Nuclear physics

Learning objectives

• Relate mass defect to binding energy.
• Use decay constant, half-life and activity.
• Explain energy release through binding-energy changes.

The big idea

A bound nucleus has less mass than its separated constituents. The difference corresponds to binding energy: energy must be supplied to separate the nucleus completely.

A nucleus of a given radionuclide has a constant probability of decay per unit time, leading statistically to exponential decay. Fission or fusion can release energy when products have greater total binding energy; neither converts all initial mass into energy.

1. Mass defect measures binding

Compare a nucleus’s mass with the total masses of its separated constituent protons and neutrons. The bound nucleus has lower mass; the difference times c² is the energy needed to separate it completely. Use nuclear masses consistently, or account carefully for electrons when using atomic masses. Binding energy per nucleon helps compare nuclei, but it is not the binding energy of each nucleon in every individual reaction.

2. Fission and fusion redistribute binding energy

A reaction can release energy if its products have lower total rest energy and greater total binding energy than its starting particles. Heavy-nucleus fission and light-nucleus fusion can both meet that condition in appropriate reactions. Conservation must include emitted neutrons, photons and other products. It is the mass difference that corresponds to released energy; the entire original rest mass is not usually converted.

3. Exponential decay describes a population

For one radionuclide, the probability per unit time is represented by decay constant λ. Expected undecayed number follows N = N₀e^(−λt), while activity is λN and half-life is ln 2/λ. Activity in becquerels counts decays per second. A detector records only some emitted radiation, so background-corrected count rate is proportional to activity only when efficiency and geometry remain stable.

4. A logarithmic plot can expose a trend

If corrected activity or count rate follows a single exponential, plotting its natural logarithm against time gives gradient −λ. Subtract background before taking logarithms; otherwise a late-time offset can flatten the graph misleadingly. Individual readings fluctuate, especially with small counts. A mixture of radionuclides or changing detection conditions can depart from one straight-line trend, so do not force all decay data into a single ideal model.

A method you can reuse

1. Balance nucleon and charge numbers for the process.
2. Use consistent mass data and calculate only the mass difference.
3. Convert the mass defect to energy with c².
4. For decay, correct count rates and choose the population or activity relationship.
5. Check half-life, logarithm conventions and statistical limitations.

Equations and relationships

Investigate and explain

Analyse supplied background-corrected activity data using a logarithmic plot. Do not obtain or handle radioactive sources independently.

Use structure and decay models alongside calculations. They do not reproduce all binding-energy data, detector efficiencies or radiation hazards; practical sources require controlled specialist procedures.

Open the interactive investigation

Check your understanding