Conservation laws in particle interactionsAQA A-Level Physics: Revision notes

In beta-minus decay a neutron changes into a proton, so a down quark changes into an up quark:

n → p + e⁻ + $\bar{\nu}_e$ (udd → uud)

In beta-plus decay a proton changes into a neutron, so an up quark changes into a down quark:

p → n + e⁺ + $\nu_e$ (uud → udd)

Quark charges are +⅔ (up) and −⅓ (down). The change in quark charge is carried away by the emitted electron or positron.

In every interaction in these questions the following are conserved:

• Charge: the total charge before equals the total after
• Baryon number: baryons +1, antibaryons −1, mesons and leptons 0
• Lepton number: leptons +1, antileptons −1. The electron family (e⁻, $\nu_e$) and the muon family (μ⁻, $\nu_\mu$) are counted separately
• Strangeness: strange quark −1, anti-strange +1. The data you need for other particles is given in the question

The energy and momentum of the whole system are also conserved.

To test whether an interaction can occur, add up each quantity on each side and compare.

Worked example: π⁻ + p → K⁰ + Λ⁰

• Charge: (−1) + (+1) = 0 before; 0 + 0 = 0 after ✓
• Baryon number: 0 + 1 = 1 before; 0 + 1 = 1 after ✓
• Strangeness: 0 + 0 = 0 before; (+1) + (−1) = 0 after ✓

All quantities balance, so the interaction is allowed. If any one quantity does not balance, the interaction cannot occur.

If one particle in an interaction is unknown, use the conservation laws to work out its properties.

Example: π⁻ → μ⁻ + ?

• Charge: −1 → −1 + ?, so the unknown is neutral
• Muon lepton number: 0 → +1 + ?, so the unknown has −1

The unknown particle is a muon antineutrino.

In beta-minus decay the electron has electron lepton number +1, so the second particle must be an electron antineutrino to keep the total at 0.

Energy is conserved in every interaction: the total energy, including rest energy and kinetic energy, is the same before and after. A decay can only happen if the products have no more rest energy than the particle that decays. A collision that creates new particles needs enough kinetic energy to supply their rest energy.

Momentum is also conserved. In beta decay the electron is emitted with a range of kinetic energies, so a third particle (the antineutrino in beta-minus decay) must share the energy and momentum.