AQA A-Level Physics: Nuclear physics (7408) – Revision Notes

For full explanations and worked examples, use the Nuclear physics study guide.

These notes cover section 3.8 Nuclear physics (3.8.1.1 to 3.8.1.8) of the AQA AS and A-level Physics (7407/7408) specification, version 1.4 (AS and A-level exams June 2016 onwards). The section is A-level only and is assessed in Paper 2. Sections 3.9-3.13 are options – you study one, assessed in Paper 3 Section B – while this section is compulsory.

Links: AQA A-level Physics hub, printable checklist, free diagnostics and the practice questions. Decay equations and nuclide notation from AS: Particles and radiation revision notes. For MeV and J conversions: SI units and prefixes.

Equations

Quantity	Equation	Notes
Inverse-square law (γ)	I = k/x²	correct for background first
Rate of decay	ΔN/Δt = −λN	λ = probability of decay per unit time
Number remaining	N = N₀e^(−λt)	same form for A and count rate
Activity	A = λN	Bq needs λ in s⁻¹
Half-life	T½ = ln2/λ
Number of nuclei	N = (mass/molar mass) × Nᴀ	molar mass in the same mass unit
Closest approach	Eₖ = Qq/(4πε₀r)	gives an upper limit for R
Nuclear radius	R = R₀A^(1/3)	R₀ about 1.2 fm
Mass-energy	E = mc²	1 u = 931.5 MeV
Binding energy	(mass of nucleons − mass of nucleus) × c²	use nuclear, not atomic, mass

3.8.1.1 Rutherford scattering

Observation	Conclusion
Most α pass straight through	atom mostly empty space
Some deflected through large angles	small, concentrated positive charge
Very few rebound	nucleus holds most of the mass

Plum-pudding model replaced by the nuclear model; later the neutron, then measured nuclear radii. Models change when evidence demands it.

3.8.1.2 α, β and γ

• α: strongly ionising, few cm in air, stopped by paper.
• β⁻: less ionising, about a metre in air, stopped by a few mm of aluminium.
• γ: weakly ionising, obeys I = k/x² in air, reduced by thick lead.
• Absorption test: background → no absorber → paper → aluminium → lead. Subtract background each time.
• Thickness gauges: β for paper and aluminium foil; γ for steel.
• Safe handling: tongs (distance), short time, lead-lined storage, never point a source at people.
• Background sources: radon, rocks and buildings, cosmic rays, food, medical, nuclear industry.
• Medicine: weigh benefit against the risk to healthy tissue.

Method in steps: Required practical 12

1. Measure background count over a long time.
2. Record counts at several measured distances d.
3. Corrected count rate C = measured − background.
4. Plot 1/√C against d: straight line confirms I ∝ 1/x².
5. The d-intercept gives −e, the offset between measured and true distance.

3.8.1.3 Radioactive decay

• Random and spontaneous; constant decay probability λ for each nucleus.
• Equal times give equal fractional falls.
• Modelling: throw many dice and remove each one showing a six. Every die has the same probability (1/6) per throw, so about 1/6 of those left go each throw and the number falls exponentially, like nuclei with constant λ.
• Decay curve: read several halvings and average.
• Log graph: ln A = ln A₀ − λt. Gradient = −λ, intercept = ln A₀.
• Uses: waste storage times, radioactive dating.

Worked reminder. T½ = 2.0 h. Fraction left after 5.0 h:

λ = ln2 / 2.0 = 0.347 h⁻¹     N/N₀ = e^(−0.347 × 5.0) = 0.177

Worked reminder. Nuclei in 1.0 g of a nuclide with molar mass 100 g mol⁻¹: N = (1.0/100) × 6.02 × 10²³ = 6.02 × 10²¹. Keep the mass and molar mass in the same unit.

3.8.1.4 Nuclear instability

Region of N–Z graph	Decay	Z	N
Above stability line (too many n)	β⁻	+1	−1
Below line (too many p)	β⁺ or electron capture	−1	+1
Very heavy	α	−2	−2

• Stable light nuclei: N ≈ Z. Heavier stable nuclei: N > Z.
• β⁺: X → Y + e⁺ + νₑ. Electron capture: X + e⁻ → Y + νₑ.
• Excited daughter → γ emission; γ energy = gap between nuclear levels.
• Technetium-99m: pure γ emitter, short half-life, used in diagnosis.

3.8.1.5 Nuclear radius

• α closest approach: overestimates R (α never touches); α can be affected by the strong force near the nucleus.
• Electron diffraction: electrons feel no strong force; intensity against angle shows a central maximum, a first minimum, weaker maxima. First minimum: sin θ ≈ 0.61λ/R.
• Typical R: 10⁻¹⁵ to 10⁻¹⁴ m.
• ln R against ln A: gradient 1/3, intercept ln R₀.
• V ∝ A and mass ∝ A, so density is constant, about 10¹⁷ kg m⁻³.

Worked reminder. Radius of a nucleus with A = 64, R₀ = 1.2 fm: R = 1.2 × 64^(1/3) = 4.8 fm.

