Atomic & nuclear physics
The idea
The nucleus packs protons and neutrons into a few femtometers, where the strong force outmuscles the enormous Coulomb repulsion between protons. Its energy ledger is the binding energy: a bound nucleus weighs measurably less than its free constituents, and that mass defect, through E = mc², is the energy that would be released in forming it. The curve of binding energy per nucleon peaks near iron, which is why fusing light nuclei and splitting heavy ones both release energy.
Unstable nuclei decay by alpha, beta, or gamma emission, and the process is genuinely random per nucleus yet exquisitely predictable in aggregate. Each nucleus has a fixed decay probability per unit time λ, giving dN/dt = −λN and the exponential law N(t) = N₀e^(−λt); the half-life t½ = ln2/λ is the time for any starting amount to halve. Activity (decays per second) inherits the same exponential, which underlies radiometric dating and dosage planning alike.
Half-life is statistics, not scheduling: after one half-life each surviving nucleus is not 'half dead' — it is unchanged, with exactly the same chances going forward. Nuclei have no memory, which is precisely why the exponential never quite reaches zero and why two half-lives leave a quarter, not nothing.
Worked example
Cobalt-60, used in radiotherapy, has a half-life of 5.27 years. A hospital source must be replaced when its activity falls to 10% of the initial value. How long after installation is replacement due?
- Convert the half-life to a decay constant: λ = ln2/t½ = 0.693/5.27 ≈ 0.132 per year — a decay rate that gives each nucleus roughly a one-in-eight chance of decaying in any given year.
- Set up the exponential decay condition for activity: A(t) = A₀e^(−λt), and replacement occurs when A/A₀ = 0.10.
- Solve by taking the natural log: e^(−λt) = 0.10 gives λt = ln10 ≈ 2.303, so t = 2.303/0.132 ≈ 17.5 years.
- Cross-check by counting half-lives: 10% lies between 1/8 (three half-lives, 15.8 yr) and 1/16 (four half-lives, 21.1 yr), and indeed log₂(10) ≈ 3.32 half-lives × 5.27 yr ≈ 17.5 yr — consistent.
- Interpret: the answer is independent of the source's initial size; a source twice as strong reaches 10% of its own initial activity at exactly the same time, because exponential decay scales every starting amount identically.
Answer. The source reaches 10% of its initial activity after about 17.5 years.
Check your understanding
- Why does exponential decay follow from each nucleus having a constant decay probability per unit time?
- How would you explain to a friend that a nucleus surviving three half-lives is no closer to decaying than a fresh one?
- Why does the binding-energy curve peaking at iron make both fusion of light elements and fission of heavy ones exothermic?
- Where does the released energy in a decay physically come from, given that total mass-energy is conserved?
Build the foundations first
Atomic & nuclear physics builds on these concepts. If any feel shaky, start there.