The exponential decay law
Radioactive decay is a quantum process: it is genuinely random and memoryless. A given nucleus has no age and no plan; in any instant it has a fixed probability per unit time, the decay constant \lambda, of decaying. You cannot say which nucleus will go or when — only the statistics of a large ensemble. That single assumption gives the radioactive decay law.
Exponential decay: N halves every half-life; the activity A (decays per second) is proportional to how many nuclei remain.
Alpha decay: tunnelling out
In alpha decay a heavy nucleus ejects a helium-4 nucleus (two protons, two neutrons), dropping its Z by 2 and A by 4. Because it is a two-body split, the alpha comes out mono-energetic — a sharp line — carrying most of the released energy. Alpha emission dominates among heavy nuclei, where mounting Coulomb repulsion makes shedding a tightly bound alpha cluster favourable.
The deep puzzle Gamow solved in 1928: the alpha is trapped behind a Coulomb barrier some 25–30 MeV high, yet it escapes carrying only ~5 MeV. Classically impossible. Quantum mechanically the alpha tunnels through the barrier, its wavefunction leaking out with exponentially small amplitude. Because the tunnelling probability depends exquisitely sensitively on the decay energy, half-lives span an astonishing range — from microseconds to billions of years — captured by the Geiger–Nuttall law.
The Geiger–Nuttall law: a tiny change in the decay energy Q swings the half-life over many orders of magnitude — the fingerprint of tunnelling, often estimated with the WKB approximation.
Beta decay: the weak force and the neutrino
Beta decay converts a neutron into a proton (or vice versa), moving a nucleus along an isobar toward the valley floor without changing A. In \beta^- decay a neutron becomes a proton, emitting an electron; the mirror processes are \beta^+ (positron) emission and electron capture. This is the work of the weak interaction, mediated by the massive W boson — the only force able to change one kind of quark into another.
Beta-minus decay. The antineutrino was the ghost demanded by conservation laws.
Beta electrons emerge with a continuous spread of energies, not a sharp line — which in the 1920s looked like energy conservation was broken. Pauli rescued it by postulating an unseen third particle sharing the energy: the neutrino. It restores conservation of energy, momentum and angular momentum, and was detected only in 1956. The decay rate itself follows from Fermi's golden rule, the workhorse of quantum transition theory.
Gamma decay and the Q-value
After an alpha or beta decay the daughter is often left in an excited state. It relaxes by emitting a high-energy photon — gamma decay — with no change in Z or A, just as an excited atom emits light, but at MeV rather than eV energies. Whether any decay can happen at all is decided by its Q-value, the energy released, computed from the masses of the ingredients minus the products.
A decay proceeds spontaneously only if it releases energy (Q > 0); the Q is shared as kinetic energy of the products.
Worked example: dating with carbon-14
A wooden artefact has a carbon-14 activity that is 25% of a living tree's. Carbon-14 beta-decays with a half-life of 5730 years. How old is it?
- Relate activity to time. Activity is proportional to the number of C-14 nuclei, so \mathcal{A}/\mathcal{A}_0 = N/N_0 = e^{-\lambda t} = 0.25.
- Count half-lives. 0.25 = (1/2)^2, so exactly two half-lives have passed — no calculator needed.
- Multiply. t = 2 \times t_{1/2} = 2 \times 5730 = 11{,}460 years.
- Sanity-check the general case. For an arbitrary fraction, invert the exponential: t = \dfrac{t_{1/2}}{\ln 2}\ln\!\dfrac{\mathcal{A}_0}{\mathcal{A}}. Plugging \mathcal{A}_0/\mathcal{A} = 4 gives back 11,460 years — good.
Two closing structural ideas. Decay chains: a heavy nucleus like uranium-238 decays through a long series of alphas and betas before ending at stable lead-206. Secular equilibrium: when a long-lived parent feeds short-lived daughters, each daughter's activity climbs until it equals the parent's — so radon in your basement is continually resupplied by the uranium in the rock beneath it.