AQA A-Level Physics: Nuclear physics (7408)

This guide teaches section 3.8 Nuclear physics of the AQA AS and A-level Physics (7407/7408) specification, version 1.4 (AS and A-level exams June 2016 onwards). It covers 3.8.1 Radioactivity, every sub-section from 3.8.1.1 to 3.8.1.8. Section 3.8 is A-level only: it is not in the AS, and at A-level it is assessed in Paper 2 (sections 6.2, 7 and 8, with sections 1 to 6.1 assumed). Sections 3.9-3.13 are options – you study one, assessed in Paper 3 Section B – but nuclear physics is compulsory for everyone.

Links: course hub, printable checklist, free diagnostics, revision notes and practice questions. Nuclide notation and α and β⁻ decay equations are in Particles and radiation.

What this topic covers

Spec	What you must be able to do
3.8.1.1	Rutherford scattering (qualitative); how models of the nucleus changed
3.8.1.2	α, β, γ properties and identification; thickness gauges; inverse-square law; background; medical risk and benefit. Required practical 12
3.8.1.3	Random decay, decay constant, activity, half-life; molar mass; half-life from graphs
3.8.1.4	N–Z graph; decay modes; changes in N and Z; excited states; technetium-99m
3.8.1.5	Radius by α closest approach and electron diffraction; R = R₀A^(1/3); nuclear density
3.8.1.6	E = mc², mass difference, binding energy, u; fission and fusion energies; binding energy per nucleon graph
3.8.1.7	Induced fission, chain reaction, critical mass; moderator, control rods, coolant
3.8.1.8	Fuel, remote handling, shielding, shut-down, waste; risk and benefit

Constants used: e = 1.60 × 10⁻¹⁹ C, ε₀ = 8.85 × 10⁻¹² F m⁻¹, Nᴀ = 6.02 × 10²³ mol⁻¹, proton mass 1.007276 u, neutron mass 1.008665 u, electron mass 0.000549 u.

3.8.1.1 Rutherford scattering

α particles were fired at thin gold foil in a vacuum, with a detector moved round the foil.

• Most α particles went straight through. Most of the atom is empty space.
• A few were deflected through large angles. There is a concentrated positive charge that repels them.
• A very small number bounced back (more than 90°). The positive charge and most of the mass are in a tiny central nucleus.

This replaced the “plum pudding” model (positive charge spread through the atom, electrons embedded in it), which could not explain large-angle scattering. Later the neutron was discovered and nuclear size measured (3.8.1.5): models change when evidence cannot be explained.

3.8.1.2 α, β and γ radiation

	α	β⁻	γ
Nature	helium nucleus	fast electron	EM radiation
Ionisation	strong	weaker	very weak
Range in air	a few cm	about a metre	follows inverse-square law
Stopped by	paper	a few mm of aluminium	reduced by thick lead

Identifying radiation. Measure background first, then the count rate with no absorber, paper, a few mm of aluminium, and lead; subtract background each time. A large drop with paper means α; a further drop to background with aluminium means β; a count above background through aluminium that falls with lead means γ.

Thickness gauges. β for paper and aluminium foil (a thicker sheet absorbs more, the count rate falls, rollers are adjusted; α would be stopped completely). γ for steel, which stops β.

Hazards. α is least dangerous outside the body (dead skin stops it) and most dangerous inside, where all its energy goes into a little tissue.

Inverse-square law for γ. γ spreads from a point source with almost no absorption in air, so I = k/x², x being the distance from the source. Doubling x quarters the intensity: use long tongs, short exposure and lead-lined storage.

Background radiation comes from radon gas, rocks and building materials, cosmic rays, food and drink, medical uses and the nuclear industry. Subtract it.

Medicine. Diagnosis and therapy also damage healthy cells, so the benefit is weighed against the risk and doses kept as low as is useful.

Required practical 12: inverse-square law for γ

Record counts from a γ source at a range of measured distances d from a GM tube, plus a background count. The true distance is x = d + e, with e an unknown offset (the source sits inside its holder). Corrected rate C = k/(d + e)², so 1/√C against d is a straight line if the law holds, with d-intercept −e.

