Quantum and nuclear physics (A Level)

23. Nuclear Physics — Cambridge AS & A Level Physics 9702 Revision Notes & Topical Questions

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📚 Cambridge AS & A Level Physics 9702 Hub / Topical Revision

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23. Nuclear Physics — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use mass–energy equivalence, mass defect and binding energy to explain fission and fusion, and describe radioactive decay using activity, the decay constant and…

Nuclide notation
Conservation
Mass defect Δm
Calculation
Binding energy per nucleon
Total binding energy

Section 1 of 9Mass–energy equivalence and nuclear equations

Equivalence
- mass and energy are interchangeable: mass can be converted into energy and energy into mass.
Equation
- E = mc², with E in joules, m in kilograms and c = 3.00 × 10⁸ m s⁻¹.
  What the symbols mean: E = energy (J) · m or Δm = mass (change) (kg) · c = 3.00 × 10⁸ m s⁻¹
Examples
- hydrogen fusing to helium in the Sun, uranium fission in power stations, nuclear weapons and high-energy particle collisions.
Nuclide notation
- ^A_ZX holds the nucleon number A and proton number Z.
Conservation
- both A and Z are conserved in a nuclear reaction, and the sums on each side are equated to find an unknown such as the number of neutrons released.

Section 2 of 9Mass defect and binding energy

Mass defect Δm = the difference between the sum of the masses of the separate protons and neutrons and the mass of the nucleus.
- Δm = Zm_p + (A − Z)m_n − m_nucleus.
- A bound system has less mass than separated nucleons, so energy is released when a nucleus forms.
- The term applies to separating all nucleons, not to the mass lost in a radioactive decay.
Binding energy = the energy required to separate a nucleus completely into its individual protons and neutrons.
- It is not energy stored in the nucleus but energy that must be put in to pull the nucleons apart.
Calculation
- E = Δmc², with 1 u = 1.661 × 10⁻²⁷ kg and 1 MeV = 1.6 × 10⁻¹³ J.
Size
- nuclear binding energies are in MeV, far greater than the few eV that bind electrons,
- so nuclear reactions release much more energy than chemical ones.

Section 3 of 9Binding energy per nucleon and the curve

Binding energy per nucleon = the binding energy of a nucleus divided by its number of nucleons.
- A higher value means a more stable nucleus, because more energy per nucleon is needed to break it apart.
Graph
- binding energy per nucleon plotted against nucleon number A.
- At low A there is a steep rise, so light nuclei have low values and can fuse, releasing a large energy per nucleon.
- The peak is at iron-56 (A = 56), the most stable nucleus.
- At high A there is a gentle fall, so the heaviest nuclei are less stable and can undergo fission, releasing less energy per nucleon than fusion.
Anomaly
- helium-4 is unusually stable and lies above the curve, and carbon-12 and oxygen-16 behave like three and four helium nuclei.
Total binding energy
- binding energy per nucleon × nucleon number.

Section 4 of 9Nuclear fusion

Fusion = two light nuclei combine into one heavier nucleus, releasing energy.
In stars
- four hydrogen nuclei fuse into a helium nucleus, providing fuel for the star to continue burning.
- The first step is ¹₁H + ¹₁H → ²₁H + e⁺ + ν_e, where a proton changes to a neutron by β⁺ decay.
On Earth
- research targets deuterium–tritium fusion: ²₁H + ³₁H → ⁴₂He + ¹₀n.
Conditions
- nuclei need very high kinetic energy to get close enough.
- Electrostatic repulsion between the protons must be overcome.
- The strong nuclear force acts only over very short distances, so the nuclei must almost touch.
- This needs extremely high temperatures, such as in a star's core.

Section 5 of 9Nuclear fission

Fission = a large nucleus such as uranium splits into smaller daughter nuclei, releasing energy.
Induced fission
- it is started by a neutron striking the nucleus, and extra neutrons are released.
Chain reaction
- released neutrons strike other nuclei, causing a cascade that lasts until the fuel is used up or a moderator halts it.
Uses
- controlled in nuclear power stations and uncontrolled in a nuclear bomb.

Section 6 of 9Why fusion and fission release energy

	Fusion (low A)	Fission (high A)
Dominant effect	Attractive nuclear forces beat proton repulsion	Proton repulsion becomes large and tends to break the nucleus
Change	Light nuclei form a more stable nucleus	An unstable nucleus forms more stable nuclei
Energy source	The product has a higher binding energy per nucleon, and the small loss of mass is released as energy	The products have a higher binding energy per nucleon, and the difference is released

Energy released
- − (total binding energy of reactants), or Δmc² from the mass difference.
  - (total binding energy of products)
Both processes move nuclei towards the peak at A = 56, where binding energy per nucleon is highest.

Section 7 of 9Random and spontaneous decay

Radioactive decay = the spontaneous disintegration of an unstable nucleus into a more stable one, with emission of an α, β or γ particle.
Spontaneous = not affected by temperature, pressure or chemical conditions.
Random = the time at which a given nucleus decays cannot be predicted, because each nucleus has the same constant probability of decay per unit time.
Evidence
- a Geiger–Müller tube near a source gives irregular, unpredictable counts, though the average over many nuclei is predictable.

Section 8 of 9Activity, decay constant and exponential decay

Decay constant λ = the probability that an individual nucleus will decay per unit time, in s⁻¹.
Activity A = the number of decays per unit time, in becquerels (1 Bq = 1 decay per second).
- A = ΔN/Δt = −λN, where N is the number of undecayed nuclei, and the minus sign can be dropped in calculations.
- A larger λ means a greater activity for the same number of nuclei, and activity depends on the undecayed nuclei remaining.
Number of nuclei in a sample
- N = × N_A.
  - (mass ÷ molar mass)
Exponential decay
- N = N₀e^−λt, falling rapidly but never reaching zero.
  What the symbols mean: N = number of nuclei left after time t · N₀ = number at the start · λ = decay constant (s⁻¹) · t = time (s)
- A = A₀e^−λt since A ∝ N, and C = C₀e^−λt for count rate.
- All decay curves start at N₀ on the y-axis, and a steeper curve means a larger λ.
Mathematics
- e ≈ 2.718, and ln is the inverse of e^x.

Section 9 of 9Half-life

Half-life t_½ = the time taken for the number of undecayed nuclei (and the activity) to fall to half.
- After two half-lives the activity is a quarter of the original, and after three an eighth.
Derivation
- with N = ½N₀ at t = t_½, ½ = e^−λt_½, so ln ½ = −λt_½.
Result
- t_½ = ln 2/λ = 0.693/λ.
  What the symbols mean: t_½ = half-life (s) · λ = decay constant (s⁻¹) · ln 2 ≈ 0.693
Relationship
- half-life and decay constant are inversely proportional, so a short half-life means a large λ and fast decay.

Last updated: 13 October 2026