Nuclear Decay & Binding Energy (IPhO Modern)

Nuclear olympiad questions usually test two things: (1) energy bookkeeping via mass differences (binding energy and Q-values), and (2) exponential decay law fluency (half-life, activity, and simple chains). The fastest solutions come from choosing a consistent mass convention and writing Q-values cleanly.

1. Definitions (Must Know)

A. Binding energy and mass defect

For a nucleus with Z protons and N neutrons: B = (Zmₚ + Nmₙ - m_nucleus)c²

The binding energy per nucleon is B/(A) where A = Z + N.

B. Q-value (energy released)

For a reaction or decay: Q = (mᵢₙᵢₜᵢₐₗ - m_final)c²

• Q > 0: energetically allowed and releases energy.
• Q < 0: requires input energy.

C. Decay law and activity

Number of undecayed nuclei: N(t) = N₀e^(-λ t)

Activity: A(t) = -dN/dt = λ N(t)

Half-life: T_(1/2) = (ln 2)/λ

D. Recoil sharing (two-body decay at rest)

If a nucleus at rest decays into two products, they have equal and opposite momentum. For heavy nuclei and MeV-scale Q-values, the recoil is usually non-relativistic and the lighter product gets most of the kinetic energy.

2. Key Ideas (What Earns Marks)

• Use mass differences ruthlessly. Q-value questions are bookkeeping, not deep nuclear structure.
• State your mass convention. Using atomic masses is often simplest because electrons cancel in many Q-values.
• Expect recoil to be small for alpha decay. Most kinetic energy goes to the alpha particle.
• Half-life fluency is free marks. Every decay question wants N(t), A(t), or T_(1/2).
• Binding energy curve intuition. Fusion and fission release energy when products move toward higher binding energy per nucleon.

3. Detailed Explanations

A. Binding energy curve and why energy can be released

Light nuclei gain binding energy per nucleon by fusing (up to iron-region nuclei). Very heavy nuclei can release energy by fissioning into mid-mass fragments. The sign comes from comparing total rest mass before and after.

B. Q-values using atomic masses (practical exam trick)

Using atomic masses (including electrons) can simplify Q-values:

• Alpha decay: electrons cancel automatically if you use atomic masses for parent and daughter atoms and the helium atom mass. Q_α = (Mₚₐᵣₑₙₜ - M_daughter - M_α)c²

• Beta minus decay (β⁻): using atomic masses, Q_(β⁻) = (Mₚₐᵣₑₙₜ - M_daughter)c² because the extra electron is accounted for in the atomic mass difference.

• Beta plus decay (β⁺): you must pay for creating a positron and effectively removing an electron from the atomic bookkeeping, giving an extra 2mₑc² penalty: Q_(β⁺) = (Mₚₐᵣₑₙₜ - M_daughter - 2mₑ)c²

C. Activity and half-life in one line

Two common forms: N(t) = N₀(1/2)^(t/T_(1/2)) A(t) = A₀(1/2)^(t/T_(1/2))

D. Why beta spectra are continuous

In β decay, the decay energy is shared between the electron (or positron), the neutrino, and the recoil. This naturally produces a continuous electron energy spectrum, unlike alpha decay where a two-body final state gives almost a single kinetic energy.

4. Common Mistakes

• Mixing nuclear masses and atomic masses mid-solution.
• Getting the sign of Q wrong (write Q = (mᵢ-m_f)c² every time).
• Forgetting the 2mₑ term in β⁺ decay when using atomic masses.
• Treating half-life as linear decay (it is exponential).
• Using A = λ N with N in moles without converting to number of nuclei.

5. Exam Tips

• If the question gives masses in u, do the u-difference first, then multiply by 931.5 MeV.
• For alpha decay at rest, the alpha gets almost all the kinetic energy: K_α ≈ Q M_daughter/(M_daughter + M_α)
• Convert between half-life and decay constant early: λ = (ln 2)/(T_(1/2))
• Use A(t) = A₀(1/2)^(t/T_(1/2)) if the question is only about ratios.

6. Worked Examples

A. Alpha decay Q-value and alpha kinetic energy (with recoil)

A nucleus X (initially at rest) undergoes alpha decay: X → Y + α Given atomic masses:

• M_X = 226.0254 u
• M_Y = 222.0176 u
• M_α = 4.0026 u

Find (1) the Q-value, and (2) the approximate kinetic energy of the alpha particle.

B. Activity after several half-lives

A radioactive sample has initial activity A₀ = 5.0 MBq and half-life T_(1/2) = 8.0 days. Find:

1. the activity after t = 24 days,
2. the decay constant λ in s⁻¹.

7. Mind Stretchers

A. Continuous beta spectrum (where does the “missing” energy go?)

In β⁻ decay, why is the emitted electron not monoenergetic, unlike alpha decay? Give a conservation-law explanation.

• The most stable nuclei have the largest binding energy per nucleon, not the largest total binding energy. Why?
• Some decays with Q > 0 are still extremely slow. What role does quantum tunneling play in alpha decay rates?

8. Practice