BetheWeizsäcker

Bethe–Weizsäcker refers to the semi-empirical mass formula (SEMF) for atomic nuclei, developed in 1935 by Hans Bethe and Carl Friedrich von Weizsäcker. It provides a simple macroscopic estimate of the binding energy and thus the mass of a nucleus with mass number A and atomic number Z, based on a liquid-drop analogy and fitted to experimental masses. The formula captures general trends in nuclear binding and explains why certain isotopes are more stable than others.

The commonly used form for the binding energy B is B(A,Z) = a_v A - a_s A^{2/3} - a_c Z(Z-1) A^{-1/3} - a_a (N - Z)^2 / A + δ(A,Z). Here the terms are the volume term (a_v A) reflecting bulk nucleon attraction, the surface term (−a_s A^{2/3}) correcting for nucleons on the surface, the Coulomb term (−a_c Z(Z-1) A^{-1/3}) accounting for proton repulsion, the asymmetry term (−a_a (N−Z)^2 / A) penalizing neutron-proton imbalance, and the pairing term δ(A,Z) that depends on the parity of N and Z to reflect even-odd effects. The constants a_v, a_s, a_c, a_a are fitted to measured masses and typical values are on the order of tens of MeV (for example a_v ≈ 15–16 MeV, a_s ≈ 17–18 MeV, a_c ≈ 0.6–0.8 MeV, a_a ≈ 23–24 MeV, with δ ∝ A^{-1/2} for even-even versus odd-A cases), though exact numbers vary by fit.

Applications include explaining the valley of stability, estimating nuclear masses and decay energetics, and guiding nucleosynthesis studies. Limitations arise from neglecting shell effects, deformation, and microscopic interactions; the formula is most reliable for medium to heavy nuclei and serves as a baseline for more sophisticated models.