PHYC10004 Chap.13 Nuclear Physics: Binding Energy, Decay and Half-Life
Nuclear Physics: Binding Energy, Decay and Half-Life
The final weeks of PHYC10004 move inside the atom's core. Rutherford's alpha-scattering experiment revealed a tiny, dense, positively charged nucleus, and Chadwick's neutron completed the picture of nuclei built from protons and neutrons, with radius 1.2 fm times A to the one-third.
A short-range strong force binds all nucleons together against the Coulomb repulsion between protons.
Because a bound nucleus is lighter than its separate parts, its binding energy can be found from masses at 931.49 MeV per atomic mass unit, and the same bookkeeping gives the energy released in decays, fission and fusion.
Unstable nuclei decay by alpha, beta-minus or beta-plus emission towards the stability line, often followed by gamma rays. Decay is statistical: the number of nuclei falls as N₀e^(−λt), with half-life ln 2/λ and activity λN in becquerels, which underpins radiocarbon dating with carbon-14's half-life of 5730 years. The chapter ends with fission chain reactions and deuterium-tritium fusion.
What this chapter covers
- 01
Rutherford Scattering
- 02
Nuclear Size and Composition
- 03
The Strong Nuclear Force
- 04
Binding Energy and Mass Defect
- 05
Alpha, Beta and Gamma Decay
- 06
Q-Values of Reactions
- 07
The Decay Law and Half-Life
- 08
Activity and Counting
- 09
Radiocarbon Dating
- 10
Fission and Fusion
Worked example · free
How much of a short-lived sample remains after a day
- 1One day is 24 h, which is 24/6.0 = 4 half-lives.
- 1Remaining mass: 100 mg × (½)⁴ = 6.25 mg.
- 1To reach 1.0 mg: (½)^(t/6.0) = 1/100, so t = 6.0 × ln 100/ln 2.
- 1Evaluating: t = 6.0 × 4.61/0.693 = 40 h, between six and seven half-lives.
Key terms
- Mass Number
- The mass number A is the total number of protons and neutrons in a nucleus, A = Z + N.
- Binding Energy
- Binding energy is the energy needed to separate a nucleus into its individual protons and neutrons, equal to its mass defect times c².
- Mass Defect
- The mass defect is the amount by which a nucleus is lighter than the sum of its separate protons and neutrons.
- Decay Constant
- The decay constant λ is the probability per unit time that a given nucleus decays, related to the half-life by λ = ln 2/t½.
- Activity
- Activity is the number of decays per second in a radioactive sample, R = λN, measured in becquerels.
- Q-Value
- The Q-value of a nuclear reaction is the energy released, equal to the decrease in total mass times c².
- Chain Reaction
- A chain reaction is a self-sustaining sequence of fissions in which, on average, at least one neutron from each fission causes another.
Nuclear Physics: Binding Energy, Decay and Half-Life FAQ
Why does a nucleus weigh less than its separate parts?
Assembling a nucleus releases energy as the strong force binds the nucleons together. By mass-energy equivalence that released energy corresponds to a loss of mass, so the bound nucleus is lighter, and the missing mass times c² is its binding energy.
How do I decide whether a nucleus decays by beta-minus or beta-plus?
Compare its neutron-proton balance with the stability line. A nucleus with too many neutrons converts a neutron into a proton by beta-minus decay, raising Z by one; one with too many protons converts a proton into a neutron by beta-plus decay, lowering Z by one.
Can we predict when a particular nucleus will decay?
No. Decay is statistical: each nucleus has a fixed probability per unit time of decaying, but no way exists to say when a given one will go. Only for large numbers of nuclei does the exponential decay law describe the sample precisely.
Why is carbon-14 used to date archaeological remains?
Living things keep the same carbon-14 to carbon-12 ratio as the atmosphere until they die, after which the carbon-14 decays with a half-life of 5730 years. That half-life is comparable with archaeological ages, so a measurable but countable fraction remains.
Exam move
Practise nuclear questions as balanced bookkeeping. Write every decay equation with A and Z balanced and the correct neutrino, then compute binding energies and Q-values from masses, converting with 931.49 MeV per atomic mass unit and watching whether masses are atomic or nuclear.
Work half-life problems both in whole half-lives and with logarithms, and activity problems from mass to number of nuclei to becquerels, converting λ to inverse seconds. Finish by explaining fission chain reactions, fusion and the choice of a dating isotope in a few sentences each.
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