Radioactive Decay & Half-Life

Unstable nuclei decay spontaneously by emitting radiation. The process is governed by a single principle: the probability of decay in any small time interval $dt$ is constant (equal to $\lambda\,dt$, independent of the nucleus's age or environment). This is the decay constant $\lambda$.

Applying this to a large population of $N$ nuclei gives the exponential law. The three main decay modes are:

• Alpha ($\alpha$) decay: nucleus emits a helium-4 nucleus; large mass and charge change.
• Beta ($\beta$) decay: a neutron converts to a proton (or vice versa) emitting an electron/positron and a neutrino.
• Gamma ($\gamma$) decay: excited nucleus releases a high-energy photon; no change in $A$ or $Z$.

Activity $A = \lambda N$ measures the number of decays per second (unit: Becquerel, $1\,\text{Bq} = 1\,\text{decay/s}$).

Decay law:

$N(t) = N_0\,e^{-\lambda t}, \quad A(t) = \lambda N_0\,e^{-\lambda t} = A_0\,e^{-\lambda t}$

Half-life (time for half the nuclei to decay):

$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$

After $n$ half-lives: $N = N_0 / 2^n$.

Carbon-14 has $T_{1/2} = 5730\,\text{years}$. A wood sample shows activity $25\%$ of a living tree's activity. How old is the sample?

$\frac{A}{A_0} = 0.25 = e^{-\lambda t} \Rightarrow \lambda t = \ln 4 = 2\ln 2 \Rightarrow t = 2T_{1/2}$

$t = 2 \times 5730 = 11{,}460\,\text{years}$

The sample is approximately 11,460 years old — this is the principle behind radiocarbon dating.

Radiocarbon dating (C-14, $T_{1/2} = 5730\,\text{yr}$) dates organic artefacts up to ~50,000 years. Uranium-lead dating (U-238, $T_{1/2} = 4.5\,\text{Gyr}$) dates rocks and meteorites — including Earth itself. Medical PET scanners use fluorine-18 ($T_{1/2} = 110\,\text{min}$), short enough to decay away after the scan. Smoke detectors contain Americium-241 ($T_{1/2} = 432\,\text{yr}$), whose alpha emission ionises air to complete a circuit that smoke interrupts.

1. A radioactive isotope has $T_{1/2} = 8\,\text{days}$ and initial activity $A_0 = 1600\,\text{Bq}$. What is the activity after 24 days?

2. Why does a decay curve look smooth for a large sample but jagged and stepwise for only a few hundred atoms?