Neutron Generation Time and Prompt Neutron Lifetime — Separate ℓp, Λ and ℓ̄ and Reactor Kinetics Falls into Place

Principle and mechanism

First, separate the three times

Textbooks and papers mix the terms "generation time", "lifetime" and "mean lifetime", so it is worth stating at the start which quantity is meant. In particular, "mean generation time" can mean either Λ (prompt) or ℓ̄ (including delayed neutrons); when a precursor mean life enters the calculation, ℓ̄ is meant.

When k > 1 the neutron population grows by a factor k in one generation, so the generation interval is shorter than the lifetime (Λ = ℓ/k). The point kinetics equation dn/dt = [(ρ−β)/Λ]n + Σλ_ic_i contains Λ; putting an ℓ̄ of order 0.08 s in place of Λ counts the delayed-neutron effect twice.

What sets ℓ_p

A prompt neutron in a thermal reactor is born at about 2 MeV, slows down, then diffuses as a thermal neutron until it is absorbed. So ℓ_p ≈ t_s + t_d. The slowing-down time t_s is short, a few μs (about 7 μs in water, constant-ξΣ_s approximation); most of the lifetime is the mean diffusion time t_d spent as a thermal neutron before absorption or leakage (Lamarsh Eq. 7.2: in an infinite thermal reactor ℓ_p ≈ t_d).

• t_d = λ_a/v = 1/(Σ_av). For 1/v absorption it is independent of energy: t_d = 1/(Σ_a(E₀)v₀).
• Moderator alone (Lamarsh Table 7.1): water 2.1×10⁻⁴ s, Be 3.9×10⁻³ s, graphite 0.017 s, heavy water (with 0.25 % light water) 4.3×10⁻² s.
• With fuel mixed in, t_d = t_dM(1−f) (f: thermal utilization). An infinite critical homogeneous U-235–water reactor (f = 0.484) has ℓ_p = 1.08×10⁻⁴ s (Lamarsh Example 7.1); a commercial PWR with f ≈ 0.9 has about 2×10⁻⁵ s.
• In a finite reactor leakage shortens the lifetime: ℓ = 1/[vΣ_a(1 + L²B²)].
• A fast reactor has almost no slowing down, so ℓ_p is of order 10⁻⁷ s.

The mean free path λ = 1/Σ is a distance (cm); the lifetime is a time. They are linked by t = λ_a/v: in water λ_a = 1/0.0197 = 50.8 cm, divided by 2,200 m/s, gives 2.3×10⁻⁴ s, the same order as Table 7.1.

With prompt neutrons only: T = ℓ_p/(k_∞−1)

In an infinite homogeneous reactor with prompt neutrons only, the fission rate grows as N_F(t) = N_F(0)e^t/T. After one lifetime ℓ_p the fission rate is k_∞ times larger, so N_F(t + ℓ_p) = k_∞N_F(t); the Taylor expansion N_F + ℓ_p·dN_F/dt ≈ k_∞N_F gives dN_F/dt = [(k_∞−1)/ℓ_p]N_F, i.e. T = ℓ_p/(k_∞−1). With ℓ_p = 2×10⁻⁵ s and k_∞ = 1.001, T = 0.02 s, and after one second the power is e⁵⁰ (about 5×10²¹) times larger.

Adding the delayed path: ℓ̄ = ℓ_p + Σβ_i/λ_i

The fraction β appears only after waiting for precursor decay, so ℓ̄ = (1−β)ℓ_p + Σβ_i(ℓ_p + 1/λ_i) = ℓ_p + Σβ_i/λ_i holds exactly. With Keepin's six groups (β = 0.0065), Σβ_i/λ_i = 0.0848 s and τ̄ = 13.0 s, so ℓ̄ ≈ 0.085 s, about 4,200 times ℓ_p. Worked example: with τ̄ = 12.5 s and ℓ_p = 2.1×10⁻⁵ s (slowing-down time 10⁻⁶ s plus diffusion time 2×10⁻⁵ s), ℓ̄ = 0.0813 s. For small reactivity T ≈ ℓ̄/ρ ≈ 85 s (ρ = 0.001), a controllable rate. This is a simple approximation; the exact period comes from the inhour equation.

What Λ does in reactor kinetics

Keeping Keepin's six-group β_i and λ_i (β ≈ 0.0065) and changing only Λ to 10⁻⁷, 2×10⁻⁵ and 10⁻³ s, the six-group inhour equation ρ = ωΛ + Σβ_iω/(ω+λ_i) shows where Λ matters. This isolates the effect of Λ; it is not a comparison of the real responses of different reactor types.

