Lesson 19: Nuclear Physics and Radioactivity

The nucleus, tiny as it is, holds nearly all the atom's mass and all of its drama: nuclei decay, transmute and, in fission and fusion, release the Binding energy that powers stars and reactors. This lesson covers radioactive decay, half-life, and the mass–energy bookkeeping of nuclear reactions.

Decay, activity and half-life

Unstable nuclei decay randomly — each nucleus has a fixed probability of decaying per unit time, but no nucleus can be predicted individually. The activity A (decays per second, in becquerels) is proportional to the number of undecayed nuclei N, so both fall exponentially. The half-life T½ is the time for half the nuclei (and half the activity) to go: after n half-lives, the fraction remaining is (½)ⁿ. Half-life is a property of the isotope, untouched by temperature, pressure or chemistry — which is what makes radioactive dating possible.

The three radiations differ sharply: alpha particles (helium nuclei, charge +2) are stopped by paper or skin but are highly ionising inside the body; beta particles (electrons ejected when a neutron converts to a proton) travel metres in air and need millimetres of aluminium to stop; gamma rays (very high-energy photons) are the most penetrating, reduced but never fully stopped by lead. In alpha decay the mass number falls by 4 and atomic number by 2; in beta decay the mass number is unchanged and atomic number rises by 1. Nuclear equations must balance both numbers.

Worked example 1: half-life of iodine-131

Iodine-131, used in thyroid medicine, has a half-life of 8.0 days. A sample's activity is initially 4.0 × 10⁸ Bq. Find its activity after 40 days, and the mass of an initial 80 g sample remaining after 24 days. 40 days = 5 half-lives: A = 4.0 × 10⁸ × (½)⁵ = 4.0 × 10⁸ ÷ 32 = 1.25 × 10⁷ Bq. 24 days = 3 half-lives: mass = 80 × (½)³ = 80 ÷ 8 = 10 g. The same factor applies to activity, atom count and mass of the isotope — they all halve together.

Worked example 2: the binding energy of helium-4

Forming a helium-4 nucleus from two protons and two neutrons releases energy because the product weighs less than the ingredients. Using atomic masses (so electron masses cancel): two hydrogen-1 atoms (1.007825 u each) plus two neutrons (1.008665 u each) total 2.015650 + 2.017330 = 4.032980 u. A helium-4 atom has mass 4.002603 u. Mass defect Δm = 4.032980 − 4.002603 = 0.030377 u. With 1 u ≡ 931.5 MeV: E = 0.030377 × 931.5 = 28.3 MeV per nucleus — about 7.1 MeV per nucleon, near the summit of the binding-energy curve, which is why fusing light nuclei (and splitting heavy ones) both release energy.

In the lab: dice — a model of random decay

Roll 200 dice; each throw, remove every die showing a 6 (probability 1/6 per throw, the dice "decay constant"). The survivors fall roughly as 200, 167, 139, 116, 96, 80 … — an exponential curve whose "half-life" is ln 2 ÷ ln(6/5) ≈ 3.8 throws; after about 4 throws roughly half are gone. Plotting ln(N) against throws gives a straight line, whose gradient yields the decay constant. Uncertainty here is *statistical*: counts fluctuate by about ±√N, so with 200 dice a single throw's removal count wavers by several dice, and the fit improves with more dice or repeated class-pooled runs. Real Geiger-counter work faces the same √N statistics — a background count of 24 per minute has an inherent scatter of about ±5, which is why nuclear measurements count for a long time.

Physics in the real world

Carbon-14 dating exploits a half-life of 5,730 years: living things maintain a constant carbon-14 fraction, which decays after death, so the remaining fraction reads the age like a clock. Radiotherapy uses cobalt-60 gamma rays and short-lived isotopes targeted at tumours — short half-lives are medically desirable because the dose stops soon after treatment. Smoke detectors contain americium-241 whose alpha particles ionise air in a sensing chamber. Nuclear power stations split uranium-235 (fission), releasing about 200 MeV per nucleus; the Sun fuses hydrogen into helium by the reaction quantified above.

Common mistakes

• Thinking that after two half-lives everything is gone — a quarter remains; decay is exponential, never a countdown to zero in a fixed number of steps.
• Confusing radiation with contamination: irradiation stops when the source is removed; contamination is radioactive material itself being present.
• In nuclear equations, changing the mass number in beta decay (it stays the same; a neutron becomes a proton).
• Believing half-life changes with temperature or sample size — it is a fixed probability per unit time per nucleus.
• Forgetting that activity halves on the same schedule as N.

