AME2020 SEMF Binding Energy Fit

What this research found

The five-term liquid-drop mass formula of Bethe and Weizsäcker was fitted to the binding energies of all 3,487 nuclei with mass number 16 or above in the AME2020 atomic mass evaluation, then interrogated through its residuals. Five coefficients reproduce total binding energies to a root-mean-square deviation of 0.0762 MeV per nucleon, but what is left over is not random noise: the residuals form positive ridges at the magic neutron numbers 28, 50, 82, and 126, where a smooth macroscopic model necessarily underpredicts binding. Doubly-magic lead-208 is the sharpest single case, underbound by 11.20 MeV.

• The best-fit coefficients are a volume term of 15.296 ± 0.018 MeV, a surface term of 16.357 ± 0.056 MeV, a Coulomb term of 0.6917 ± 0.0012 MeV, an asymmetry term of 22.30 ± 0.04 MeV, and a pairing term of 11.39 ± 0.87 MeV, all within accepted published ranges. The Coulomb term is pinned most tightly, to 0.17%, and pairing least, at about 8%.
• Five parameters get within a few parts per thousand of the mass surface: the fit gives a coefficient of determination of 0.99995 on total binding energy, a root-mean-square error of 3.67 MeV on totals, and 0.0762 MeV per nucleon on the per-nucleon quantity. The per-nucleon residuals are nearly unbiased, with a mean of -0.004 and a standard deviation of 0.076 MeV per nucleon.
• The two benchmark nuclei fail in opposite directions. Iron-56, near the peak of the binding curve but at no shell closure, is overpredicted by 2.54 MeV (0.52%), while doubly-magic lead-208 is underpredicted by 11.20 MeV (0.68%), at 1625.23 MeV predicted against 1636.43 MeV measured.
• For nuclei sitting exactly at each magic neutron number, the mean residual exceeds a local background of nuclei within four neutrons by 0.0344 MeV per nucleon at N = 28, 0.0226 at N = 50, 0.0149 at N = 82, and 0.0084 at N = 126. The enhancement shrinks with mass, as a shell gap of similar energy is spread over more nucleons, and N = 20 is the sole exception, falling in the noisy light-nucleus region.
• Mapping residuals across the neutron-proton plane shows underbound ridges along the magic neutron numbers, with the strongest signal at doubly-magic intersections, confirming the pattern is driven by shell closures in both nucleon species rather than by mass number alone.
• The evaluation's extrapolated masses barely move the answer. Dropping all 1,003 of them and refitting on the 2,484 measured nuclei shifts coefficients by at most about 0.5 MeV, concentrated in the surface and pairing terms, while volume, Coulomb, and asymmetry move by 0.1 to 0.2 MeV or less.

How it was done

Binding energies per nucleon were parsed from the AME2020 mass table distributed by the IAEA Atomic Mass Data Center, release dated 3 March 2021, using its documented fixed-width record format, with proton, neutron, and mass numbers checked for consistency on every record. Restricting to mass number 16 or above removed the 71 lightest nuclides and left 3,487 nuclei, of which 2,484 carry experimentally determined masses and 1,003 are evaluation extrapolations, retained but flagged. Because total binding energy is linear in the five coefficients, the fit reduces to a single unweighted linear least-squares solve with standard errors from the covariance matrix, and a second fit restricted to measured masses served as a sensitivity check. The parser was validated against rounded published binding energies for oxygen-16, iron-56, and lead-208, agreeing to within 0.001 MeV per nucleon in each case. Residuals were then plotted per nucleon against neutron number, mapped across the nuclear chart, and compared at each magic number against a local background of nuclei within four neutrons.

Data sources

• AME2020 atomic mass evaluation, IAEA Atomic Mass Data Center, release dated 3 March 2021 - 3,487 nuclei with mass number 16 or above
• Huang, Wang, Kondev, Audi and Naimi, Chinese Physics C 45:030002 (2021) - AME2020 input data and adjustment procedures
• Wang, Huang, Kondev, Audi and Naimi, Chinese Physics C 45:030003 (2021) - AME2020 tables and graphs
• von Weizsäcker, Zeitschrift für Physik 96:431 (1935) - the semi-empirical mass formula
• Goeppert Mayer, Physical Review 75:1969 (1949) and Haxel, Jensen and Suess, Physical Review 75:1766 (1949) - the nuclear magic numbers