AQA A-Level Physics: Nuclear Physics Complete Guide
AQA A-Level Physics: Nuclear Physics Complete Guide
Nuclear Physics is Section 3.8 of the AQA A-Level Physics specification. It is one of the more mathematical Year 2 topics, with three core equations — for half-life and decay constant, for nuclear radius, and for mass-energy equivalence — that have to be combined fluently across exam questions. It is also one of the most contextual, because the same physics underpins medical imaging, smoke detectors, carbon dating, the energy released by the Sun, the design of every operating power reactor, and the prospects for fusion. A well-prepared candidate is not only fluent in the calculations but can also describe the practical consequences with confidence.
This guide covers all of Section 3.8 in roughly the order an exam paper is likely to test it: atomic structure and Rutherford scattering, the types of radioactive decay, decay equations and half-life, nuclear radius and density, binding energy per nucleon, fission and fusion, induced fission and reactor physics, and Required Practical 7. Throughout we point out the conceptual moves examiners reward — particularly the careful use of mass-energy units — and the small mistakes that lose marks.
Atomic Structure and Rutherford Scattering
The modern picture of the atom — a tiny dense positively-charged nucleus surrounded by orbiting electrons — emerged from the Geiger-Marsden experiment of 1909, often referred to as Rutherford scattering. Alpha particles from a radioactive source were directed at a very thin gold foil, and the angles at which they scattered were measured using a movable detector.
The crucial observations were:
- Most alpha particles passed straight through the foil with little deflection.
- A small fraction were deflected through significant angles.
- A very small fraction (about 1 in 8000) were deflected backwards at angles greater than 90°.
The conclusions, when combined with the earlier raisin-pudding model, were that an atom is mostly empty space (most alphas pass through); that almost all the mass and all the positive charge is concentrated in a very small volume at the centre — the nucleus — that occasionally a positive alpha gets close enough to be repelled strongly; and that the closest-approach distance gives an upper bound on the nuclear radius, around 10⁻¹⁴ m.
A typical exam question asks you to discuss why each observation rules out the raisin-pudding (plum-pudding) model. The mark scheme rewards specific reasoning rather than vague "the nucleus was small."
Types of Radioactive Decay
There are four types of decay you must know:
- Alpha decay (α). A helium-4 nucleus (2 protons, 2 neutrons) is emitted. A → A−4, Z → Z−2. Range in air: a few centimetres; stopped by paper or skin. Highly ionising. Symbol: ⁴₂He or ⁴₂α.
- Beta-minus decay (β⁻). A neutron in the nucleus converts to a proton, emitting an electron and an antineutrino. A unchanged, Z → Z+1. Range in air: tens of centimetres; stopped by a few millimetres of aluminium. Moderately ionising. Symbol: ⁰₋₁e or ⁰₋₁β.
- Beta-plus decay (β⁺). A proton in the nucleus converts to a neutron, emitting a positron and a neutrino. A unchanged, Z → Z−1. Range similar to β⁻. The positron annihilates with an electron almost immediately, producing two 511 keV gamma photons in opposite directions — the basis of PET scanning.
- Gamma decay (γ). A high-energy photon emitted from an excited nucleus. A and Z both unchanged — it is a transition between nuclear energy levels, not a transmutation. Range: many metres in air; significantly attenuated only by thick lead or concrete. Weakly ionising.
You will need to be able to write balanced decay equations for each type. The two rules are conservation of nucleon number (A) and conservation of charge (Z). Beta-minus and beta-plus decay also require the emission of the appropriate neutrino or antineutrino — examiners often dock marks if these are missing.
Activity, Decay Constant and Half-Life
Radioactive decay is a random process — you cannot predict when any individual nucleus will decay — but it is statistically predictable for large numbers of nuclei.
The decay constant λ is the probability per unit time that a given nucleus will decay. It has units of s⁻¹.
The activity A of a sample is the rate at which nuclei in the sample are decaying:
A = λN
where N is the number of undecayed nuclei. Activity is measured in becquerels (Bq) — one becquerel is one decay per second.
The number of undecayed nuclei follows exponential decay:
N = N₀ e^(−λt)
with corresponding equations for activity (A = A₀ e^(−λt)) and for the count rate measured by a detector.
