Nuclei
Nuclei covers nuclear size and density, mass defect and binding energy, the binding-energy-per-nucleon curve, radioactivity and decay law, and fission and fusion. Half-life and binding-energy numericals are frequent.
Core rules
- Nucleus contains Z protons and A − Z neutrons; nuclear radius R = R₀A^(1/3), R₀ ≈ 1.2 fm, so density is nearly constant across nuclei.
- Mass defect Δm = (Z m(p) + (A − Z)m(n)) − M(nucleus); binding energy = Δm × 931.5 MeV per u.
- BE/A rises to about 8.75 MeV near iron-56 then falls, which is why both fission of heavy nuclei and fusion of light nuclei release energy.
- Nuclear force is short-range, charge-independent, strongly attractive and saturates.
- Decay law N = N₀e^(−λt); half-life T = 0.693/λ; mean life τ = 1/λ = 1.44T. Activity A = λN, measured in becquerel.
- Alpha decay reduces A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged; gamma emission changes neither.
Worked example
A helium nucleus has 2 protons and 2 neutrons. Constituent mass = 2(1.007276) + 2(1.008665) = 4.031882 u, while the helium nucleus is 4.001506 u, so Δm = 0.030376 u. Binding energy = 0.030376 × 931.5 = 28.3 MeV, giving BE/A ≈ 7.07 MeV. A sample with half-life 5 years retains (1/2)³ = 1/8 of its activity after 15 years.
Traps
- Using the atomic mass of hydrogen in place of the proton mass without accounting for electron masses.
- Confusing half-life with mean life; they differ by the factor 1.44.
- Forgetting that gamma decay alters neither A nor Z.