Radioactive Decay and Half-Life

A constant probability per nucleus, per instant

Every unstable nucleus in a sample has exactly the same probability of decaying in the next small interval of time, regardless of how long it has already existed — radioactive nuclei have no memory and do not "age" toward decay. That single assumption, formalised as a constant decay rate λ per nucleus per unit time, is the entire physical content behind the simulation on this page, which starts with N₀ nuclei and lets each one decay stochastically, exactly like flipping N₀ independent, heavily biased coins every tick.

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From one nucleus's coin flip to the exponential law

With N nuclei present, the expected number decaying per unit time is λN, which gives the differential equation dN/dt = −λN. Solving it produces the familiar exponential decay law:

N(t) = N₀ · e^(−λt)

λ = ln(2) / T½

N₀    initial number of nuclei
λ    decay constant, probability per nucleus per unit time
T½    half-life, the time for half the sample to decay

The half-life T½ is the more intuitive quantity most people know, and it's directly tied to λ by ln(2) ≈ 0.693. It is a property of the isotope alone, unaffected by temperature, pressure, or chemical bonding, which is what makes decay such a reliable clock — carbon-14 dating, uranium-lead dating of rocks, and medical isotope dosing all rest on λ being a true physical constant for a given nuclide.

Why the real curve wobbles around the smooth theory line

N(t) = N₀e^(−λt) is the expected count, the average over infinitely many repeated experiments. Any single real run, including the one animating on this page, decays a finite, integer number of discrete nuclei, and the actual count at each moment follows a binomial process (each surviving nucleus independently has probability e^(−λdt) of surviving another interval dt), which fluctuates around the smooth exponential with a relative spread that shrinks as 1/√N. With thousands of nuclei the wobble is small and the simulated curve tracks theory closely; with only a handful remaining near the end of a run, the discreteness becomes visually obvious, exactly the effect you can watch in the simulation as the count runs down.

Why decay counting is a Poisson process

Because each of the very large number of nuclei has a tiny probability of decaying in a short interval, the number of decays actually observed in a fixed short window follows a Poisson distribution with mean λN·Δt, the standard statistical model used to size error bars on any real Geiger-counter measurement: the uncertainty on a count of n decays is √n, which is why radiation measurements always improve by counting longer, trading time for statistical precision.

Frequently asked questions

Why can't you predict exactly when a specific nucleus will decay?

Because decay is fundamentally probabilistic at the quantum level: every unstable nucleus has the same constant decay probability per unit time regardless of its history, with no internal clock counting down. You can only predict the probability distribution of when it decays, not the specific instant, which is why the simulation models each nucleus as an independent random event rather than a scheduled one.

Does a nucleus become more likely to decay the longer it exists?

No. Radioactive decay is memoryless: the decay probability per unit time is exactly the same for a nucleus that formed a microsecond ago and one that has existed since the birth of the solar system. This is a direct consequence of quantum mechanical tunnelling being the underlying decay mechanism for most radioactive processes.

Why does the simulated count deviate more from the theoretical curve near the end of a run?

Because the theoretical curve N(t) = N₀e^(−λt) describes the average over many repeated experiments, while any single run decays a finite integer number of nuclei. The relative size of random statistical fluctuations grows as the remaining count shrinks, roughly as 1/√N, so the last few dozen nuclei visibly wobble around the smooth theoretical line.

Try it live

Everything above runs in your browser — open Radioactive Decay Simulator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Radioactive Decay Simulator simulation

Radioactive Decay: From a Single Coin Flip to the Half-Life Law