Exam Experience

1. During beta-minus decay, a nucleus: A) loses 4 in mass number and 2 in atomic number. B) keeps the same mass number and gains 1 in atomic number as a neutron becomes a proton. C) emits a helium nucleus. D) loses mass number 1 and atomic number 1. Answer: B. A neutron converts to a proton plus an electron (the beta particle); A is unchanged, Z rises by 1.

2. A sample has a half-life of 6 hours. Starting from 64 g of the pure isotope, the mass remaining after 24 hours is: A) 16 g. B) 8 g. C) 4 g. D) 2 g. Answer: C. 24 hours is 4 half-lives: 64 → 32 → 16 → 8 → 4 g.

3. Which radiation is the most penetrating but least ionising per unit path? A) Alpha. B) Beta. C) Gamma. D) All are equal. Answer: C. Gamma rays interact least often per metre travelled, so they penetrate furthest; alpha is the most intensely ionising but is stopped by paper.

Constructed response (5 marks). A wooden artefact contains carbon-14 with activity 1/8 of that in living wood. Carbon-14's half-life is 5,730 years. (a) How many half-lives have elapsed since the tree was felled? (b) Calculate the age of the artefact. (c) State one assumption of the method and one reason the practical dating limit is about 50,000 years. Scoring guide: (a) 1 mark for recognising 1/8 = (½)³, so 3 half-lives. (b) 1 mark for 3 × 5,730 = 17,190 years ≈ 17,000 years. (c) 1 mark for an assumption (e.g., the atmospheric carbon-14 fraction was the same in the past, or no contamination by younger/older carbon); 1 mark for the limit reason: after ~9 half-lives the remaining activity is under 0.2% of living levels and is swamped by background/counting statistics. 1 further mark for coherent presentation with units. Common error: dividing 5,730 by 8 or multiplying 5,730 × 8 instead of counting halvings.

Going further: a kilogram of uranium

Lesson 19's binding-energy logic scales to industrial size. One fission of uranium-235 releases about 200 MeV = 200 × 1.602 × 10⁻¹³ J = 3.2 × 10⁻¹¹ J. A kilogram of pure U-235 contains (1000 ÷ 235) × 6.022 × 10²³ = 2.56 × 10²⁴ nuclei. Fissioning them all releases 2.56 × 10²⁴ × 3.2 × 10⁻¹¹ = 8.2 × 10¹³ J — the output of a 1 GW power station running for nearly a full day, from a lump of metal the size of a tennis ball. The same mass of coal yields about 3 × 10⁷ J: nuclear fuel is roughly three million times more energy-dense, which is why power stations measure their uranium in tonnes per year and their coal in trainloads per day. The price of that density is the waste and the control problem. Fission fragments are neutron-rich, radioactive isotopes with half-lives from seconds to decades; their decay heat continues after shutdown (about 7% of full power at the moment of shutdown, falling over days), which is why reactor cooling can never simply be switched off — the central lesson of the Fukushima accident. And because each fission releases 2–3 neutrons, the chain reaction must be actively throttled by control rods absorbing neutrons to hold the multiplication factor at exactly 1.0: the difference between a power station and a bomb is not the fuel but the neutron bookkeeping.

In focus: shielding and the half-value layer

Radiation shielding obeys the same halving logic as decay, which makes it easy to reason about. For a given gamma source and material, the half-value layer is the thickness that halves the intensity. If lead's half-value layer for a particular gamma energy is 1.0 cm, then 3.0 cm of lead passes (½)³ = 1/8 of the intensity, and 10 cm passes about (½)¹⁰ ≈ 1/1000. Real protection combines three independent strategies, each from this course: time (dose is proportional to exposure duration), distance (inverse-square law — doubling distance quarters the dose rate, Lesson 12) and shielding (half-value layers). Radiation workers budget all three, and their personal dosimeters integrate the result. A sanity anchor for public discussions: the average person absorbs about 2–3 millisieverts per year from natural background — cosmic rays, rocks, and the potassium-40 in their own body (your bones are measurably radioactive; a banana's potassium makes it a standard joking unit, the "banana equivalent dose" of about 0.1 microsieverts). Medical CT scans add roughly 10 mSv in one procedure. Risk judgements need these comparisons, and they are exactly the kind of quantitative framing this topic exists to teach — physics as a defence against both complacency and panic.

Formula card

Exam technique: count halvings rather than reaching for logarithms: write the chain 1 → ½ → ¼ → ⅛ and match the given fraction. In nuclear equations, balance the top numbers (mass) and bottom numbers (charge) as two separate audits — an equation that balances only one of them is wrong.