Four Factor Formula

η (neutron yield per absorption in fuel) is the number of neutrons produced per neutron absorbed in the fuel. ε (fast fission factor) accounts for fissions caused by fast neutrons in U-238. p (resonance escape probability) is the fraction of neutrons that slow down to thermal energies without being absorbed in U-238 resonances. f (thermal utilization factor) is the fraction of thermal neutrons absorbed in the fuel rather than in non-fuel materials (moderator, coolant, structural materials). Their product gives the infinite multiplication factor k∞.

For a typical PWR: η ≈ 1.8–2.0 (depending on enrichment and fuel type), ε ≈ 1.02–1.03 (small due to low fast fission in U-238), p ≈ 0.88–0.92 (significant resonance absorption in U-238), and f ≈ 0.80–0.85 (some thermal neutrons absorbed in moderator, poisons, and structural materials). Their product gives k∞ ≈ 1.3–1.5, which is then reduced by leakage (non-leakage probability) to achieve k_eff ≈ 1.0 for a critical core.

The resonance escape probability p is typically 0.85–0.92 because U-238 has many strong absorption resonances in the epithermal energy region (several eV to keV). As neutrons slow down, they have a significant probability of being captured by U-238 resonances, especially at 6.7 eV, 20.9 eV, and higher resonances. This loss is unavoidable in thermal reactors using U-238 as the fertile material. Higher enrichment reduces the U-238 fraction, increasing p slightly, but it remains the most sensitive factor to fuel composition and moderator-to-fuel ratio.

In a thermal reactor, the fast fission factor ε is typically 1.02–1.05, meaning only 2–5% of fissions are caused by fast neutrons (mostly in U-238). In a fast reactor, where there is no moderation, all fissions occur at fast energies, so ε is not used as a separate factor; instead, the multiplication factor is determined by the neutron spectrum and the fission cross-sections. The concept of ε is specific to thermal reactor analysis, where a small fraction of fissions occur before neutrons are thermalized.

The thermal utilization factor f is the fraction of thermal neutrons absorbed in the fuel. It depends on the ratio of the fuel absorption cross-section to the total absorption cross-section (fuel + moderator + poisons + structural materials). Increasing the moderator-to-fuel ratio (more water relative to fuel) increases the moderator absorption, reducing f. Conversely, a tighter lattice (more fuel per unit volume) increases f but also reduces moderation. Optimizing f is a key design trade-off in thermal reactors.

Higher enrichment increases the U-235 concentration, which: (1) raises η because the fission-to-capture ratio in fuel improves, (2) reduces the U-238 fraction, increasing p (less resonance absorption), (3) increases f because more thermal neutrons are absorbed in U-235 rather than U-238, and (4) slightly increases ε. The net effect is a larger k∞, which provides more excess reactivity to compensate for burnup and fission product poisoning. This allows longer cycle lengths before the reactor becomes subcritical.

k∞ is the product of the four factors (η × ε × p × f) and represents the multiplication factor for an infinite, non-leaking reactor (no neutron leakage). k_eff is the effective multiplication factor for a finite reactor, which includes leakage: k_eff = k∞ × (fast non-leakage probability) × (thermal non-leakage probability). In a real reactor, k_eff is always less than k∞ because some neutrons leak out. For a critical reactor, k_eff = 1, while k∞ is typically 1.3–1.5.

The four-factor formula assumes a homogeneous, well-moderated, thermal reactor and is based on neutron energy groups (fast, epithermal, thermal). For fast reactors, there is no thermalization, so the concept of thermal utilization and resonance escape is not applicable. For heterogeneous cores (e.g., with burnable poison rods, water gaps, or complex lattice geometries), the assumptions of the four-factor formula break down. Modern reactor design uses multi-group diffusion or transport theory (like MCNP) to account for spatial and energy effects directly, though the four-factor formula remains a useful pedagogical tool.

Burnable absorbers reduce the thermal utilization factor f initially because they absorb thermal neutrons that would otherwise go to the fuel. This reduces k∞ at the beginning of the cycle, allowing the reactor to operate with a higher fuel enrichment. As the absorber burns out, f increases, compensating for the depletion of fissile material. The other factors (η, ε, p) are largely unaffected by burnable absorbers, except for small changes in the neutron spectrum. The net effect is a flatter reactivity curve over the cycle.

η = ν × σ_f / σ_a, where ν is the average number of neutrons produced per fission, σ_f is the fission cross-section, and σ_a is the absorption cross-section. A high η (>2.0) indicates that more neutrons are available for breeding (capture in U-238 to form Pu-239). For U-235, η ≈ 2.07; for Pu-239, η ≈ 2.88 in a thermal spectrum. This is why plutonium is a better fuel for breeding: it produces more excess neutrons. The four-factor formula shows that a higher η directly increases k∞, making breeding reactors possible.

The moderator's slowing-down power and absorption cross-section affect p and f. Light water (H₂O) has high moderating power but also high absorption, reducing f. Heavy water (D₂O) has lower absorption, so f is higher, but its moderating power is lower, affecting the slowing-down process and p. Graphite has low absorption and moderate moderating power. The choice of moderator influences the resonance escape probability p (because moderation speed affects resonance absorption) and the thermal utilization factor f (through absorption in the moderator). This is why CANDU reactors (D₂O) can use natural uranium—they have higher f and p, achieving k∞ > 1 without enrichment.

The four-factor formula gives k∞, which assumes no leakage. Leakage is accounted for in the full reactor analysis by multiplying k∞ by the non-leakage probabilities (fast and thermal): k_eff = k∞ × P_FNL × P_TNL. These probabilities depend on the core geometry and the neutron diffusion lengths. In a critical reactor, the leakage losses are balanced by the excess reactivity provided by the four factors. The formula is often extended to the six-factor formula to explicitly include leakage.

As moderator temperature increases: (1) η remains approximately constant (depends on fuel temperature), (2) ε may slightly increase or decrease depending on spectrum shift, (3) p decreases because Doppler broadening of U-238 resonances increases resonance absorption, and (4) f decreases because the moderator density decreases, reducing moderation and increasing absorption in the fuel relative to the moderator? Actually, in LWRs, the MTC is negative: f decreases because the moderator density drops, reducing neutron slowing-down, which shifts the spectrum to higher energies and lowers f. The combined effect is a negative temperature coefficient, which is a key safety feature.

In MOX fuel (plutonium-uranium oxide): (1) η is higher (for Pu-239, η ≈ 2.88 vs. 2.07 for U-235), which increases k∞, (2) ε is slightly higher because of faster neutron spectrum, (3) p is lower because Pu-239 has resonances in the epithermal range, increasing resonance absorption, and (4) f is similar but depends on the plutonium isotopics. The net effect is that MOX reactors can have similar or slightly higher k∞, but they have a lower delayed neutron fraction (β), requiring careful control. The four-factor formula helps in understanding these trade-offs.

The four-factor formula is used in the early design phase to estimate the required enrichment, moderator-to-fuel ratio, and core dimensions to achieve criticality. It provides a quick, analytical way to understand the effect of design changes on k∞. However, for modern, heterogeneous cores with burnable poisons, control rods, and complex geometries, the formula is not accurate enough. Designers use detailed neutron transport codes (like MCNP, SCALE, or CASMO) that solve the neutron transport equation in 3D with hundreds of energy groups. The four-factor formula is still valuable for teaching the fundamental neutron life cycle and for initial scoping studies.