Neutron Multiplication k-effective Calculator

The effective multiplication factor k_eff is the ratio of neutrons produced by fission in one generation to the neutrons lost (by absorption or leakage) in the previous generation. When k_eff = 1, the reactor is critical and maintains a steady chain reaction. When k_eff > 1, the reactor is supercritical and power rises. When k_eff < 1, the reactor is subcritical and the chain reaction dies out. k_eff differs from k∞ by including neutron leakage from the finite reactor boundary.

In the one-group diffusion approximation, k_eff = k∞ / (1 + M²B²), where k∞ is the infinite multiplication factor, M² is the migration area (cm²), and B² is the geometric buckling (cm⁻²). The factor 1/(1 + M²B²) equals the non-leakage probability P_NL. This formula applies to bare, homogeneous thermal reactors and is accurate when M²B² is small compared to 1.

Geometric buckling B² measures how rapidly the neutron flux varies across the reactor and depends only on geometry. For a sphere of radius R: B² = (π/R)². For a finite cylinder of radius R and height H: B² = (2.405/R)² + (π/H)². For a rectangular box with sides a, b, c: B² = (π/a)² + (π/b)² + (π/c)². The unit is cm⁻² when dimensions are in cm. Larger reactors have smaller buckling and lower leakage.

Migration area M² = L² + τ, where L² is the diffusion area (cm²) and τ is the Fermi age (cm²). M² quantifies the mean squared distance a neutron travels from birth to absorption. For light water with UO2 fuel, M² is roughly 50-70 cm². For heavy water with UO2 fuel, M² is 5,000-7,000 cm². For graphite-moderated reactors, M² is 300-500 cm². Larger M² means neutrons travel farther, so leakage is more significant for a given core size.

k∞ is the multiplication factor in a hypothetical infinite reactor where no neutrons leak. k_eff accounts for the actual finite size of the reactor: k_eff = k∞ × P_NL, where P_NL is the non-leakage probability. For a critical reactor, k_eff = 1.000. Large power reactors have P_NL of 0.95 to 0.99, so k_eff is only slightly below k∞. Smaller reactors lose proportionally more neutrons through leakage.

Reactivity ρ = (k_eff - 1) / k_eff. It is zero at criticality, positive when supercritical, and negative when subcritical. The unit pcm (percent-milli) equals 10^-5, so 1 pcm corresponds to ρ = 0.00001. A typical power reactor at beginning-of-life has excess reactivity of 15,000-25,000 pcm, which is controlled by soluble boron, control rods, and burnable poisons.

The optimum height-to-diameter ratio that minimizes critical volume for a finite cylinder is H = 2R × 0.924, or H/D ≈ 0.924. At this ratio, the radial and axial components of buckling are equal, meaning neither direction contributes disproportionately to leakage. Real reactor designs deviate from this optimum for engineering reasons (containment dimensions, coolant flow), but it serves as the baseline for bare-reactor lattice calculations.

As reactor dimensions increase, geometric buckling B² decreases (proportional to 1/R² for a sphere), so the term M²B² decreases and k_eff approaches k∞. In the limit of an infinite reactor, B² approaches zero and k_eff = k∞. This means that for a given fuel composition, there is a minimum critical size at which k_eff = 1. Reactors smaller than this critical size cannot sustain a chain reaction regardless of fuel enrichment.

For a bare spherical reactor, criticality (k_eff = 1) requires k∞/(1 + M²B²) = 1, so B² = (k∞ - 1)/M². Since B² = (π/R)² for a sphere, the critical radius is R_crit = π / √((k∞ - 1)/M²) = π × √(M²/(k∞ - 1)). For a typical LWR with k∞ = 1.30 and M² = 60 cm², R_crit = π × √(60/0.30) = π × √200 = π × 14.14 ≈ 44.4 cm.

The two-group model separates neutrons into fast and thermal energy groups, each with its own diffusion and absorption properties. It gives k_eff = k∞ × P_FNL × P_TNL, where P_FNL = exp(-B²τ) is the fast non-leakage probability and P_TNL = 1/(1 + L²B²) is the thermal non-leakage probability. The one-group model combines these into the single factor 1/(1 + M²B²) = 1/(1 + (L² + τ)B²). The two-group model is more accurate for small reactors and large M² systems like heavy-water or graphite reactors.

During normal operation, a critical reactor maintains k_eff = 1.000 exactly (controlled by operator). During startup, k_eff is brought slightly above 1 to allow controlled power increase. For safe shutdown, control rods are inserted to achieve a deeply subcritical k_eff (often 0.95 or below), providing a shutdown margin of at least 5,000 pcm. Regulatory requirements typically demand a shutdown margin of at least 1,000-2,000 pcm with the most reactive control rod stuck out (single-failure criterion).

Xenon-135 is the most powerful neutron absorber in reactor operations, with a thermal neutron absorption cross-section of 2.65 million barns. At equilibrium in a high-flux reactor, Xe-135 reduces k_eff by several thousand pcm (typically 2,000-3,000 pcm for LWRs). After a reactor shutdown or power reduction, Xe-135 concentration peaks ('xenon peak') within 6-12 hours due to I-135 decay, causing a temporary additional negative reactivity that can prevent immediate restart. This phenomenon, known as xenon poisoning or xenon override, must be managed in reactor operations.