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Semi-Empirical Mass Formula

What you are seeing: the binding energy per nucleon B/AB/A along the valley of stability (A,Z∗(A))(A, Z^*(A)), computed from the Bethe-Weizsacker formula B=aVA−aSA2/3−aCZ(Z−1)/A1/3−aA(N−Z)2/A+δ.B = a_V A - a_S A^{2/3} - a_C Z(Z-1)/A^{1/3} - a_A (N-Z)^2/A + \delta. The peak near A≈60A \approx 60 at B/A≈8.8B/A \approx 8.8 MeV is iron-peak territory; fusion releases energy below it, fission releases above.

Hover (slide) the mass number AA slider to see the four terms individually. The Coulomb term grows like Z2/A1/3Z^2/A^{1/3} and dominates the fall-off above iron; the asymmetry term grows quadratically with N−ZN-Z (neutron excess); the pairing term oscillates by parity.

Figure 1. Bethe-Weizsacker B/A vs A.
A56

WHAT TO TRY

  • Slide the mass number A along the curve: binding energy per nucleon rises, peaks near iron-56, then falls, the reason fusion releases energy below the peak and fission above it.
  • The terms compete: the volume term wants more nucleons while the surface and Coulomb terms penalise them, and the balance carves the valley of stability.
  • Watch the kinks at magic numbers: the smooth liquid-drop formula misses the extra binding from closed nuclear shells, where the real curve jumps above the fit.