• Low reactivity (delayed supercritical): at 0.5$ the periods are 5.72, 5.73 and 6.51 s, a small difference, because the six delayed groups set the period. With Λ as long as 1 ms, however, the period at 0.8$ is already 1.72 s, about 1.9 times longer than the 0.92–0.95 s of the other two.
• Near and above prompt critical: at 1.2$ the periods split widely, 0.08 ms, 15 ms and 0.47 s. Well above 1$ the chain is sustained by prompt neutrons alone, so T ≈ Λ/(ρ−β), proportional to Λ. Just above 1$ this approximation does not hold.
• Λ's share even below 1$: the prompt jump right after a step insertion ends with time constant Λ/(β−ρ) (0.5$, PWR about 6 ms).

What it means in a plant

• Normal operation and startup: reactivity is kept far below 1$, so the period is set by β_eff and the precursors. In startup-rate management Λ is practically invisible.
• Reactivity accidents: when about 1$ or more is inserted, as in a rod ejection (SRP 15.4.8), the rate of power rise and the peak width depend on Λ, and Doppler feedback ends the transient. Safety analyses enter β_eff and Λ as conservative beginning- and end-of-cycle values. An example cited for large-PWR accident-analysis input is ℓ ≈ 19.5 μs with β_eff 0.0051 (AP1000 first cycle).
• Burnup effects: as boron concentration falls, thermal absorption decreases and Λ lengthens somewhat toward end of cycle, while Pu build-up reduces β_eff. The same Δk becomes a larger dollar value, so end-of-cycle reactivity accidents are the conservative case.
• Reactor-type comparison: in a heavy-water reactor (CANDU), heavy-water absorption is small, so Λ is about 0.9–1 ms, about 50 times that of a PWR, and even prompt-supercritical transients are relatively slow. A fast reactor has Λ ~10⁻⁷ s and a small Pu-239 β (about 0.0021), so the design of prompt negative feedback (Doppler, fuel expansion) is central to safety.
• Measurement and calculation: at delayed criticality, Λ is measured from the prompt decay constant α₀ = β_eff/Λ (Rossi-α, pulsed-neutron methods); for a PWR it is about 0.0065/2×10⁻⁵ ≈ 325 s⁻¹. The point kinetics equations are stiff, with Λ (10⁻⁵ s) and 80 s time constants side by side, so implicit integration is used.

Regulatory and design criteria

The following US references are examples of how kinetics parameters enter design review and accident analysis.

• NUREG-0800 SRP 4.3 (Nuclear Design): reviews the calculation methods and uncertainties of kinetics parameters such as β_eff and the prompt neutron lifetime. SRP 15.4.1 and 15.4.2 (uncontrolled rod withdrawal) and 15.4.8 (rod ejection) use these values as analysis inputs.
• 10 CFR 50 Appendix A: GDC 11 (reactor inherent protection), GDC 28 (reactivity limits).
• Reactivity-accident acceptance criteria: RG 1.77 (1974) was replaced by RG 1.236 in 2020. The acceptance criteria have changed over time; the physics of the generation time has not.
• FSAR Chapter 15: tabulates event-specific kinetics parameters (minimum and maximum β_eff, prompt neutron lifetime); values are unit- and cycle-specific.

Easily confused pairs

Key points at a glance

1. Prompt neutron lifetime ℓ_p: mean time from birth of a prompt neutron to absorption or leakage; in a thermal reactor ℓ_p ≈ t_s + t_d ≈ t_d.
2. Diffusion time: t_d = 1/(vΣ_a); with fuel t_dM(1−f); finite reactor 1/[vΣ_a(1+L²B²)].
3. Generation time Λ: birth to absorption causing fission, Λ = ℓ_p/k_eff; the time constant of the point kinetics equation.
4. Typical values: fast reactor ~10⁻⁷ s, LWR ~2×10⁻⁵ s, PHWR ~1 ms.
5. Prompt only: T = ℓ_p/(k_∞−1) = 0.02 s (k_∞ = 1.001) → uncontrollable.
6. Mean generation time: weighted mean including the precursor wait, ℓ̄ = ℓ_p + Σβ_i/λ_i ≈ 0.085 s (PWR example, β = 0.0065, τ̄ ≈ 13 s), about 4,000 times longer.
7. Delayed supercritical (low ρ): the period is governed by the six delayed groups and Λ matters little → controllable.
8. Prompt supercritical (ρ > β): well above, T ≈ Λ/(ρ−β), proportional to Λ → key parameter in reactivity-accident analysis.
9. Measurement: Rossi-α, α₀ = β_eff/Λ (PWR about 325 s⁻¹).
10. Operation and safety: Λ grows and β_eff falls toward end of cycle; conservative values applied under SRP 4.3 and 15.4.8.