Reactor Physics and Nuclear Criticality
Multiplication and reactivity
k_eff = neutrons produced / neutrons lost; criticality is k = 1. Reactivity is ρ = (k − 1)/k, and 1 pcm = 10⁻⁵ Δk/k. The six-factor formula (k∞ = ε p f η, then multiplied by fast and thermal non-leakage probabilities) links enrichment, moderator-to-fuel ratio, and geometry to k. Control is by boron (chemical shim), control rods, and burnable poisons.
Feedback and kinetics
Doppler broadening of U-238 resonances gives a prompt negative fuel-temperature coefficient, the dominant self-limiting feedback. The moderator temperature coefficient depends strongly on boron: typical PWR Technical Specifications limit MTC to ≤ +5 pcm/°F at hot zero power and require it be negative at hot full power. Void coefficients are negative in both PWRs and BWRs because cores are undermoderated relative to the optimum ratio. Delayed neutrons (β_eff ≈ 650 pcm for fresh U-235, decreasing as Pu builds in) make the reactor controllable; reaching ρ = β_eff is prompt criticality.
Traps
- Using (k − 1) as reactivity instead of (k − 1)/k.
- Forgetting that post-shutdown xenon peaks roughly 9–11 hours later, and that samarium builds in slowly and does not burn out.
- Confusing reactor period with startup rate, or assuming β_eff is constant over core life.
Worked example
If k = 1.0025, ρ = 0.0025/1.0025 = 0.002494 ≈ 249 pcm. Near unity, the shortcut ρ(pcm) ≈ (k − 1) × 10⁵ is within about 0.3% - fine for estimating rod worth or boron changes.