Binding energy and the fission barrier

Write the binding energy as a budget. The volume term a_V A says the strong force saturates: a nucleon deep inside binds only to its neighbours, so binding grows with the nucleon count, not with the number of pairs. The surface term −a_S A²⁄³ corrects for nucleons at the skin, which are short of neighbours; it scales as area, and a sphere minimises it. The Coulomb term −a_C Z²/A¹⁄³ is the electrostatic self-energy of a uniformly charged sphere, 3Z²e²/20πε₀R with R = r₀A¹⁄³; it is the only term that does not saturate. Its coefficient is not free — (3/5)(e²/4πε₀)/r₀ = (0.6)(1.44 MeV fm)/(1.2 fm) = 0.72 MeV, against the fitted 0.711. The asymmetry term −a_A(A−2Z)²/A is not electrostatic at all but the Pauli cost of filling neutron and proton Fermi seas to different depths, and δ is the residual pairing of like nucleons in time-reversed orbits.

Divide the budget by A. The volume term is a constant 15.75 MeV; the surface penalty falls off as A^(−1/3), so light nuclei suffer most; the Coulomb penalty grows as Z²/A⁴⁄³, so heavy nuclei suffer most. The two crossing penalties give B/A a steep rise to a broad maximum and a slow decline after it. Measured, the peak is ⁶²Ni at B/A = 8.795 MeV, with ⁵⁸Fe at 8.792 and ⁵⁶Fe at 8.790 — a plateau, not a spike, and the ranking flips if you order by mass per nucleon instead, where ⁵⁶Fe wins because it carries fewer of the heavier neutrons. Both downhill sides release energy: run up the light side by fusing, the heavy side by splitting. ²³⁶U sits at B/A = 7.59 MeV and its fragments land near 8.4 MeV, not on the 8.79 plateau: at A ≈ 95 and 140 they sit on the slow decline past the peak, and they inherit the parent's neutron excess besides. That gain of about 0.8 MeV per nucleon over 236 nucleons is the ≈190 MeV scale of fission energy.

Take a volume-conserving quadrupole, R(θ) = R₀[1 + αP₂(cos θ)], with R₀ rescaled at order α² so the enclosed volume never changes. That choice matters: it freezes the volume and asymmetry terms and isolates the two that care about shape. Compute the surface area and the electrostatic self-energy of that shape and expand. A prolate spheroid has more area than the sphere of equal volume, so E_S = E_S⁰(1 + (2/5)α² + …). Its charge sits further apart, so E_C = E_C⁰(1 − (1/5)α² + …). Both series start at α², never at α: a sphere is already an extremum, the first derivative vanishes, and the second derivative decides. Adding them gives ΔE = (α²/5)(2E_S⁰ − E_C⁰). For ²³⁶U, E_S⁰ = 17.8 × 236²⁄³ = 680 MeV and E_C⁰ = 0.711 × 92²/236¹⁄³ = 974 MeV, so ΔE = (α²/5)(1360 − 974) = 77α² MeV — positive, and worth only 0.77 MeV at α = 0.1.

Divide the bracket by 2E_S⁰ and the geometry collapses. ΔE = (2/5)E_S⁰(1 − x)α² with x = E_C⁰/2E_S⁰ = (a_C/2a_S)(Z²/A). With a_C = 0.711 and a_S = 17.8 MeV that prefactor is 0.711/35.6 = 0.01997, so x ≈ (Z²/A)/50.1 and nothing else about the nucleus enters: two nuclei with the same Z²/A have the same small-amplitude stability whatever their size. ⁵⁶Fe has Z²/A = 12.1 and x = 0.24, deeply stable. ²³⁵U has 36.0 and x = 0.72; ²³⁹Pu has 37.0 and x = 0.74. The condition x = 1 needs Z²/A = 50.1, and it is approached long before it is reached: ²⁹⁴Og has Z²/A = 47.4 and x = 0.95, leaving the liquid drop a barrier of a fraction of an MeV — yet it survives about a millisecond. Shell corrections, absent from this model entirely, supply what the drop does not.

This is where the expansion stops, and the stopping point is the lesson. ΔE = (2/5)E_S⁰(1 − x)α² is a parabola through the origin: for x < 1 it rises without bound, for x > 1 it falls without bound. Neither has a stationary point away from α = 0, so the model as expanded supplies no saddle height — and therefore no WKB tunnelling exponent, no spontaneous-fission half-life, and no threshold for induced fission. For ²³⁶U it predicts 77α² MeV, which passes 6 MeV at α = 0.28 and climbs to 77 MeV at α = 1, where the polar radius is already four times the equatorial one — plainly absurd beside a measured barrier of about 6 MeV. The barrier appears only once the expansion is abandoned: full liquid-drop energies over a multi-parameter shape family that includes necking, plus Strutinsky shell corrections added back onto the smooth drop. That gives ²³⁶U a saddle near 6 MeV, splits it into two humps with a shallow second minimum between them — the fission isomers — and makes the mass yield asymmetric.

Carried to all orders and over necked shapes, the smooth drop gives an actinide one broad hump and a symmetric split; the measured barrier has two humps and the measured split is lopsided. What the drop lacks is shell structure. Strutinsky's method keeps it for the bulk energy and adds the missing piece, E(shape) = E_LD(shape) + δEₛₕₑₗₗ(shape), where δEₛₕₑₗₗ is the sum of the occupied single-particle energies in the deformed well minus the same sum over a level density smoothed across several MeV. It is negative where levels near the Fermi surface are sparse and positive where they bunch, and as the well stretches it oscillates by a few MeV, comparable to the barrier itself. In the actinides it deepens the deformed ground state and carves a second well near the top of the smooth barrier, at an axis ratio near 2:1 and roughly 2–3 MeV above the ground state. States trapped there are the fission isomers, such as the 14 ms shape isomer of ²⁴²Am, which fission quickly because only the outer hump remains to cross. Shells act again at scission: closures near doubly magic ¹³²Sn stabilise a heavy fragment of A ≈ 135–140, so the thermal-fission yield of ²³⁵U peaks near A ≈ 95 and 140 and the symmetric split near A = 118 is suppressed several-hundred-fold.