For small ρ, what is the simple expression for the reactor period P in terms of Λ and ρ?

• A. P ≈ Λ / (β_eff - ρ)
• B. P ≈ Λ / (ρ - β_eff)
• C. P ≈ Λ × (ρ - β_eff)
• D. P ≈ Λ / ρ

Answer: (B)

The period reflects how fast the neutron population grows or decays when reactivity is small. In one-group kinetics with delayed neutrons, the net effect of reactivity is (ρ − β_eff). The mean generation time Λ sets the time scale for a generation of neutrons. For small ρ, the population changes approximately as dN/dt ≈ (ρ − β_eff) N / Λ. Solving N(t) = N0 exp(t/P) gives 1/P = (ρ − β_eff)/Λ, so P ≈ Λ / (ρ − β_eff).

This means the sign of P tells you whether the reactor is supercritical (positive P) or subcritical (negative P) and how quickly the population changes. If ρ is less than β_eff, the period is negative, indicating decay with a characteristic time |P|. The other options either misplace the sign, ignore the delayed-neutron contribution, or mix units, so they don’t correctly describe the small-ρ behavior.