the semi-empirical mass formula
/ Bethe: BAY-tuh; Weizsäcker: VITE-zek-er /
Suppose you want to predict the binding energy of any nucleus without solving the impossible many-body quantum problem. The semi-empirical mass formula does it with a handful of physically motivated terms, each capturing one thing a nucleus 'cares about', with numerical coefficients fitted to measured masses. It is half theory, half curve-fit, and it works remarkably well across the whole chart of nuclides.
The formula writes the binding energy as B(Z,A) = a_V A - a_S A^(2/3) - a_C Z(Z-1)/A^(1/3) - a_A (A-2Z)^2/A + delta. The five pieces are: a volume term a_V A (each nucleon is bound by its neighbours, so binding grows with the number of nucleons); a surface term -a_S A^(2/3) (nucleons on the surface have fewer neighbours, a correction proportional to surface area); a Coulomb term -a_C Z(Z-1)/A^(1/3) (mutual electrostatic repulsion of the protons); an asymmetry term -a_A (A-2Z)^2/A (a quantum-mechanical penalty for departing from equal neutron and proton numbers, rooted in the Pauli principle); and a pairing term delta (nuclei prefer even numbers of like nucleons). The first two terms come straight from picturing the nucleus as a liquid drop.
With coefficients on the order of a_V approximately 15.8, a_S approximately 18.3, a_C approximately 0.71, a_A approximately 23.2 MeV, the formula reproduces the binding-energy curve, predicts the most stable charge Z for a given A (the valley of stability), and estimates the energy released in fission. Its honest limitation is that it is a smooth, collective, liquid-drop picture: it completely misses the shell structure that makes magic-number nuclei extra stable. Those bumps require the shell model on top.
For a fixed mass number A, minimizing the formula over Z gives a parabola of masses; the bottom picks out the most stable isobar. This is why, for example, along the A = 127 chain nuclei beta-decay toward iodine-127.
The mass parabola from the SEMF is the map that tells an unstable nucleus which way to decay.
It is called semi-empirical for a reason: the coefficients are fitted, not derived from first principles, and the smooth formula cannot reproduce the discrete extra stability at magic numbers. Do not use it to explain shell closures.