Radioactive decay and half-life

What is radioactive decay?

Some atomic nuclei are unstable. At a random moment, with no warning, one breaks apart and gives off a burst of radiation. You can never predict when a single atom will go. It might be in the next second, or not for a thousand years.

So how is any of this useful? Because when you have a huge number of these atoms, a very reliable thing happens. After a fixed stretch of time, called the half-life, exactly half of them are left. Wait one more half-life and half of those are gone too, so a quarter remains. Another, and you are down to an eighth. The pile shrinks by half each time, like clockwork.

Different atoms have wildly different half-lives, from a fraction of a second to billions of years. That range is a gift. Living things take in a slightly radioactive form of carbon, and once they die it slowly ticks away with a known half-life. Measure how much is left and you know how long ago they died. That is carbon dating, and the same trick on other atoms tells us the age of rocks and of the Earth itself.

What is a half-life?

The key idea is that decay is memoryless. A nucleus does not age, wear out, or get closer to the edge. At every instant it has the same fixed chance of decaying, and a nucleus that has waited a million years is no more likely to go in the next second than a brand new one. Each nucleus carries a constant probability per unit time of decaying, written \(\lambda\) and called the decay constant.

For one atom that means pure chance. For a large number \(N\) it means something very predictable. In a short time, a fixed fraction of whatever is left decays, so the amount lost is always proportional to the amount you still have. That is the signature of exponential decay:

\[ N(t) = N_0\, e^{-\lambda t}. \]

Here \(N_0\) is how many you started with. The time for the count to fall to half is the half-life, and it is tied to \(\lambda\) by a short piece of algebra:

\[ t_{1/2} = \frac{\ln 2}{\lambda}. \]

The activity, the number of decays per second that a Geiger counter actually clicks out, is proportional to \(N\), so it follows the very same falling curve. A source with a short half-life is fierce but brief. One with a long half-life is faint but lasts a very long time.

The radiation comes in three main kinds. Alpha decay throws out a heavy chunk, two protons and two neutrons bound together, which is a helium nucleus. Beta decay converts a neutron into a proton (or the reverse) and flings out a fast electron. Gamma decay is the nucleus shedding spare energy as a high-energy photon, usually just after one of the others. Alpha is stopped by paper, beta by a sheet of metal, gamma needs thick lead or concrete.

Radiometric dating puts the exponential law to work. An unstable parent atom decays into a stable daughter, and both stay locked in the sample. Measure the ratio of parent left to daughter built up, and the exponential curve reads straight off the elapsed time. Carbon-14, with a half-life of about 5730 years, dates once-living material like wood, bone and cloth over tens of thousands of years. For rocks, which are far older, geologists use pairs like uranium decaying to lead, whose half-lives run to billions of years.

The exponential decay law

The law from a differential equation

The exponential form is not an assumption but a consequence. If each nucleus decays independently with constant rate \(\lambda\), then the expected number lost in a time \(dt\) is \(\lambda N\,dt\), giving \(\frac{dN}{dt} = -\lambda N\). Integrating with \(N(0)=N_0\) returns \(N(t)=N_0 e^{-\lambda t}\), and setting \(N=N_0/2\) yields \(t_{1/2}=\ln 2/\lambda\). The mean lifetime of a nucleus is \(\tau = 1/\lambda = t_{1/2}/\ln 2\), the time constant of the exponential.

Why it is only exact for large N

Decay is a Poisson process. Over any interval each surviving nucleus decays with a fixed probability, independently of the others, so the number of decays is Poisson-distributed. Its mean and variance are equal, which means the count of decays fluctuates about its expectation by roughly \(\sqrt{N}\). The smooth exponential is the large-\(N\) limit; with a handful of atoms left you see ragged, discrete jumps, and the \(\sqrt{N}\) scatter is exactly the counting noise that dominates a weak measurement.

Chains and equilibrium

Many nuclei decay to daughters that are themselves unstable, forming a chain, and when a long-lived parent feeds a short-lived daughter the daughter reaches secular equilibrium, decaying as fast as it is produced so that their activities become equal.

Why the clock can be trusted

The decay constant \(\lambda\) is set by nuclear physics alone. Alpha decay is quantum tunnelling of the alpha particle out through the Coulomb barrier, and its ferocious sensitivity to barrier height is what spreads half-lives across more than twenty orders of magnitude (the Geiger–Nuttall relation). Beta decay is governed by the weak interaction. Both processes live inside the nucleus, essentially untouched by temperature, pressure or chemical bonding, which sit at energies millions of times too small to matter. That near-total immunity to the environment is precisely why a half-life measured today held four billion years ago, and why radiometric dates are reliable.