Semi-Empirical Mass Formula

Semi-Empirical Mass Formula

The Bethe-Weizsäcker formula (1935) models nuclear binding energy as a sum of five physical terms, treating the nucleus as a liquid drop.

The Formula

$B(Z, A) = a_V A - a_S A^{2/3} - a_C \frac{Z(Z-1)}{A^{1/3}} - a_A \frac{(A-2Z)^2}{A} + \delta$

Volume term $a_V A$: Each nucleon binds to its neighbours. Energy grows with volume ($\propto A$).

Surface term $-a_S A^{2/3}$: Nucleons on the surface have fewer neighbours — a correction proportional to surface area ($\propto A^{2/3}$).

Coulomb term $-a_C Z(Z-1)/A^{1/3}$: Electrostatic repulsion between protons reduces binding.

Asymmetry term $-a_A (A-2Z)^2/A$: The Pauli exclusion principle favours equal numbers of protons and neutrons.

Pairing term $\delta$: Nucleon pairs (spin-up/down) have extra stability.

$\delta = \begin{cases} +11.2/\sqrt{A} & \text{if } Z \text{ and } N \text{ both even} \\ -11.2/\sqrt{A} & \text{if } Z \text{ and } N \text{ both odd} \\ 0 & \text{if } A \text{ odd} \end{cases}$

Empirical Coefficients

Your Task

Implement three functions. All constants must be defined inside each function.

• pairing_term(Z, A) — returns $\delta$ in MeV (+11.2/√A, −11.2/√A, or 0)
• semf_binding_energy(Z, A) — returns total binding energy $B(Z,A)$ in MeV
• semf_binding_per_nucleon(Z, A) — returns $B(Z,A)/A$ in MeV/nucleon