Radioactive Decay: Alpha, Beta, Gamma & Half-Life

Every atom of carbon-14 in your body is slowly decaying. Not might decay — will decay, on a timescale of thousands of years. Every gram of uranium in Earth's crust is doing the same, on a timescale of billions. What makes radioactive decay so strange is that it's fundamentally probabilistic: you cannot predict when any individual nucleus will decay, only the probability per unit time. That randomness isn't ignorance — it's a feature of quantum mechanics that has been tested to extraordinary precision.

The mathematics that emerges from that randomness is remarkably clean: exponential decay, with a half-life that is constant regardless of temperature, pressure, or chemical environment. That independence from external conditions is what makes radioactive dating so reliable — and so different from any other clock available to science.

Why Nuclei Decay

Nuclear stability depends on the ratio of protons to neutrons and on the total number of nucleons. Light stable nuclei tend to have roughly equal proton and neutron numbers (N ≈ Z). Heavy nuclei require more neutrons than protons to offset proton-proton electrostatic repulsion. Beyond bismuth-209 (Z = 83), no stable nuclei exist — all nuclei with Z > 83 are radioactive.

A nucleus decays when it can reach a lower energy state by doing so. The excess energy is carried away by the emitted radiation. Which type of decay occurs depends on what imbalance exists in the nucleus.

Alpha Decay (α)

An alpha particle is a helium-4 nucleus: two protons + two neutrons, written ⁴₂He or α. Alpha decay occurs in heavy nuclei (typically A > 200) where the nucleus can reduce both its proton-proton repulsion and total mass by ejecting a tightly bound helium-4 cluster.

The daughter nucleus has Z reduced by 2 and A reduced by 4. Alpha particles are emitted with discrete kinetic energies (typically 4–8 MeV) and have very short range in matter: a few centimetres in air, stopped by a sheet of paper or skin. However, alpha emitters inside the body are extremely dangerous — the dense ionisation they produce in tissue causes severe radiation damage.

Alpha decay is explained by quantum tunnelling — classically, the alpha particle cannot escape the nuclear potential well (the Coulomb barrier is too high). Quantum mechanically, the wavefunction has a non-zero probability of extending beyond the barrier, giving a finite tunnelling probability per unit time. This explains why alpha decay has a characteristic half-life that can vary from microseconds to billions of years depending on the barrier height.

Beta Decay (β)

Beta decay converts a neutron to a proton (β⁻) or a proton to a neutron (β⁺), emitting a fast electron or positron.

Beta-minus decay (β⁻): a neutron transforms into a proton, emitting an electron (e⁻) and an anti-neutrino (v̄_e):

Example: ¹⁴₆C → ¹⁴₇N + e⁻ + v̄_e (carbon-14 → nitrogen-14)

Beta-plus decay (β⁺): a proton transforms into a neutron, emitting a positron and a neutrino:

Unlike alpha particles, beta particles have a continuous energy spectrum (the energy is shared between the electron and neutrino). Beta particles have greater range than alpha particles — several metres in air, stopped by a few mm of aluminium — but are less ionising.

Beta decay is mediated by the weak nuclear force — one of the four fundamental forces. The discovery of the neutrino in beta decay (postulated by Pauli in 1930, confirmed in 1956) was a triumph of particle physics.

Gamma Decay (γ)

Gamma rays are high-energy photons emitted when a nucleus in an excited state drops to a lower energy state — exactly analogous to atomic emission of visible light photons, but with much higher energies (typically 0.1–10 MeV). Gamma decay usually follows alpha or beta decay, which often leave the daughter nucleus in an excited state.

Gamma decay does not change Z or A — only the nuclear energy state changes. Gamma rays are the most penetrating radiation: they require centimetres of lead or metres of concrete to reduce intensity significantly. They interact with matter through the photoelectric effect, Compton scattering, and pair production.

The Decay Law and Half-Life

Radioactive decay is a random quantum process: each nucleus has a fixed probability λ (the decay constant) of decaying per unit time, independent of its history or the presence of other nuclei. For a large population N of identical nuclei, the rate of decay (activity A) is:

Solving this differential equation gives the exponential decay law:

where N₀ is the initial number of nuclei and λ is the decay constant (s⁻¹). The half-life T½ is the time for half the nuclei to decay:

After n half-lives: N = N₀ × (½)ⁿ. After 1 half-life: N₀/2. After 2: N₀/4. After 10: N₀/1024 ≈ 0.1% of N₀.

Worked Examples

Example 1: Remaining activity after multiple half-lives

A sample of iodine-131 (T½ = 8 days) has initial activity 400 MBq. What is its activity after 24 days?

Example 2: Using the exponential law

Carbon-14 has T½ = 5,730 years. What fraction remains after 17,190 years?

(Equivalent to 3 half-lives: (½)³ = 1/8 = 12.5% ✓)

Example 3: Radiocarbon dating

A wood sample has 25% of the C-14 activity of living wood. How old is it?

(Two half-lives: (½)² = 0.25, so 2 × 5,730 = 11,460 years ✓)

Activity and the Becquerel

Activity A measures the number of decays per second:

Unit: becquerel (Bq) = 1 decay per second. 1 curie (Ci) = 3.7 × 10¹⁰ Bq (the activity of 1 gram of radium-226). The activity of a radioactive source decreases exponentially at the same rate as N: A(t) = A₀ e^(−λt).

Applications of Radioactive Decay

Radiocarbon dating: living organisms maintain C-14 at atmospheric equilibrium (~1 part in 10¹²). After death, no new C-14 is incorporated — the ratio of C-14 to C-12 decreases with T½ = 5,730 years. Measuring the residual ratio dates organic material up to ~50,000 years old with high precision.

Nuclear reactions and radioactive decay are both governed by the same mass-energy equivalence from special relativity. Nuclear medicine: Tc-99m emits gamma rays detectable outside the body and has a 6-hour half-life — long enough for imaging, short enough to minimise radiation dose. It is used in ~40 million medical scans per year worldwide. I-131 treats thyroid cancer: it is preferentially absorbed by the thyroid, where its beta radiation destroys tumour cells.