Mass defect and separation energies

Take a neutral ⁵⁶Fe atom, Mₐₜ = 55.9349363 u. Take 26 hydrogen ATOMS and 30 free neutrons: 26(1.00782503) + 30(1.00866492) = 26.2034508 + 30.2599476 = 56.4633984 u. The parts outweigh the whole by Δm = 0.5284621 u, and at 931.494 MeV per u that defect is B = 492.26 MeV — the work needed to disassemble the nucleus into free nucleons at rest. Atomic rather than nuclear masses are used on purpose. Each m(¹H) carries one electron, so Z of them stand on the parts side against the Z electrons already inside Mₐₜ, and the rest masses cancel exactly. What does not cancel is the difference between the atom's total electron binding Bₑ(Z) and Z times hydrogen's 13.6 eV: roughly 1 keV for carbon, about 34 keV for iron, some 0.5 MeV for lead against 1636 MeV of nuclear binding. Standard tables ignore it. Say so, rather than pretending it is zero.

Nobody types ten-digit masses. Tables list the mass excess Δ ≡ (Mₐₜ − A u)c² in keV, and because Z + N = A the A u terms cancel identically, leaving B = Z Δ_H + N Δₙ − Δ(Z, A) with Δ_H = 7.2890 MeV and Δₙ = 8.0713 MeV. For ¹⁴C, Δ = 3.0199 MeV, so B = 6(7.2890) + 8(8.0713) − 3.0199 = 43.734 + 64.570 − 3.020 = 105.284 MeV. In NumPy the table is one array and every binding energy is a dot product with the integer vector (Z, N, −1), so a whole isotope chain costs one line. The precision this preserves is worth naming: B is a small difference of large numbers — 0.53 u out of 56 u for ⁵⁶Fe — so it inherits the absolute, not the relative, uncertainty of the masses. A few tenths of a keV on Mₐₜ(⁵⁶Fe) lands as a few tenths of a keV on 492.26 MeV, better than one part in 10⁶. Every fit and every drip-line criterion later in this unit rests on that.

Divide by A and you get the only binding number comparable between different nuclides. B/A climbs steeply and unevenly through the light nuclei — ²H 1112, ⁴He 7074, ¹²C 7680 keV per nucleon — flattens across the iron–nickel region and falls slowly to 7867 keV at ²⁰⁸Pb. The maximum is ⁶²Ni at 8794.5 keV per nucleon, with ⁵⁸Fe at 8792.2 and ⁵⁶Fe at 8790.3: the top three lie within 4.2 keV per nucleon of one another, 0.05% of B/A, yet with sub-keV mass uncertainties that margin is hundreds of standard deviations. The ordering is measurement, not model. Never rank total B across different A: ⁶²Ni's 545.26 MeV beats ⁵⁶Fe's 492.26 MeV by 53 MeV solely because it has six more nucleons. And that ⁵⁶Fe still dominates the observed iron peak is a statement about nuclear statistical equilibrium and the neutron-to-proton ratio at freeze-out, not about which nuclide is most bound.

How much does it cost to pull one neutron off? Not B/A. The removal energy is the first difference Sₙ = B(Z, N) − B(Z, N−1), or Sₙ = Δ(Z, A−1) + Δₙ − Δ(Z, A) directly from the table; the proton version is Sₚ = B(Z, N) − B(Z−1, N). These are nuclear ionisation energies: the departing nucleon sits in the highest occupied level, not at the average depth. Doubly magic ¹⁶O has B/A = 7976 keV but Sₙ = 127.619 − 111.955 = 15.66 MeV and Sₚ = 127.619 − 115.492 = 12.13 MeV, both far above that average. Add one neutron and ¹⁷O's Sₙ collapses to 131.762 − 127.619 = 4.14 MeV, a factor of 3.8 down, while B/A barely moves, from 7976 to 7751 keV per nucleon. The average is smooth because it averages; the difference is where the level structure shows.

Plot Sₙ along an isotope chain and it zigzags. Carbon: 18.72 MeV at N = 6, 4.95 MeV at N = 7, 8.18 MeV at N = 8. Even N is always the expensive one, because removing a neutron from an even-N nucleus breaks a J = 0 coupled pair while removing the odd neutron from an odd-N nucleus does not. The amplitude of that zigzag is the pairing energy, and it comes off measured masses before any model is fitted — the semi-empirical mass formula's ±a_P A^(−1/2) term exists to reproduce this feature, not to explain it. The one-sided estimate ½[Sₙ(N+1) − Sₙ(N)] gives 1.62 MeV for carbon but is biased by whatever smooth slope Sₙ carries; the symmetric four-point Δ⁽⁴⁾(N) = ¼[Sₙ(N−1) + Sₙ(N+1) − 2 Sₙ(N)] at odd N cancels a linear trend exactly. When you want shell structure without the pairing noise, difference by two instead: S₂n = B(Z, N) − B(Z, N−2) = Sₙ(N) + Sₙ(N−1) removes a whole pair, runs smoothly with N, and drops sharply where N crosses 8, 20, 28, 50, 82 or 126.

Every number in this topic is a difference of two ground-state masses. That fixes what is energetically permitted and nothing else. Sₙ < 0 means the nucleus cannot hold that neutron and it leaves in about 10⁻²¹ s; Sₙ > 0 means neutron emission is closed, and the mass table stops there. Alpha decay makes the point sharply: ²³²Th has Q_α = +4.08 MeV and a half-life of 1.4 × 10¹⁰ years, ²¹²Po has Q_α = 8.95 MeV and 0.29 µs. Both are allowed by the same table, and their rates differ by 24 orders of magnitude, because a rate needs the Coulomb barrier and a tunnelling exponent that no mass difference contains. The same discipline applies in reverse: a large Sₙ does not make a nuclide long-lived against beta decay, which is governed by an entirely different mass difference and by a matrix element the masses know nothing about.