Nuclear Physics

This guide covers Topic 23, Nuclear physics, in full — subtopics 23.1 Mass defect and nuclear binding energy and 23.2 Radioactive decay — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content, extending the atomic and nuclear structure introduced at AS Level.

Before studying this

This resource assumes the nuclear atom, isotopes and radioactive decay from Particle Physics: Atoms, Nuclei and Fundamental Particles, and photon energy from Quantum Physics.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 23

23.1 Mass defect and nuclear binding energy — understanding the equivalence between energy and mass as represented by E = mc², and recalling and using this equation; understanding the concept of nuclear binding energy, and its relation to mass defect; using data to calculate the mass defect and the binding energy for a nucleus; sketching and using the graph of binding energy per nucleon against nucleon number; explaining the relevance of binding energy per nucleon to nuclear fusion and to nuclear fission; representing simple nuclear reactions, including fission and fusion, by nuclear equations, applying conservation of nucleon number and proton number; calculating the energy released in a nuclear reaction from the change in mass using ΔE = Δmc².

23.2 Radioactive decay — understanding that fluctuations in count rate provide evidence for the random and spontaneous nature of radioactive decay; representing alpha- and beta-decay by radioactive decay equations, using nuclide notation and conserving nucleon number and proton number; understanding and using the term activity and its unit, the becquerel; recalling and using A = λN; understanding that the decay constant λ is the probability of decay of a nucleus per unit time; recalling and using x = x₀e^(−λt), where x could represent activity, number of undecayed nuclei or received count rate; recalling and using the relation between decay constant and half-life, and applying this relation to solve problems.

Mass-energy equivalence and mass defect

Einstein’s mass-energy equivalence states that mass and energy are interchangeable, related by:

E = mc²

The mass of a nucleus is always slightly less than the sum of the masses of its individual, separated protons and neutrons — this difference is the mass defect. Using E = mc² on this mass defect gives the nuclear binding energy: the energy that would be needed to separate the nucleus completely into its individual nucleons (or, equivalently, the energy released when the nucleons come together to form the nucleus).

Worked example. A nucleus has a mass defect of 3.0 × 10⁻²⁸ kg. Its binding energy:

E = mc² = 3.0 × 10⁻²⁸ × (3.0 × 10⁸)² = 2.7 × 10⁻¹¹ J

Binding energy per nucleon

Plotting binding energy per nucleon (binding energy divided by nucleon number, A) against nucleon number produces a characteristic curve that rises steeply for light nuclei, peaks around iron (A ≈ 56), then falls slowly for heavier nuclei. Since a higher binding energy per nucleon means a more stable nucleus, this graph explains both major nuclear energy release processes:

• Nuclear fusion, combining light nuclei into a heavier one, moves up the steep left-hand side of the curve toward the peak, releasing energy.
• Nuclear fission, splitting a heavy nucleus into lighter ones, moves up the shallow right-hand side of the curve toward the peak, also releasing energy.

Nuclear equations

A nuclear reaction — radioactive decay, fission or fusion — is represented by a nuclear equation written with nuclide notation, ₐᶻX, where A is the nucleon (mass) number and Z is the proton (atomic) number. In any such equation, nucleon number and proton number are each conserved (the totals on each side must match), even though mass and energy are not separately conserved:

Alpha decay:   ₉₂²³⁸U  →  ₉₀²³⁴Th + ₂⁴He
Beta decay:    ₆¹⁴C   →  ₇¹⁴N  + ₋₁⁰e + antineutrino
Fission:       ₉₂²³⁵U + ₀¹n → ₅₆¹⁴¹Ba + ₃₆⁹²Kr + 3₀¹n
Fusion:        ₁²H + ₁³H → ₂⁴He + ₀¹n

In each case, check the equation balances: the nucleon numbers (top, superscripts) sum to the same total on both sides, and the proton numbers (bottom, subscripts) sum to the same total on both sides.

Energy released in a nuclear reaction

Because mass and energy are equivalent (E = mc²), a nuclear reaction that releases energy must produce products whose total mass is less than the total mass of the reactants — the “missing” mass, Δm, has been converted into the energy released:

ΔE = Δmc²

Worked example. In a fusion reaction, the total mass of the reactants exceeds the total mass of the products by 3.0 × 10⁻²⁹ kg. The energy released:

ΔE = Δmc² = 3.0 × 10⁻²⁹ × (3.0 × 10⁸)² = 2.7 × 10⁻¹² J

This is the same equation used for binding energy above — the only difference is which masses are being compared (reactants and products of a whole reaction, rather than separated nucleons and an assembled nucleus).

Radioactive decay is random and spontaneous

Radioactive decay is random (it is impossible to predict exactly when any individual nucleus will decay) and spontaneous (unaffected by external conditions such as temperature or pressure). Fluctuations observed in the count rate of a radioactive source, even when the average rate is constant, are direct experimental evidence of this randomness.

Activity, decay constant, and exponential decay

The activity A of a sample is its rate of decay, measured in becquerels (1 Bq = 1 decay per second):

A = λN

where N is the number of undecayed nuclei present, and λ is the decay constant — the probability that any individual nucleus decays in unit time. Because each nucleus has a constant probability of decay per unit time, the number of undecayed nuclei (and correspondingly the activity and measured count rate) falls exponentially:

x = x₀ e^(−λt)

where x may represent N, A, or measured count rate.

Half-life

The half-life is the time taken for the activity (or number of undecayed nuclei) to fall to half its original value, related to the decay constant by:

λ = ln 2 / t½

Worked example. A radioactive isotope has a decay constant of 0.020 s⁻¹. Its half-life:

t½ = ln 2 / λ = 0.693 / 0.020 ≈ 34.7 s

Common mistakes

• Confusing mass defect with the total mass of the nucleus — mass defect is specifically the difference between the mass of separate nucleons and the actual nucleus mass.
• Assuming binding energy per nucleon keeps increasing indefinitely with nucleon number — it peaks around iron and decreases for heavier nuclei, which is why both fusion (light nuclei) and fission (heavy nuclei) release energy.
• Treating radioactive decay as predictable for an individual nucleus — it is only statistically predictable in large numbers; any single nucleus’s decay time is genuinely random.
• Forgetting the relationship λ = ln 2/t½ or misapplying it — this is the essential link between the decay constant and the more commonly quoted half-life.
• Forgetting to balance both nucleon number and proton number when writing or checking a nuclear equation — both totals must match on each side, not just one.
• Using the mass of a whole atom instead of a bare nucleus (or vice versa) inconsistently when calculating Δm — as long as the same convention is used for every mass in the calculation, the electron masses cancel out, but mixing conventions gives a wrong answer.

Quick revision checklist

• E = mc², mass defect, and nuclear binding energy
• The binding-energy-per-nucleon curve, and why it explains both fusion and fission
• Writing balanced nuclear equations for decay, fission and fusion, conserving nucleon and proton number
• ΔE = Δmc² for the energy released in a nuclear reaction, from the mass difference between reactants and products
• A = λN, and x = x₀e^(−λt) for exponential decay
• λ = ln 2/t½ relating decay constant and half-life

Related resources

• Quantum Physics — the previous A Level topic
• Medical Physics — the next A Level topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.