Fission barriers, yields and reactor kinetics

Deform a nucleus at fixed volume into a spheroid of elongation ε. The surface term grows, a_S A²⁄³(1 + 2ε²/5); the Coulomb term relaxes, a_C Z²A^(−1/3)(1 − ε²/5). Their difference is ΔE = (ε²/5)[2a_S A²⁄³ − a_C Z²A^(−1/3)], and the bracket turns negative when Z²/A > 2a_S/a_C = 50.1. Nothing in nature reaches that, so define the fissility x = (Z²/A)/50.1. For 236U, Z²/A = 8464/236 = 35.86, so x = 0.716 and a barrier survives. But the parabola is only the opening move: the real barrier of 236U is about 6.2 MeV, a third of a percent of its 1790 MeV binding energy, and it is exquisitely sensitive to x. Spontaneous-fission partial half-lives make the point. 236U at x = 0.716 lasts 2.4 × 10¹⁶ y; 240Pu at x = 0.735 lasts 1.2 × 10¹¹ y; 252Cf at x = 0.761 lasts 86 y. Six percent in x buys fourteen orders of magnitude of lifetime, because x sits inside the exponent of a tunnelling integral.

A thermal neutron brings no useful kinetic energy — 0.025 eV against a barrier of megaelectronvolts. All it delivers is its separation energy in the compound nucleus. For 235U + n → 236U that is 6.5 MeV against an activation energy of 6.2 MeV, so the compound nucleus is born above its own barrier. For 238U + n → 239U it is 4.8 MeV against 6.6 MeV, a deficit of 1.8 MeV the neutron must bring as kinetic energy; the measured threshold sits near 1 MeV once tunnelling through the real double-humped barrier is allowed for. The 1.7 MeV gap between those two separation energies is almost all pairing: 235U has an odd neutron number, so the arriving neutron pairs up and 236U is even-even, collecting δ ≈ 12/√A ≈ 0.8 MeV where 239U pays it instead. That one sign flip is the whole difference between fissile, meaning fissionable by neutrons of any energy, and merely fissionable.

The liquid drop, left alone, prefers a symmetric split — that is where the Coulomb repulsion driving the fragments apart is largest. Thermal fission of 235U does the opposite. The mass-yield curve is double-humped: about 6.4% per fission near A = 95 and near A = 139, with the symmetric valley near A = 117 down at 0.01%, a peak-to-valley ratio of roughly 600. The heavy peak barely moves when the fissioning nucleus changes, staying near A = 140, while the light peak floats to take up the remainder — near A = 100 for 239Pu. The anchor is shell structure near Z = 50 and N = 82, worth several MeV at the scission point and entirely absent from the mass formula. Raise the excitation with 14 MeV neutrons and the valley fills in to within a factor of ten of the peaks: shell corrections wash out with temperature, and the drop's symmetric preference reasserts itself.

Price the split on the B/A curve. 235U sits at 7.59 MeV per nucleon and the fragments' β-stable descendants near 8.49, so repackaging some 235 nucleons releases about 210 MeV, of which 12 MeV leaves with antineutrinos and about 200 MeV is recoverable — 3.1 × 10¹⁰ fissions per second per watt. Most of it, 168 MeV, is fragment kinetic energy, and a Coulomb estimate shows where that comes from. Split 236U into 95Rb and 139Cs: E_C = Z₁Z₂(1.44 MeV fm)/d with d = 1.2(95¹⁄³ + 139¹⁄³) = 11.69 fm gives 251 MeV for touching spheres. The measured 168 MeV needs d = 17.4 fm, so at scission the fragments are already elongated and half again as far apart. Charge density is roughly conserved, so the A = 139 fragment is born with Z ≈ 139 × 92/236 ≈ 54, as 139Xe or, in this split, 139Cs with Z = 55, while the stable A = 139 nuclide is 139La with Z = 57. The fragments are born neutron-rich: they evaporate 2.42 prompt neutrons, then β-decay down the chain.

Follow one generation of neutrons. η is the number produced per neutron absorbed in fuel, ε the fast-fission bonus, p the resonance escape probability past the 238U resonances, f the thermal utilisation — the share of thermal absorptions landing in fuel — and P_FNL, P_TNL the fractions that fail to leak while fast and while thermal. Multiply them: k_eff = η f p ε P_FNL P_TNL. For a light-water core at 3.2% enrichment, η = 1.85, f = 0.71, p = 0.75, ε = 1.05 and the two non-leakage terms together 0.97 give k_eff = 1.003, a reactivity of ρ = 0.0033, or half a dollar. Notice the discipline that demands: six numbers each known to a percent or so must multiply out to within a tenth of a percent of unity. No core is designed to be exactly critical. It is built with excess reactivity for burnup and then held down by rods and soluble boron, which is what the operator actually moves.

Point kinetics reads dn/dt = [(ρ − β)/Λ]n + ΣᵢλᵢCᵢ. The prompt generation time Λ is about 10⁻⁴ s in a light-water core, so if ρ alone drove the exponent the period would be Λ/ρ — 0.1 s at ρ = 0.0010, hopeless for any mechanical control. It does not, because a fraction β = 0.0065 of the neutrons from 235U are emitted not at scission but seconds later, by precursors such as 87Br and 137I whose weighted mean life is 13.0 s, giving λ̄ = 0.0767 s⁻¹. While ρ < β the prompt chain alone is subcritical, every extra generation must wait on a precursor, and the stable period is T = (β − ρ)/(λ̄ρ) = 72 s at ρ = 0.0010 — a factor of 717 slower. Quote reactivity in dollars, ρ/β, and the cliff sits at exactly 1 $. Above it the prompt chain closes by itself and T → Λ/(ρ − β), tens of milliseconds. 239Pu has β = 0.0021, so a plutonium core keeps under a third of that margin.