This simulation models the stochastic decay of a population of unstable nuclei. It begins with N₀ nuclei and, each frame, gives every surviving nucleus an independent chance of decaying. The per-step decay probability is 1 − e^(−λ·dt), where the decay constant λ = ln2 / T½. The purple trace plots the actual surviving count, while the amber dashed curve shows the smooth exponential law N(t) = N₀·e^(−λt).
The N₀ slider sets the starting population (50–800), the Half-life slider sets T½ in seconds (2–30), and the Speed slider multiplies how fast simulation time advances (1×–8×). Live readouts show N (nuclei left), A (activity, decays per second) and t (elapsed time). Radioactive decay underpins carbon-14 dating, medical imaging tracers, nuclear power and smoke detectors, so understanding half-life is genuinely useful.
What does this simulation actually show?
It shows a fixed population of unstable nuclei decaying at random over time. The purple dots on the left are individual nuclei, fading as they decay, and the graph on the right plots the surviving count N against time. A theoretical exponential curve is overlaid so you can compare random reality with the textbook law.
How is each decay decided?
Every surviving nucleus is tested independently each frame. It decays if a random number falls below the probability 1 − e^(−λ·dt), where dt is the simulation time elapsed that frame. This is a Monte Carlo method: decay is truly random per nucleus, not scheduled in advance.
What is half-life?
The half-life T½ is the time for half of the nuclei in a sample to decay. After one half-life roughly N₀/2 remain, after two about N₀/4, and so on. It is set by the Half-life slider here, anywhere from 2 to 30 seconds.
N₀ sets the initial number of nuclei (50 to 800). Half-life sets T½ in seconds (2 to 30), which determines the decay constant λ = ln2/T½. Speed multiplies how quickly simulation seconds pass relative to real time (1× to 8×) without changing the physics.
The smooth amber curve follows N(t) = N₀·e^(−λt), the exponential decay law, with decay constant λ = ln2 / T½. The simulation does not use this formula directly; instead it lets the law emerge from many independent random decays.
Because decay is stochastic. With a finite number of nuclei the random sample fluctuates around the ideal exponential, especially when only a few nuclei remain. The mismatch is statistical noise, and it shrinks as N₀ grows larger.
Activity is the rate of decays per second, displayed as A in the toolbar. It equals the number of nuclei that decayed in a step divided by the time of that step. Like N itself, activity falls off exponentially as the sample is depleted.
The statistical behaviour is faithful: independent per-nucleus probabilities reproduce true exponential decay on average, which is exactly how real radioactivity works. The timescales (seconds) and small populations are scaled for visualisation; real samples contain astronomically more atoms and span half-lives from fractions of a second to billions of years.
Each nucleus has a constant probability of decaying per unit time, independent of its age or its neighbours. When every member of a population decays at the same fixed rate, the total count shrinks proportionally to its current size, and that condition mathematically produces exponential decay.
Exponential decay governs radiocarbon and radiometric dating, dosimetry and the choice of medical tracers in PET scans, the design of nuclear reactors and radioactive waste storage, and even smoke detectors using americium-241. Knowing the half-life lets scientists predict how long material stays active.