3.8.1.6 Mass and energy

• Atomic mass unit: 1 u is one-twelfth of the mass of a carbon-12 atom (1.661 × 10⁻²⁷ kg); 1 u = 931.5 MeV.
• Mass difference: mass of separate nucleons minus mass of nucleus.
• Binding energy: energy needed to separate the nucleus into nucleons.
• Binding energy per nucleon peaks near A ≈ 56 (about 8.8 MeV).
• Fusion (left of peak) and fission (right of peak) both increase binding energy per nucleon, so release energy.
• Energy released = total binding energy after − total binding energy before, or mass lost × 931.5 MeV.
• Atomic mass → nuclear mass: subtract Z × 0.000549 u.

Method in steps: binding energy per nucleon

1. Convert the atomic mass to nuclear mass (subtract Z electron masses).
2. Add Z proton masses and (A − Z) neutron masses.
3. Mass difference = step 2 − step 1, in u.
4. Multiply by 931.5 to get MeV (or by c² after converting to kg, for J).
5. Divide by A for binding energy per nucleon.

Method in steps: energy released in a reaction

1. Add the masses on each side (atomic masses are fine if the electrons balance).
2. Mass lost = mass before − mass after.
3. Energy released = mass lost × 931.5 MeV.
4. Check that nucleon number and charge balance in the equation first.

3.8.1.7 Induced fission

• Thermal neutron absorbed by U-235 → two fragments + 2 or 3 fast neutrons + energy.
• Chain reaction: each fission causes at least one more.
• Critical mass: minimum mass for a self-sustaining chain reaction.
• Moderator (water, graphite): slows neutrons by elastic collisions; light nuclei, low absorption.
• Control rods (boron, cadmium): absorb neutrons; moved in or out to keep one further fission per fission.
• Coolant (water, carbon dioxide): removes heat; high specific heat capacity, flows easily.
• Head-on elastic collision, neutron mass m, nucleus M: v/u = (m − M)/(m + M).

3.8.1.8 Safety aspects

• Enriched uranium fuel; spent fuel moved by remote handling.
• Steel and concrete shielding absorbs neutrons and γ.
• Emergency shut-down: control rods fully inserted.
• High-level waste: cooled under water, then sealed and stored long term. Low-level waste: sealed and buried.
• Weigh low-carbon, reliable output against accident risk and waste.

Must-know distinctions

• Decay constant (probability per unit time) vs activity (decays per second).
• Moderator slows neutrons; control rods absorb them.
• Mass difference (mass, u or kg) vs binding energy (energy, MeV or J).
• β⁺ decay emits a positron; electron capture absorbs an electron. Both: Z − 1, N + 1.
• Closest approach gives an upper limit; electron diffraction gives the radius.

Quick self-test

1. Corrected γ count rate is 960 min⁻¹ at 0.10 m. Find it at 0.30 m.
2. T½ = 12 h. What fraction remains after 2 days?
3. T½ = 30 s. Find λ.
4. Find the ratio of nuclear radii for A = 216 and A = 27.
5. Convert a mass difference of 0.0050 u to MeV.
6. How do Z and N change in α decay?
7. Which radiation is used to monitor steel thickness?
8. Why does electron diffraction give a better radius than α scattering?
9. Find the activity of 3.0 × 10¹⁸ nuclei with λ = 2.0 × 10⁻¹⁰ s⁻¹.
10. Find the mass equivalent of 1.0 MeV in kg.
11. A neutron hits an oxygen-16 nucleus head-on elastically. What fraction of its kinetic energy does it keep?
12. Which side of the binding energy per nucleon peak releases energy by fusion?

Answers

1. Distance × 3 → rate ÷ 9: 960/9 = 107 min⁻¹.
2. 48 h = 4 half-lives: (1/2)⁴ = 1/16 (0.0625).
3. λ = ln2/30 = 0.0231 s⁻¹.
4. (216/27)^(1/3) = 8^(1/3) = 2.
5. 0.0050 × 931.5 = 4.66 MeV.
6. Z − 2, N − 2.
7. γ (β is stopped by steel).
8. Electrons are leptons, so no strong-force interaction; α results only give an upper limit.
9. A = λN = 6.0 × 10⁸ Bq.
10. m = E/c² = 1.6 × 10⁻¹³ / (3.00 × 10⁸)² = 1.78 × 10⁻³⁰ kg.
11. ((1 − 16)/17)² = (15/17)² = 0.779.
12. The low-A side (A below about 56).

Where marks are usually lost

• Leaving background in the count rate before testing the inverse-square law.
• Mixing time units: λ in day⁻¹ or h⁻¹ used with A = λN to give “Bq”.
• Forgetting molar mass is usually in g mol⁻¹ when the sample mass is in kg.
• Giving the gradient of a ln A graph as λ without saying λ = −gradient.
• Using atomic masses in binding-energy sums without subtracting electron masses.
• Treating closest-approach distance as the radius rather than an upper limit.
• Saying control rods “slow” neutrons, or the moderator “absorbs” them.
• Writing β⁺ or electron-capture equations without the neutrino, or with an antineutrino.
• Naming fission and fusion regions the wrong way round on the binding energy per nucleon graph.

Official syllabus

AQA AS and A-level Physics (7407/7408) specification, version 1.4, July 2026 (AS and A-level exams June 2016 onwards), published by AQA. Section 3.8 Nuclear physics (A-level only).