Radioactive Decay: Random Atoms, Clockwork Crowd
By Lucian — builder & engineer, LK Forge
Not one radioactive atom knows when it will decay — the moment is pure chance, and a survivor is as good as new. Yet a lump of them thins out on a schedule so exact we date the Earth by it. Both facts are true at once, and the bridge between them is a crowd. We ran the decay simulator’s random atoms and watched the smooth law appear.
· 6 min read · exact law + a measured Monte-Carlo run
The law, and one noisy run under it
The smooth curve is 2^(−t/t½): every step of one half-life along the axis halves what is left. Laid over it is a single Monte-Carlo run of just 200 atoms — the same per-nucleus coin-flips the simulator makes. It tracks the law, but visibly jitters, and gets ragged in the tail where only a handful of atoms remain and each one’s luck shows.
The exact law and one random 200-atom run. Plotted in half-lives, the smooth curve is identical for every isotope and every starting amount — only the axis’s real-world scale changes.
Smoothness is a headcount
Why does the jagged run become a glassy curve for real materials? Because the statistical noise falls with the square root of the number of atoms. We measured the spread of the surviving fraction at one half-life across hundreds of independent runs, at each sample size: it lands almost exactly on the textbook 0.5/√N, straight as a ruler on a log–log plot. A hundred atoms wobble by 5%; ten thousand by 0.5%; and a real sample of 10²³ atoms is smooth past any hope of measuring the noise.
Measured noise (dots) sits on the 0.5/√N line: σ = 5.2% at N=100, 1.6% at 1,000, 0.46% at 10,000, 0.15% at 100,000. Ten times the atoms, a third of the jitter.
Four clocks, one law
Only the decay constant λ = ln2/t½ — the per-atom chance of decaying each unit of time — changes from isotope to isotope. It spans an astonishing range, and with it the jobs these isotopes do.
| Isotope | Half-life | Decay constant λ | What it clocks |
|---|---|---|---|
| Iodine-131 | 8.02 d | 8.64×10⁻² /d | medical imaging & therapy |
| Cobalt-60 | 5.27 yr | 1.32×10⁻¹ /yr | cancer radiotherapy |
| Carbon-14 | 5,730 yr | 1.21×10⁻⁴ /yr | radiocarbon dating |
| Uranium-238 | 4.47 Gyr | 1.55×10⁻¹⁰ /yr | dating rocks & the Earth |
Half-life ladder (any isotope): 1→50%, 2→25%, 3→12.5%, 4→6.25%, 5→3.13%, 6→1.56%, 7→0.78% remaining. λ spans nine orders of magnitude from cobalt-60 to uranium-238, yet the shape of the curve is identical.
Reproduce it yourself
The law is one line; the crowd effect is a loop of coin-flips.
const remaining = (t, tHalf) => Math.pow(2, -t / tHalf); // exact law
const lambda = tHalf => Math.LN2 / tHalf; // decay constant
// stochastic: flip every surviving atom each half-life step
function survivorsAfterOneHalfLife(N) {
let alive = N;
for (let i = 0; i < alive; i++) if (Math.random() < 0.5) alive--;
return alive / N; // scatter across runs ≈ 0.5/sqrt(N)
}The exponential law and the per-nucleus step come from the decay solver; the noise-vs-N ensemble is the small script behind this post.
Watch atoms flip
Open the lab, pick an isotope, set the atom count low to see the jitter, then crank it up and watch the curve smooth into the ideal exponential.