Worked example. Measured rate at 0.20 m is 510 min⁻¹; background is 30 min⁻¹. Corrected rate = 480 min⁻¹. At 0.40 m the corrected rate is 480 ÷ 4 = 120 min⁻¹, so the measured rate is 150 min⁻¹.

3.8.1.3 Radioactive decay

Decay is random: you cannot predict which nucleus decays next. Each nucleus has the same constant probability of decay per unit time, the decay constant λ, so the rate of decay is proportional to the number present:

ΔN/Δt = −λN        N = N₀e^(−λt)
A = λN             A = A₀e^(−λt)        T½ = ln2 / λ

Activity is in becquerels (decays per second), so use λ in s⁻¹.

Worked example (molar mass). A sample contains 2.0 µg of an isotope with molar mass 131 g mol⁻¹ and half-life 8.0 days. Find its activity, and how long until the activity falls to 1.0 × 10⁸ Bq.

N = (2.0 × 10⁻⁶ / 131) × 6.02 × 10²³ = 9.19 × 10¹⁵
λ = ln2 / (8.0 × 86 400 s) = 1.00 × 10⁻⁶ s⁻¹
A = λN = 9.2 × 10⁹ Bq
t = ln(A₀/A) / λ = ln(9.22 × 10⁹ / 1.0 × 10⁸) × 8.0 / ln2 = 52 days

Half-life from graphs. On a decay curve, read the time to fall from any value to half of it, from several starting points, and average. On a log graph, ln A = ln A₀ − λt, so a plot of ln A against t is a straight line with gradient −λ and intercept ln A₀.

Worked example (log graph). A best-fit line of ln(A/Bq) against t falls from 9.20 to 7.12 in 50 min.

gradient = (7.12 − 9.20) / 50 = −0.0416 min⁻¹ → λ = 0.0416 min⁻¹
T½ = ln2 / 0.0416 = 16.7 min

Applications. Long-half-life waste must be stored for a very long time. Radioactive dating compares the remaining fraction with the start and solves N = N₀e^(−λt) for t.

3.8.1.4 Nuclear instability

On a graph of N against Z, stable light nuclei lie close to N = Z. Heavier stable nuclei bend above that line (N > Z): extra neutrons add strong-force attraction without adding repulsion.

Unstable nucleus	Decay mode	Change in Z	Change in N
Neutron-rich (above the line)	β⁻	+1	−1
Proton-rich (below the line)	β⁺	−1	+1
Proton-rich	electron capture	−1	+1
Very heavy	α	−2	−2

Equations, for a parent X and daughter Y:

β⁺:               ᴬ_Z X → ᴬ_(Z−1) Y + e⁺ + νₑ
electron capture: ᴬ_Z X + e⁻ → ᴬ_(Z−1) Y + νₑ

After α or β decay the daughter is often in an excited state. It drops to the ground state by emitting a γ photon, so γ energies equal gaps on a nuclear energy level diagram.

Technetium-99m is a long-lived excited state used in medical diagnosis. It emits only γ, which leaves the body to be detected and ionises little, and its short half-life means the activity in the patient falls quickly after the scan.

3.8.1.5 Nuclear radius

Closest approach. An α particle aimed head-on at a nucleus stops when all its kinetic energy has become electric potential energy:

Eₖ = Qq / (4πε₀r)

This gives an upper limit for the radius.

Worked example. A 5.0 MeV α particle approaches a gold nucleus (Z = 79) head-on.

Eₖ = 5.0 × 10⁶ × 1.60 × 10⁻¹⁹ = 8.0 × 10⁻¹³ J
r = (2e)(79e) / (4πε₀ Eₖ) = 4.5 × 10⁻¹⁴ m

Electron diffraction. High-energy electrons, with de Broglie wavelength about the size of a nucleus, diffract round it. Intensity against angle falls from a central maximum to a minimum, then a weaker maximum. For a circular object the first minimum is at sin θ ≈ 0.61λ/R. Electrons are leptons, so the strong force does not act on them: the method is more accurate than α scattering.

