What is the correct form of the basic two-group point-kinetics equations for neutron density n(t) and precursor concentrations C_i?

• A. dn/dt = [(ρ - β)/Λ] n + sum over groups of λ_i C_i; dC_i/dt = β_i/Λ n - λ_i C_i
• B. dn/dt = [(ρ + β)/Λ] n - sum over groups of λ_i C_i; dC_i/dt = β_i/Λ n + λ_i C_i
• C. dn/dt = [(β - ρ)/Λ] n + sum over groups of λ_i C_i; dC_i/dt = β_i/Λ n - λ_i C_i
• D. dn/dt = [ρ/Λ] n + sum over groups of β_i C_i; dC_i/dt = -λ_i C_i

Answer: (A)

These equations describe how neutron density and delayed-neutron precursors evolve together in a two-group point-kinetics model. The prompt-neutron production is governed by the reactivity ρ and the prompt-generation time Λ, but because a fraction β of neutrons are delayed, the net prompt contribution is scaled by (ρ − β)/Λ times the current neutron density n. The delayed-neutron precursors, grouped with decay constants λ_i and fractional yields β_i, feed neutrons as they decay, giving a source term sum over groups of λ_i C_i in the neutron equation.

Simultaneously, each precursor group is produced by fission at a rate (β_i/Λ) n and decays at rate λ_i C_i. Put together, the correct form is:

dn/dt = [(ρ − β)/Λ] n + sum_i λ_i C_i

dC_i/dt = (β_i/Λ) n − λ_i C_i

This structure reflects the immediate prompt neutrons and the slower release from delayed-neutron precursors. If you omit the minus β in the first term, or misplace signs or terms in the precursor equations, the balance between prompt production and delayed neutron sources would be incorrect, leading to the wrong dynamic behavior.