Point kinetics, inhour roots and feedback

A prompt fission neutron in a thermal core is born, slows and is absorbed in a mean generation time Λ ≈ 1.0 × 10⁻⁴ s — 2 × 10⁻⁵ s in an LWR lattice, 10⁻³ s in graphite, 4 × 10⁻⁷ s in a fast core. About 0.65% of U-235 fission neutrons instead wait inside a precursor nucleus, Br-87 and I-137 and their relatives, sorted into six groups running from λ₁ = 0.0124 s⁻¹ to λ₆ = 3.01 s⁻¹. Weight the delays by yield and the picture inverts: 1/λ̄ = Σ(βᵢ/λᵢ)/β = 13.0 s, so the mean generation time over all neutrons is ℓ̄ = Λ + Σβᵢ/λᵢ = 0.085 s, some 850 times Λ. The same ratio appears as an inventory — at steady state the core holds Σβᵢ/(Λλᵢ) ≈ 850 precursor nuclei for every prompt neutron in flight. A 0.65% minority owns the clock, and every number in this topic follows from that.

Stack the state as x = (n, C₁, …, C₆)ᵀ and the model is ẋ = Ax, with A₀₀ = (ρ − β)/Λ, A₀ᵢ = λᵢ, Aᵢ₀ = βᵢ/Λ and Aᵢᵢ = −λᵢ. At fixed ρ the matrix is constant, so x(t) = Σⱼ aⱼ vⱼ exp(ωⱼt) is exact and np.linalg.eig(A) returns the seven ωⱼ without the inhour equation ever being written down. Substituting the ansatz by hand instead gives ρ = ωΛ + Σβᵢω/(ω + λᵢ), whose poles at ω = −λᵢ force exactly one root into each interval between them, one below −λ₆ and one above −λ₁. That last one is ω₀. The spread of the roots is also the numerical warning: at ρ = 0 the root ω₀ is zero and the rest run from −0.0143 s⁻¹ to −65.4 s⁻¹, close to −β/Λ, a stiffness ratio near 4600. An explicit RK45 step is stability-limited to a few hundredths of a second, so a day-long xenon transient would need millions of steps; integrate with BDF or Radau, or exponentiate A directly.

Because Λ is 10⁻⁴ s while the precursors move on seconds, eliminate the fast variable: set ṅ ≈ 0 to get n ≈ ΛΣλᵢCᵢ/(β − ρ). A step in ρ cannot move the precursor inventory in 10⁻⁴ s, so the numerator is unchanged and the power jumps by β/(β − ρ) = 1/(1 − ρ$) on the time constant Λ/(β − ρ). At 20 cents that is ×1.25 in 19 ms; at 50 cents, ×2.00; at 90 cents, ×10. None of this is a new equilibrium. Reactivity is still positive, so ΣλᵢCᵢ keeps growing and the power tracks it on the period of the largest inhour root — 37 s at 20 cents, 5.8 s at 50 cents, 0.49 s at 90 cents. The one-group shortcut T = (β − ρ)/(λ̄ρ) + Λ/ρ gives 52 s, 13 s and 1.5 s. It is exact only as ρ → 0: written as ρ = ω₀[Λ + Σβᵢ/(λᵢ + ω₀)], the inhour equation says each group holds its neutrons for 1/(λᵢ + ω₀), never longer than the period, and the two slowest groups, which carry three quarters of the 13 s, lose their hold first. One lumped λ̄ misses that and overstates the period, threefold by 90 cents. The prompt jump is a change of level; the inhour root is the change of rate, and only returning ρ to zero stops it.

Drive ρ up toward β and ω₀ must outrun every λᵢ. Once ω ≫ λ₆ = 3.01 s⁻¹ the delayed sum saturates at β and the inhour equation collapses to ρ − β = ω₀Λ. At exactly ρ = β with Λ = 1.0 × 10⁻⁴ s the root is ω₀ = 4.6 s⁻¹, a 0.22 s period; at 1.10 $ it is 9.0 s⁻¹, T = 0.11 s, still well above (ρ − β)/Λ = 6.5 s⁻¹ because a root only three times λ₆ has not yet saturated the delayed sum. Do the same in a Pu-fuelled fast core, β_eff = 0.0035 and Λ = 4 × 10⁻⁷ s, and 1.10 $ gives (ρ − β)/Λ = 875 s⁻¹, within 1% of the inhour root, T = 1.1 ms. Downward there is no matching cliff, because every root lies above −λ₁ = −0.0124 s⁻¹: a scram worth ten dollars leaves an asymptotic period of −80.9 s and one worth a hundred −80.7 s, both pressed against the −80.6 s floor, after which decay heat takes over. That asymmetry is the whole argument for quoting reactivity in dollars. Prompt critical is 1.00 $ in every core; in pcm it is 650 in fresh U-235, about 500 in a burnt LWR, 350 in a fast oxide core and 210 in pure Pu-239.

Point kinetics is linear only while ρ is externally fixed. Close the loop with ρ(t) = ρₑₓₜ + α_f ΔT_f + α_c ΔT_c and it becomes nonlinear. The fuel term is Doppler: hotter fuel broadens the U-238 capture resonances, the Doppler width Γ_D = √(4E₀kT/A) growing from 54 meV at 300 K to 98 meV at 1000 K for the 6.67 eV resonance. Broadening conserves ∫σ dE but spoils self-shielding, so the resonance escape probability p falls, giving α_f ≈ −2.5 pcm K⁻¹ and scaling as −K/√T. It counts as prompt because it tracks fuel temperature, which tracks power within milliseconds. Coolant and void terms wait for the heat to cross the gap, seconds later, and their sign is a lattice choice: an undermoderated LWR runs near −150 pcm per percent void, while graphite-moderated water-cooled channels ran strongly positive, and a large sodium-cooled core is positive in its interior. Not one of these coefficients falls out of point kinetics — each is computed from transport or fitted to a test and inserted by hand.

The slowest feedback of all is a nuclide. I-135 (6.57 h) decays to Xe-135 (9.14 h), whose thermal absorption cross-section is 2.6 × 10⁶ b — worth about −2700 pcm at full power in an LWR. Its running level follows X∞ = (γ_I + γ_Xe)Σ_f φ/(λ_Xe + σₐ φ), so after a scram the burnout term σₐ φ vanishes while the iodine keeps feeding: xenon peaks roughly 11 h later, well above its operating value, and the core can be locked out for a day. That transient is a thermal-spectrum phenomenon only; at 100 keV the xenon cross-section is millibarns and a fast reactor never meets it. Line the discriminators up and the reactor types sort themselves: Λ (4 × 10⁻⁷ s fast, 2 × 10⁻⁵ s LWR, 10⁻³ s graphite), β_eff (0.0065 fresh U-235, drifting toward 0.005 as Pu-239 builds in, 0.0021 for pure Pu-239), the sign of the void coefficient, and whether xenon exists at all.