Nuclei - Quick Revision
Atomic mass and composition
- Atomic mass unit: 1 u = 1/12 mass of carbon-12 = 1.660539 x 10^-27 kg; 1 u = 931.5 MeV/c^2.
- Nucleus made of protons (mp = 1.00727 u) and neutrons (mn = 1.00866 u), together called nucleons.
- Z = number of protons (atomic number), N = number of neutrons, A = Z + N = mass number.
- Isotopes: same Z, different N. Isobars: same A. Isotones: same N.
Size of the nucleus
- R = R0 A^(1/3), with R0 = 1.2 fm. Volume is proportional to A, so nuclear density is constant (~2.3 x 10^17 kg/m^3), independent of A.
Mass-energy and binding energy
- E = mc^2 (mass-energy equivalence).
- Mass defect: delta M = [Z mp + (A - Z) mn] - M.
- Binding energy: Eb = (delta M) c^2; Eb/nucleon = Eb/A.
- BE/nucleon is nearly constant (~8 MeV) for 30 < A < 170, peaking ~8.75 MeV near A = 56; lower for very light and very heavy nuclei.
Nuclear force
- Short-range (a few fm), charge-independent, much stronger than the Coulomb force, attractive beyond ~0.8 fm and repulsive below it; saturates, giving constant BE/nucleon.
Radioactivity
- Three decays: alpha (helium nucleus emitted), beta (electron/positron), gamma (high-energy photons).
- Decay law: N = N0 e^(-lambda t); half-life T = 0.693/lambda; mean life tau = 1/lambda; activity A = lambda N (Bq).
Nuclear energy
- Energy is released when loosely bound nuclei become more tightly bound.
- Fission: a heavy nucleus (e.g. U-235) splits into two intermediate fragments, releasing ~200 MeV per fission.
- Fusion: light nuclei combine into a heavier one (energy source of stars); requires very high temperature to beat the Coulomb barrier.
- Q = (sum of initial masses - sum of final masses) c^2; Q > 0 exothermic, Q < 0 endothermic.