Fission energy, decay heat and waste inventory

Per thermal fission of U-235 the ledger reads 169.1 MeV of fragment kinetic energy, 4.8 MeV of prompt neutrons, 7.0 MeV of prompt gammas, 6.5 MeV of fission-product betas, 6.3 MeV of delayed gammas and 8.8 MeV of antineutrinos: 202.5 MeV in all. The split matters more than the sum, because the channels deposit in different places on different clocks. The fragments stop within a few micrometres, so essentially all 169 MeV appears inside the pellet where the fission happened — that is what sets the fuel centreline temperature. Neutrons and prompt gammas travel centimetres and heat moderator and structure. The 12.8 MeV of beta plus delayed gamma is not prompt at all: fission products emit it over seconds to years. While the core runs, that inventory sits in equilibrium, so the delayed 12.8 MeV is delivered at the same rate it is created and you cannot tell it apart — until you scram.

Fragments are born far on the neutron-rich side of stability, so each fission chain runs through about six β⁻ decays, and every one emits an antineutrino of mean energy near 1.5 MeV. With an interaction cross-section around 10⁻⁴³ cm², the mean free path in water is of order a hundred light-years, so all 8.8 MeV leaves the site: recoverable energy is 193.7 MeV, not 202.5. In an operating core roughly 8–9 MeV per fission comes back as capture gammas from (n, γ) on U-238, structure and control material, which is why 200 MeV per fission is the working number rather than 193. The escaping flux is not wasted information: a 1 GWₜₕ core emits about 2 × 10²⁰ antineutrinos per second, detected by inverse beta decay at KamLAND and Daya Bay, and because Pu-239 yields fewer antineutrinos above the 1.8 MeV threshold than U-235, the counting rate drifts down as the fissile vector shifts — a safeguards signal no operator can screen.

Write one balance equation per nuclide: dNᵢ/dt = −(λᵢ + σₐ, i φ) Nᵢ + Σⱼ (branch or yield) × (λⱼ or σⱼ φ) Nⱼ. Stack them and it is dN/dt = A N, with A the burnup matrix — a few thousand square for a real library. Hold the flux φ constant over a step and the solution is exactly N(t) = e^{At} N(0). Two facts about A govern the numerics. It is not Hermitian and generally not normal, so there is no orthonormal eigenbasis to hide behind; the object you want is the exponential itself, not the spectrum. And its eigenvalues, essentially −(λᵢ + σₐ, i φ), run from about 5 × 10⁻¹⁸ s⁻¹ for U-238 up past 10⁶ s⁻¹ for the shortest-lived precursors. A step short enough for the fastest nuclide would need of order 10¹⁴ steps to cross an 18-month cycle, so explicit integration is out. Production codes evaluate e^{At} directly, usually by CRAM, a rational approximation tuned to the negative real axis, which reaches roughly 10⁻¹⁴ relative accuracy at order 16.

Fire one fission at t′ = 0 and its β-plus-γ afterglow falls empirically as f(t) ≈ 2.66 t⁻¹·² MeV s⁻¹, with t in seconds, from about 10 s to 100 days. That power law is not fundamental; it is what a few hundred exponentials look like when the fission-product half-lives are spread roughly log-uniformly from milliseconds to decades. Now run the core at constant fission rate R for a time T and stop it. The decay power is the superposition of every burst since startup: P(t) = R ∫₀ᵀ 2.66 (t + t′)⁻¹·² dt′ = 13.3 R [t⁻⁰·² − (t + T)⁻⁰·²]. Integrating a power law raised the exponent by one. Divide by P₀ = 200 R MeV s⁻¹ and the coefficient is 13.3/200 ≈ 0.066 — the 12.8 MeV delayed share of the ledger reappearing to within 4%, since at equilibrium the delayed energy is emitted as fast as it is made. For t ≪ T the second term falls away, though only as (t/T)⁰·² relative to the first, and the core cools as t⁻⁰·²; for t ≫ T the bracket collapses to 0.2 T t⁻¹·² and the burst law returns.

At 48 GW⋅d per tonne of heavy metal, about 5% of the heavy-metal atoms have fissioned — nineteen in twenty leave the core intact. By mass a discharged PWR assembly is roughly 93% U-238, 0.8% U-235, 0.5% U-236, 1.2% plutonium, 0.1% minor actinides (Np-237, Am-241, Am-243, Cm-244) and 4–5% fission products. Which of those carries the heat depends entirely on when you ask. For the first century it is Cs-137 with its Ba-137m daughter (30.1 y) and Sr-90 with Y-90 (28.8 y); those two pairs set cask loading and repository drift spacing. Then the picture changes owner. Am-241 is not a fission product at all: it is the β⁻ daughter of Pu-241, whose 14.3-year half-life feeds it far faster than its own 433-year half-life removes it, so the Am-241 inventory grows after discharge and peaks near 73 years out. Design a repository on the discharge inventory and you have designed it for the wrong isotopes.

The usual figure of merit is radiotoxicity — activity weighted by ingestion dose coefficient — compared against the uranium ore the fuel came from. Fission products dominate only for the first several decades, and by about 300 years they have died with Cs-137 and Sr-90 to below the ore level; after that the curve is entirely actinide, first Am-241, then Pu-239 and Pu-240, then Np-237, and it does not fall back to the ore level for some 200 000 years. Strip out the plutonium and minor actinides and the crossover drops to centuries, which is the whole case for partitioning and transmutation. But the dose most repository performance assessments actually predict at the surface comes from none of these. It comes from long-lived fission and activation products that form anions and barely sorb on rock: Tc-99 as pertechnetate (2.1 × 10⁵ y), I-129 (1.6 × 10⁷ y), Se-79, and Cl-36 activated from chlorine impurity. They carry a negligible share of the radiotoxicity and move with the groundwater. Different nuclides set the two different constraints.