The half-life t_½ is the time for half the original nuclei to decay. Setting N = N₀/2 in the exponential gives:
t_½ = ln 2 / λ
so t_½ × λ = ln 2 ≈ 0.693 — a very useful sanity check.
A common exam question asks you to find the activity of a sample at a given time, or to find how long it takes for the activity to fall below a stated value. Use the exponential equation, take logarithms, and solve for t. Watch units: λ in s⁻¹ pairs with t in s, while λ in y⁻¹ pairs with t in years.
Nuclear Radius and Density
Electron diffraction experiments — where high-energy electrons are scattered from nuclei and the diffraction minimum is detected — give direct measurements of nuclear radii. The empirical relationship is:
R = R₀ A^(1/3)
with R₀ ≈ 1.2 × 10⁻¹⁵ m (1.2 fm). The cube-root dependence is exactly what you would expect if nuclei behaved as incompressible spheres of constant density — doubling the number of nucleons doubles the volume, so the radius scales as A^(1/3).
The nuclear density is therefore essentially constant across all nuclei, around 2.3 × 10¹⁷ kg m⁻³. This is roughly 10¹⁴ times the density of ordinary matter. Examiners often ask candidates to calculate it for a specific nuclide and then comment on what the answer implies about the structure of matter — almost all the mass of an atom is in the nucleus, and the nucleus itself is essentially "packed" at the limit of strong-force-bound matter.
The diffraction angle for the first minimum in an electron-diffraction pattern is given by sin θ ≈ 1.22 λ / (2R), where λ here is the de Broglie wavelength of the incident electrons. This is a synoptic link back to wave-particle duality from Section 3.2 of the specification.
Mass Defect and Binding Energy
The mass of any nucleus is less than the sum of the masses of its constituent protons and neutrons. The difference is the mass defect Δm. By Einstein's mass-energy equivalence:
ΔE = Δm c²
the energy equivalent of this mass defect is the binding energy of the nucleus — the energy that would be required to separate the nucleus completely into individual nucleons, or equivalently the energy that was released when those nucleons came together to form the nucleus.
Binding energies are normally quoted in MeV. A useful conversion factor: 1 unified atomic mass unit (u) is equivalent to about 931.5 MeV / c². Many AQA mark schemes work directly in u; you should be fluent in both u and SI units.
The binding energy per nucleon (BE / A) is more useful than the total binding energy when comparing nuclei. Plotted against A, it produces the famous binding-energy curve:
- Very light nuclei (H, He, Li): BE/A rises rapidly with A.
- A peak around A ≈ 56 (iron, Fe-56) at about 8.8 MeV per nucleon.
- A slow decline from iron upwards, falling to about 7.6 MeV per nucleon for uranium.
The shape of this curve is the single most important fact in nuclear physics. The fact that iron sits at the peak means:
- Fusion of light nuclei (towards iron) releases energy.
- Fission of very heavy nuclei (towards iron) releases energy.
Neither process can release energy from iron itself, which is one reason why the cores of the most massive stars stop fusing when they have produced an iron core. It is also why the iron-group elements are unusually abundant in the universe relative to their immediate neighbours.
Nuclear Fission and Fusion
In nuclear fission, a heavy nucleus splits into two lighter nuclei plus several neutrons. The most common example is the induced fission of uranium-235 by a thermal neutron:
n + U-235 → Ba-141 + Kr-92 + 3n (a representative product split)
The specific daughter nuclei vary, but each fission of U-235 releases about 200 MeV — overwhelmingly as kinetic energy of the daughter nuclei and the neutrons, which then thermalises as heat. The released neutrons can induce further fissions, leading to a chain reaction.
In nuclear fusion, two light nuclei combine to form a heavier nucleus. The most accessible example is the deuterium-tritium reaction:
D + T → He-4 + n
with a release of about 17.6 MeV per reaction. Per nucleon, fusion releases far more energy than fission, but achieving it on Earth requires temperatures of order 10⁷ K so that the colliding nuclei have enough kinetic energy to overcome their mutual electrostatic repulsion. This is the engineering hurdle that fusion reactor projects like ITER are designed to overcome.
