Fission and fusion from the B/A curve

A strong-interaction rearrangement moves nucleons; it never creates or destroys them. Z and N are separately conserved in fission and fusion, so every term depending only on nucleon count cancels and the reaction energy collapses to Q = ΣB(final) − ΣB(initial). Two bookkeeping rules make that reliable. First, a free nucleon is unbound: B = 0. The neutron entering a U-235 fission and the three leaving it contribute nothing to either sum, so they may be dropped from both without changing Q. Second, watch the sign. Binding energy is a positive number that enters the mass negatively, M = Zm(¹H) + Nmₙ − B/c², so the mass ledger runs the other way: Q = [ΣM(initial) − ΣM(final)]c². Products that are more tightly bound are products that are less massive. Reverse one of those two conventions and you have built a refrigerator instead of a reactor.

The vertical axis is B/A, not B, so a height read off the curve must be multiplied by a mass number before it is an energy. Because the products' mass numbers sum to the parent's, the subtraction rearranges into Q = Σ_f A_f[(B/A)_f − (B/A)ₚₐᵣₑₙₜ]: each product's mass times its height above the parent's level. Take U-236 at 7.586 MeV per nucleon splitting into Ba-144 at 8.265 and Kr-90 at 8.591, plus two free neutrons. The two bound fragments give 144(0.679) + 90(1.005) = 97.78 + 90.45 = 188.23 MeV. Now charge the neutrons: each leaves the parent's 7.586 MeV per nucleon and lands at B = 0, a height of −7.586, costing 2(7.586) = 15.17 MeV. Q = 188.23 − 15.17 = 173.1 MeV. The identity only closes when every outgoing nucleon is counted, bound or not.

Run n + U-235 → Ba-141 + Kr-92 + 3n through mass excesses, which is the same subtraction with the A u terms already cancelled: Q = (8.071 + 40.921) − (−79.733 − 68.785 + 3 × 8.071) = 48.992 + 124.305 = 173.30 MeV. That is the prompt release for this channel — fragment kinetic energy plus the excitation that comes off as prompt neutrons and gammas. But Ba-141 and Kr-92 sit at Z/A ≈ 0.40, well left of stability, because fission preserves the parent's neutron excess. Repeat the subtraction with the stable end-points Pr-141 and Zr-92 and you get 199.25 MeV. The extra 26 MeV is the two β-decay chains, of which roughly 9 MeV leaves as antineutrinos and heats nothing. That is how 173 becomes the ~200 MeV per fission you see quoted, and why a reactor collects only about 194.

Fusion is the same subtraction taken from the light side. For ²H + ³H → ⁴He + n, the neutron is unbound, so Q = 28.2957 − (2.2246 + 8.4818) = 17.589 MeV. Per nucleon that is 17.589/5 = 3.518 MeV, nearly five times the 0.734 MeV per nucleon of a U-235 fission — the light end of the curve is far steeper than the heavy end is. Per event it is the reverse, and that is what sets the engineering. A 1 GW thermal core needs 3.2 × 10¹⁹ fissions per second at 194 MeV each; the same power from D–T needs 3.5 × 10²⁰ reactions, eleven times as many. With negligible initial momentum the two products share Q inversely with mass, Eₙ ≈ Q m_α/(m_α + mₙ) = 14.05 MeV; exact relativistic kinematics trim that to 14.03 MeV, leaving 3.56 MeV to the alpha. Only that charged 20% stays in the plasma, which is the whole ignition problem in one number.

The maximum of B/A is Ni-62 at 8.7945 MeV, with Fe-58 at 8.7922 and Fe-56 at 8.7903 — a spread of 4 keV per nucleon, so 'iron peak' is a region, not a point, and which nuclide wins depends on the quantity you rank. Rank instead by atomic mass per nucleon and Fe-56 jumps from last to first: Fe-56 gives 0.9988382 u, Ni-62 gives 0.9988443 u, Fe-58 gives 0.9988496 u. Fe-56 is the least massive per nucleon even though it is not the most bound per nucleon, because converting a neutron into a proton plus an electron is worth mₙ − m(¹H) = 0.782 MeV, and Fe-56 carries a higher proton fraction (Z/A = 0.464) than Ni-62 (0.452). Silicon burning reaches nuclear statistical equilibrium, which minimises mass per nucleon at the core's electron fraction — so the ash sits in the Fe-56/Ni-56 region, not on the B/A champion.

Symmetric splitting turns exothermic once Z²/A passes (2¹⁄³ − 1)/(1 − 2^(−2/3)) × a_S/a_C = 17.6, which on the stability line is A ≈ 90. Zr-90 sits at Z²/A = 17.8 and is perfectly stable, because at that fissility the liquid-drop barrier is tens of MeV. The barrier only disappears when the surface and Coulomb responses to quadrupole deformation cancel, at Z²/A = 2a_S/a_C ≈ 50.1. U-236 sits between the two at 35.9, fissility x = 0.716, with a barrier near 5.8 MeV. Now the reactor physics falls out: capturing a neutron on U-235 releases Sₙ = 6.55 MeV, which clears that barrier, so thermal neutrons fission it; capturing on U-238 releases only 4.81 MeV against a U-239 barrier near 6.3 MeV, so U-238 needs fast neutrons. Everything past this point — WKB penetration, cross-sections, half-lives — belongs to the next unit.