Worked example. Electrons of wavelength 3.0 × 10⁻¹⁵ m give a first minimum at 37°.

R = 0.61 × 3.0 × 10⁻¹⁵ / sin 37° = 3.0 × 10⁻¹⁵ m

Typical nuclear radii are a few fm (10⁻¹⁵ to 10⁻¹⁴ m). Data fit R = R₀A^(1/3), R₀ about 1.2 fm; ln R against ln A has gradient 1/3. See also estimation of physical quantities.

Constant density. Volume ∝ R³ ∝ A, and mass ∝ A, so density is the same for all nuclei:

ρ = m / (4/3 π R₀³) = 1.67 × 10⁻²⁷ / (4/3 π (1.2 × 10⁻¹⁵)³) = 2.3 × 10¹⁷ kg m⁻³

3.8.1.6 Mass and energy

E = mc² applies to every energy change. The mass difference (mass defect) is the mass of the separate nucleons minus the mass of the nucleus. The binding energy is the energy needed to separate a nucleus into its nucleons, equal to the mass difference × c². Use 1 u = 931.5 MeV.

Worked example. Helium-4 atomic mass 4.002603 u.

nuclear mass = 4.002603 − 2(0.000549) = 4.001505 u
Δm = 2(1.007276) + 2(1.008665) − 4.001505 = 0.030377 u
binding energy = 0.030377 × 931.5 = 28.3 MeV  → 7.07 MeV per nucleon

Binding energy per nucleon against A rises steeply for light nuclei, peaks near A ≈ 56 (about 8.8 MeV per nucleon) and falls slowly for heavy nuclei. Energy is released when products have higher binding energy per nucleon:

• Fusion of light nuclei (left of the peak).
• Fission of heavy nuclei (right of the peak).

Worked example (fission). A nucleus with A = 236 (7.6 MeV per nucleon) splits into fragments with A = 140 (8.4 MeV) and A = 94 (8.6 MeV) plus 2 neutrons.

energy released = 140(8.4) + 94(8.6) − 236(7.6)
                = 1176 + 808.4 − 1793.6 ≈ 190 MeV

This physics lets society base energy decisions on evidence.

3.8.1.7 Induced fission

A thermal (slow) neutron absorbed by uranium-235 makes it split into two fragments and two or three fast neutrons. If at least one of these, on average, causes another fission, a chain reaction runs. The critical mass is the smallest mass of fuel that sustains it; with less, too many neutrons escape.

Part	Function	Material choice	Examples
Moderator	slows fast neutrons to thermal speeds by elastic collisions	light nuclei (similar mass to a neutron), low neutron absorption	water, graphite
Control rods	absorb neutrons so each fission causes one more	absorb neutrons strongly	boron, cadmium
Coolant	carries heat to the heat exchanger	high specific heat capacity, flows easily	water, carbon dioxide gas

Moderation model. In a head-on elastic collision with a stationary nucleus (mass M), momentum and kinetic energy are conserved and a neutron (mass m) leaves with (m − M)/(m + M) × its original velocity.

Worked example. Neutron (1 u) hits a carbon-12 nucleus (12 u) head-on.

v/u = (1 − 12)/13 = −0.846
fraction of Eₖ kept = 0.846² = 0.716 → 28% lost per collision

With hydrogen (M ≈ m) it can lose all its energy at once: light nuclei moderate best.

3.8.1.8 Safety aspects

• Fuel: uranium enriched in U-235. Spent fuel is highly radioactive, so it is moved by remote handling.
• Shielding: thick steel and concrete round the core absorb neutrons and γ.
• Emergency shut-down: control rods drop fully in and stop the chain reaction.
• Waste: high-level waste (spent fuel) is hot and very active; it is cooled under water, then sealed and stored long term. Low-level waste (clothing, tools) is sealed and buried.
• Risk and benefit: low-carbon, steady output against accident risk and waste.

Common errors

• Applying I = k/x² before subtracting background.
• Using λ in h⁻¹ with N to get activity in Bq.
• Using atomic mass where nuclear mass is needed.
• Saying the moderator absorbs neutrons; the control rods do.

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).