You should be able to calculate the energy released in any given fission or fusion reaction from the mass defect using Δm and Δm × c² (or Δm × 931.5 in MeV / u). A typical exam calculation gives you the masses of all the species, asks you to compute Δm, and then asks you to convert to MeV.
Induced Fission and Reactor Physics
A nuclear reactor is a controlled chain reaction. The key components are:
- Fuel. Typically uranium oxide enriched to a few percent U-235. Pure natural uranium is 99.3% U-238 and only 0.7% U-235, so most reactors require enrichment.
- Moderator. Slows down the neutrons released by fission to thermal speeds (~ 2200 m s⁻¹). U-235 fission is strongly favoured by slow neutrons. Common moderators are graphite (used in older UK Magnox reactors and in the Chernobyl RBMK), light water (used in PWRs and BWRs), and heavy water (used in CANDU reactors).
- Control rods. Absorb neutrons to regulate the reaction rate. Materials include boron, cadmium, and hafnium. Inserting the rods reduces the reaction rate; withdrawing them increases it.
- Coolant. Removes heat from the fuel to a heat exchanger and turbine. Common coolants are water, heavy water, helium gas, and (in fast breeder reactors) liquid sodium.
- Shielding. Thick concrete and steel surrounding the core to absorb neutrons and gamma radiation.
The critical mass is the minimum mass of fissile material needed to sustain a chain reaction. Below the critical mass, too many neutrons escape; above it, the reaction grows exponentially unless controlled.
Examiners commonly ask candidates to explain how the moderator and control rods work together to maintain a steady reaction rate. The full answer requires identifying both their roles and how they interact, and should also mention the role of the coolant in transferring energy out of the core.
Required Practical 7: The Inverse-Square Law for Gamma
Required Practical 7 asks you to investigate how the count rate of gamma radiation from a sealed source varies with distance from the detector — verifying the inverse-square law:
C ∝ 1 / d²
The procedure is:
- Measure and subtract the background count rate with no source present.
- Place the gamma source at successive measured distances d from a Geiger-Muller tube.
- Record the count rate at each distance, for a suitable counting time (typically 30 s or 60 s), and subtract background.
- Plot the corrected count rate C against 1/d². The graph should be a straight line through the origin.
Several points reward marks:
- Background count rate must be measured at the start and subtracted from every reading — it is non-negligible and varies from day to day.
- Distance d should be measured from the source to the front face of the GM tube; the effective zero of d may differ slightly because of finite source and detector size.
- Safety: handle the source with tongs; minimise time of exposure; store in a lead-lined container.
Examiners reward candidates who explicitly state both the safety steps and the reasoning behind them (minimise dose by maximising distance, minimising time, and using shielding).
How to Study This Topic
Nuclear physics is one of the more equation-heavy A-Level topics, but the equations are largely independent of each other and can be drilled in isolation. A good revision sequence is:
- Memorise the alpha, beta-minus, beta-plus and gamma decay rules, including the antineutrino / neutrino emissions that examiners often look for.
- Memorise the four exponential-decay relationships and the half-life formula t_½ = ln 2 / λ.
- Memorise R = R₀A^(1/3) and the cube-root reasoning for constant nuclear density.
- Memorise the binding-energy-per-nucleon curve, including the iron peak at about A = 56 and BE/A ≈ 8.8 MeV.
- Practise Required Practical 7 with a fake dataset, including background subtraction and a plot of C against 1/d².
- Drill fission-and-fusion mass-defect calculations until u → MeV conversions feel automatic.
Related LearningBro Courses
LearningBro's AQA A-Level Physics: Nuclear Physics course covers all of the above in eight lessons:
- Atomic Structure and Rutherford Scattering — the Geiger-Marsden experiment and its conclusions.
- Radioactive Decay Types — α, β⁻, β⁺ and γ, ranges and properties.
- Radioactive Decay Equations — half-life, decay constant, exponential laws.
- Nuclear Binding Energy — mass defect, ΔE = Δmc², BE / A curve.
- Nuclear Fission and Fusion — energy release calculations, comparison of the two processes.
