Radioactivity Decay Laws

13.6 Radioactivity — Discovery and Three Decay Modes

In 1896, the French physicist Henri Becquerel made one of physics' great accidental discoveries. He had wrapped uranium-potassium sulphate in black paper, placed a photographic plate underneath separated by a thin silver sheet, and stored everything in a dark drawer for several days. When he finally developed the plate, it was darkened — something invisible was passing through paper and metal. The compound was emitting nothing it was being given; the energy was coming from inside the atomic nucleus itself.

Marie and Pierre Curie soon isolated polonium and radium, and Rutherford named the three components of the mysterious radiation by how easily they were absorbed: α, β, γ. We now know that radioactivity is a nuclear phenomenon — an unstable nucleus spontaneously transforms by emitting a particle (or photon) and rearranging itself into a more stable configuration.

α-Decay

A nucleus emits an α-particle (helium-4). The general scheme:

Example: \(^{238}_{92}\mathrm{U} \to {}^{234}_{90}\mathrm{Th} + {}^{4}_{2}\mathrm{He}\) (Q ≈ 4.27 MeV). α-decay is energetically possible only if the parent's mass exceeds the combined mass of daughter + α.

β-Decay

In β⁻-decay a neutron in the nucleus converts to a proton, ejecting an electron and an antineutrino:

Example: \(^{14}_{6}\mathrm{C} \to {}^{14}_{7}\mathrm{N} + e^{-} + \bar\nu\). Useful in carbon dating.

In β⁺-decay a proton converts to a neutron, ejecting a positron and a neutrino:

γ-Decay

After α or β emission, the daughter nucleus is often left in an excited state. It drops to its ground state by emitting one or more γ-ray photons (energies ranging from a few keV to several MeV). Z and A do not change. γ-rays are simply nuclear EM radiation with very short wavelength.

Law of Radioactive Decay

Suppose we have a sample containing N(t) radioactive nuclei at time t. The radioactive decay of an individual nucleus is a statistical, spontaneous event — we cannot predict which nucleus will decay next or when. But for a large number of nuclei, the rate of decay is found experimentally to be proportional to N:

Here \(\lambda\) is the decay (or disintegration) constant — a characteristic property of each radioactive species. Solving the differential equation with the initial condition \(N(0) = N_0\):

This is the law of radioactive decay. The number of radioactive nuclei falls exponentially with time.

Activity (R)

The activity R = |dN/dt| is the rate at which decays occur in the sample:

The SI unit of activity is the becquerel (Bq): 1 Bq = 1 decay per second. An older unit still common in medicine is the curie (Ci): 1 Ci = 3.7 × 10¹⁰ Bq.

Half-life T₁/₂

The half-life is the time after which exactly half of the nuclei present at any instant have decayed.

Setting \(N = N_0/2\) at \(t = T_{1/2}\):

Mean life τ

The mean (average) life of the nuclei in the sample is

Mean life is always slightly longer than half-life.

Worked Examples

Worked Example 13.4 — Half-life ↔ decay constant

Worked Example 13.5 — How many remain after time t?

Worked Example 13.6 — Activity

Interactive — Half-Life Simulator