chain reaches secular equilibrium (parent ≫ daughter), transient equilibrium, or none.
Curves: NA parent, NB daughter, NC stable end. Dashed = daughter peak (tpeak); vertical line = current t. Everything is integrated live, never looked up.
What the machine proves, live: the daughter builds from zero, peaks at tpeak=ln(λB/λA)/(λB−λA), then decays — RK4 matching Bateman to 1e−6; total nucleon count A+B+C is conserved; and for λA≪λB the activity ratio λBNB/λANA → 1 — secular equilibrium.
The blue team's witness (left) confirms these live; the red team (right) tries to make the daughter never build up.
Real chains branch (Bi-212 splits α and β−), leak (radon escapes rock, breaking equilibrium), and for small N the smooth curve is a lie — decay is Poisson, and counts fluctuate. Bateman gives the expectation, not the sample path.
“The daughter starts decaying immediately.” Cut. With NB(0)=0 the daughter first grows: dNB/dt = λANA > 0 until production equals decay at tpeak.
“Secular equilibrium means the daughter stops decaying.” Cut. It decays as fast as ever — its activity equals the parent's (λANA=λBNB), replenished exactly as fast as it is lost.
“Equilibrium happens for any chain.” Kept, corrected. Only when the parent outlives the daughter (λA<λB). If λA>λB the parent vanishes first — no equilibrium.
The red team's move: drop the production term — make dNB/dt = −λBNB with no feed from A. Now the daughter, starting at zero, can only decay, so it never builds up. The witness (window 7) recomputes and catches it.
Sever A→B and the daughter flatlines at zero — buildup fails, conservation fails, the witness turns red. Nothing is faked; the attack is real and it is caught.