- Nuclear Radius and Density — R = R₀A^(1/3) and the constant-density picture.
- Induced Fission and Reactor Physics — moderator, control rods, coolant, critical mass.
- Required Practical 7: Inverse-Square Law — gamma count rate vs distance, background subtraction.
Common Exam Pitfalls
Nuclear-physics questions reward systematic working and disciplined unit handling. The following errors recur every series.
- Decay constant versus half-life. lambda = ln 2 / t_(1/2) and N = N_0 exp(-lambda t). Plugging the half-life into the exponential in place of lambda is a textbook trap. Always rewrite the equation in lambda before substituting.
- Activity, count rate and number of undecayed nuclei. Activity A = lambda N is in becquerels (decays per second). Measured count rate C is less than A because the detector subtends only a small solid angle. Background must be subtracted from the measured count rate before any comparison with theory; even a small fraction of background corrupts the inverse-square-law plot.
- Atomic mass unit confusion. 1 u = 1.66 x 10^-27 kg = 931.5 MeV / c^2. Binding-energy questions are tidiest if you keep masses in u, compute Δm in u, and convert at the end via Δm x 931.5 to get the binding energy in MeV. Mixing kilograms and atomic mass units mid-calculation is a frequent source of factor-of-1000 errors.
- Binding-energy-per-nucleon graph misread. The peak sits near iron-56 at about 8.8 MeV per nucleon. Fission of heavy nuclei (uranium-235, plutonium-239) releases energy because the products lie nearer the peak; fusion of light nuclei (hydrogen isotopes) releases energy because the products also lie nearer the peak. A common error is to claim that fission releases more energy per nucleon than fusion — it is fusion that releases more, but fission releases more per event because the parent has many more nucleons.
- Mass numbers in decay equations. Alpha decay reduces A by 4 and Z by 2; beta-minus increases Z by 1 with A unchanged; beta-plus decreases Z by 1 with A unchanged; gamma changes neither. A six-mark "complete this decay chain" question requires every intermediate Z and A to balance.
- Nuclear radius formula. R = R_0 A^(1/3) with R_0 ~ 1.2 fm. This makes nuclear density constant — a key result. A question asking for the density of a nucleus has the same answer (~2.3 x 10^17 kg m^-3) regardless of which nucleus you choose.
- Half-life from a graph. Read off any pair of times t_1 and t_2 such that the activity at t_2 is exactly half the activity at t_1; then t_(1/2) = t_2 - t_1. Reading from the initial value to half of the initial value is one valid choice but not the only one — and on a log-linear plot it is much easier to take any horizontal step that corresponds to a factor-of-two drop.
Worked Example: Closest Approach in Rutherford Scattering
An alpha particle of kinetic energy 5.0 MeV is fired head-on at a gold-197 nucleus (Z = 79). Find the closest distance of approach, assuming the nucleus is stationary.
At the closest point, the alpha particle has zero kinetic energy and all of its initial KE has become electric potential energy. Equating the two:
E_k = (1 / (4 pi epsilon_0)) (Z_alpha Z_Au e^2) / r_min
Rearranging for r_min and substituting E_k = 5.0 x 10^6 x 1.6 x 10^-19 = 8.0 x 10^-13 J, Z_alpha = 2, Z_Au = 79, e = 1.6 x 10^-19 C, and 1 / (4 pi epsilon_0) = 8.99 x 10^9 N m^2 C^-2:
r_min = (8.99 x 10^9 x 2 x 79 x (1.6 x 10^-19)^2) / (8.0 x 10^-13) ~ 4.5 x 10^-14 m = 45 fm.
The radius of a gold nucleus from R = R_0 A^(1/3) with R_0 = 1.2 fm and A = 197 is about 7.0 fm. The closest approach is therefore comfortably outside the nucleus, which means the alpha particle never quite "touches" the nucleus at 5 MeV — and this is exactly why the inverse-square Coulomb picture holds and gives the Rutherford differential cross-section. Higher-energy alpha particles (above about 30 MeV) penetrate inside the nuclear radius and deviate from the Coulomb prediction, which was the experimental signature of the strong nuclear force.