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Magic numbers, read from masses

Two extra neutrons bind strongly while they fill a shell, and weakly once they have to start the next one. The size of that drop, read off measured masses, is an empirical indicator of the shell gap, not the gap itself: a mass difference also carries pairing, deformation and other correlations, and it can even be negative (⁶He). A reduced peak can indicate a weakened closure, but pairing and deformation must be checked with spectroscopy; masses alone do not say which orbitals moved.

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Definitions

B(Z,N) = Z Δ(¹H) + N Δ(n) − Δ(Z,N) from the atomic mass excesses Δ.
S_2n(Z,N) = B(Z,N) − B(Z,N−2), S_2p(Z,N) = B(Z,N) − B(Z−2,N).
Δ_2n(Z,N) = S_2n(Z,N) − S_2n(Z,N+2): the drop in S_2n just after N. At a magic N it is large; between closures in heavy nuclei it is 0.3 to 1.1 MeV (tin, lead), and in light nuclei it is larger throughout. Δ_2p likewise along Z.
Δ⁽³⁾_n(N) = ½(−1)^N+1[B(N−1) − 2B(N) + B(N+1)], the three-point odd-even staggering. It is quoted as a pairing indicator at odd N only: there it is dominated by pairing, though it also contains smooth mass curvature, polarization and, near N = Z, neutron-proton correlations; at even N it also carries the spacing between the last filled and first empty orbital (Satula, Dobaczewski, Nazarewicz, PRL 81, 3599 (1998)), so it spikes at closed shells. The five-point difference smooths further but averages the two parities together, which is why it is not used here.

Along N = Z the Wigner energy adds extra binding, so Δ_2n peaks there too (²⁴Mg, ³²S, ⁴⁰Ca): an N = Z ridge is not by itself a shell gap.

Uncertainties: for S_n, S_p, S_2n, S_2p the AME2020 tabulated ones, which include the correlations between masses; for Δ_2n, Δ_2p, Δ⁽³⁾ the mass uncertainties added in quadrature, treating the masses as independent. # marks a value that rests on at least one mass estimated from systematics rather than measured; on the chart those columns are drawn narrow and grey.

Data snapshot: AME2020, not a live mass evaluation. Statements about missing or estimated masses refer to this snapshot. Masses at full precision, Wang et al., Chin. Phys. C 45, 030003 (2021), IAEA AMDC mass_1.mas20; spins and parities: NUBASE2020, Kondev et al., Chin. Phys. C 45, 030001 (2021). Separation energies built from measured masses agree with the AME2020 rct1/rct2 tables to 10⁻³ keV; those resting on an estimated (#) mass agree to 1.4 keV, the precision AME prints such masses to.

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