Nuclear Physics: Binding Energy & SEMF
Why iron is the most stable nucleus — and where fission and fusion get their energy.
Nuclei consist of $Z$ protons and $N$ neutrons; mass number $A = Z + N$. The binding energy $B(Z,A)$ is the energy required to dissociate the nucleus into free nucleons. The semi-empirical mass formula (von Weizsäcker):
$$B(Z,A) = a_V A - a_S A^{2/3} - a_C \frac{Z^2}{A^{1/3}} - a_A \frac{(A - 2Z)^2}{A} + \delta(Z, A),$$with volume ($a_V \approx 15.8$ MeV), surface, Coulomb, asymmetry, and pairing terms.
$B/A$ peaks near $^{56}\text{Fe}$ at about $8.79$ MeV. Lighter nuclei release energy by fusing (sun's energy from $4p \to {}^4\text{He}$), heavier ones by fissioning ($^{235}\text{U}$). Nuclear forces are short-range (~ fm) and ~ 1000× stronger per nucleon than electromagnetic.