Mass defect and binding energy per nucleon

Nuclear masses are measured, not calculated: Penning traps compare the cyclotron frequencies of ions, and the best reach a part in 10¹⁰. The tables list neutral atoms, electrons included, on a scale fixed by a bound system: 1 u is one-twelfth of a neutral ¹²C atom, 1.660539 × 10⁻²⁷ kg. Its rest energy follows from the constants: 1.660539 × 10⁻²⁷ kg × (2.997925 × 10⁸ m s⁻¹)² = 1.492418 × 10⁻¹⁰ J, and dividing by 1.602177 × 10⁻¹³ J per MeV gives 931.494 MeV. Because the anchor is bound, free constituents weigh more than 1 u: M(¹H) = 1.007825 u and mₙ = 1.008665 u. That excess is carbon's binding in disguise. Six hydrogen atoms and six neutrons average 1.008245 u per nucleon, against exactly 1 u inside the carbon atom, and 0.008245 u × 931.494 MeV u⁻¹ = 7.680 MeV is B/A for ¹²C. The deficits are near 1% of the mass, so carry masses to six decimal places: 10⁻⁶ u is 0.93 keV.

To pull a nucleus apart into free nucleons at rest you must supply an energy B. Energy is conserved and rest energy is energy, so the separated nucleons have a total rest energy larger than the nucleus's by exactly B: Σmᵢc² = Mc² + B. The rest energy of a composite counts the kinetic and potential energy of its parts, and for a bound state that sum is negative. The deuteron lets you test this without a mass table. Its deficit, 1.00782503 + 1.00866492 − 2.01410178 = 0.00238817 u, gives B = 2.2246 MeV. A slow neutron captured by a proton emits a 2.2233 MeV photon, short of B by the 1.3 keV recoil of the deuteron, and no photon below 2.2259 MeV can break a deuteron apart. Scale supplies the correspondence check. Nuclear binding is 1.2 × 10⁻³ of the deuteron's rest energy; the 4.48 eV bond of H₂ is 2.4 × 10⁻⁹ of the molecule's, below what any chemical balance can resolve. Conservation of mass in chemistry is the B/Mc² → 0 limit of the same rule.

A neutral atom's rest energy is M(ᴬX)c² = m_nuc c² + Z mₑ c² − Bₑₗ, where m_nuc is the bare nucleus and Bₑₗ the total binding energy of the atom's electrons. Write B with bare proton masses, Z mₚ + N mₙ − M(ᴬX), and the atom's Z electrons are left uncancelled: B comes out low by Z mₑ c² = Z × 0.511 MeV, which is 13.3 MeV for ⁵⁶Fe, 2.7% of its 492.3 MeV. Use Z hydrogen atoms instead and each brings its own electron, so Z M(¹H) + N mₙ − M(ᴬX) counts nuclear binding alone. What survives is the mismatch in atomic binding: the formula overstates B by Bₑₗ − Z × 13.6 eV, about 34 keV for iron and 0.76 MeV for uranium, at most a few parts in 10⁴. That is invisible at the precision of this unit, and precision mass work corrects for it. Converting to nuclear masses first, m_nuc ≈ M − Z mₑ, and then using mₚ works equally well. Mixing the two, stripping the electrons from atom X while keeping the hydrogen atoms, overstates B by Z × 0.511 MeV.

Divided by A, nuclei of every size share one scale. B/A climbs steeply through the lightest nuclei, 1.112 MeV for ²H and 2.573 MeV for ³He, with ⁴He leaping to 7.074 MeV, far above ⁶Li at 5.332 MeV. From ¹²C at 7.680 MeV it rises more slowly to a broad maximum of 8.8 MeV at A ≈ 56 to 62, then falls gently: 8.504 MeV at ¹²⁰Sn, 7.867 MeV at ²⁰⁸Pb, 7.570 MeV at ²³⁸U. The flatness is informative. If every nucleon attracted every other, B would grow with the number of pairs, A(A − 1)/2, and B/A would rise in proportion to A; instead it stays between 7.5 and 8.8 MeV from A = 16 to 238, so each nucleon binds only to near neighbours and the attraction saturates. The slow fall at large A is what a long-range repulsion among Z protons would produce. These are readings of data, not a model, and this survey does not derive the curve. B/A is also an average. The marginal cost of one nucleon is the separation energy, a difference of two binding energies: ⁵⁶Fe gives Sₙ = 11.20 MeV against its mean of 8.79 MeV.

When Z and N are each conserved, every Z M(¹H) + N mₙ term cancels between the two sides, so the energy released is the gain in total binding, Q = ΣB(products) − ΣB(reactants), and the rest mass falls by Q/c². ¹²C + ¹²C → ²⁴Mg releases 198.257 − 2 × 92.162 = 13.93 MeV. Energy comes out whenever the nucleons end up higher on the curve on average, which is why fusing light nuclei and splitting heavy ones both pay: a nucleus near A = 236, at 7.59 MeV per nucleon, split into halves near 8.5 MeV, frees roughly 236 × 0.9 ≈ 210 MeV. Read the curve with care. The total decides, so one product high on the curve guarantees nothing if another sits low. The curve plots the most tightly bound nucleus of each A, while real products keep the reactants' Z and N and may lie below it, so a curve reading estimates direction and size, not Q. And if weak decays can change Z, B is no longer the whole ledger: a neutron outweighs a hydrogen atom by 0.782 MeV/c², so ⁶²Ni has the largest B/A but ⁵⁶Fe the smallest mass per nucleon.

The curve says whether energy is available, not whether it is released. ²³⁸U would release roughly 200 MeV by splitting in two, yet its spontaneous-fission half-life is near 10¹⁶ years; it decays by alpha emission instead, with a half-life of 4.5 × 10⁹ years. Deuterium in seawater could fuse with a Q of 3 to 4 MeV per reaction and never does at room temperature. Rates are set by Coulomb barriers and tunnelling, which later topics in this unit treat. The smooth trend also hides structure. Nuclei with even Z and even N are more tightly bound than their odd-numbered neighbours, and extra binding at particular proton and neutron numbers lifts ⁴He, ¹⁶O and ²⁰⁸Pb above the trend. Nor does the curve say which fragments a nucleus splits into, how often each channel occurs, or how fast a product decays: those are cross sections, branching ratios and lifetimes, and they need rate theory. The curve is a measured fact, and explaining its shape needs a model of the nuclear force that this survey does not build.