E.3 Radioactive decayIB Physics HL: Revision notes

Isotopes share a proton number but differ in neutron number. A nucleus has less mass than its separate nucleons: the mass defect Δm. The binding energy (energy to separate a nucleus into its nucleons) is E = Δmc², or Δm (u) × 931.5 MeV. Binding energy per nucleon peaks at about 8.8 MeV near iron-56; both fusion of light nuclei and fission of heavy nuclei release energy by moving towards the peak.

Decay is random and spontaneous. Alpha: A − 4, Z − 2. β⁻: n → p + e⁻ + $\bar{\nu}$ (Z + 1). β⁺: p → n + e⁺ + ν (Z − 1). Gamma: nucleus drops to a lower energy state, A and Z unchanged. Alpha is most ionising and least penetrating (stopped by paper); beta is intermediate (a few mm of aluminium); gamma is least ionising and most penetrating (reduced by cm of lead).

Activity A is decays per second (Bq). Half-life T½ is the time for N or A to halve; after n half-lives the fraction left is (1/2)ⁿ. Always subtract background from count rates before analysing them.

Evidence for the strong force: nuclei with many protons are stable despite huge electrostatic repulsion; scattering deviates from Rutherford's prediction only at very close approach, so the force is short range.

Light stable nuclei have N/Z ≈ 1; heavy ones need N/Z up to about 1.5, because extra neutrons add attraction without adding repulsion. Nuclides with too many neutrons tend to β⁻ decay; too few, β⁺ decay; very heavy ones, alpha decay.

Each nucleon attracts only its nearest neighbours, so above A ≈ 60 the binding energy per nucleon is approximately constant (about 8 MeV), falling slowly for heavy nuclei as long-range proton repulsion grows.

Alpha particles and gamma photons from a given nuclide have discrete energies, so nuclei have discrete energy levels. An alpha decay can leave the daughter in an excited state, which then emits a gamma photon whose energy matches the gap between alpha energies.

Beta particles have a continuous spectrum up to a maximum, although each decay releases a fixed energy. This shows a third particle shares the energy: the neutrino (β⁺) or antineutrino (β⁻).

N = N₀e^(−λt) and A = λN = λN₀e^(−λt), where the decay constant λ is the probability of decay per unit time. T½ = ln 2 / λ.

The probability that one nucleus decays in time t is 1 − e^(−λt). This is approximately λt only when λt is small (λt ≪ 1).

To find λ from two activities: λ = ln(A₀/A)/t.