Formula & Calculator
Six-Factor Formula (Infinite Multiplication Factor)
Combines six physical factors describing the neutron life cycle to determine the reactor's multiplication factor, accounting for both material properties and leakage.
Interpretation
k∞ = η f p ε P_FNL P_TNL. Product of factors accounting for neutron losses. η=fission neutrons per absorption in fuel; f=thermal utilization; p=resonance escape; ε=fast fission; P_FNL & P_TNL non‑leakage. Used in reactor design.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| k∞ | Infinite (or effective) multiplication factor | |
| η | Reproduction factor | |
| f | Thermal utilization factor | |
| p | Resonance escape probability | |
| ε | Fast fission factor | |
| P_FNL | Fast non-leakage probability | |
| P_TNL | Thermal non-leakage probability |
What it means
The six‑factor formula gives the infinite multiplication factor (k∞) – the ratio of neutrons produced to neutrons lost in an infinite reactor medium. The factors are: η = neutron yield per absorption in fuel, f = thermal utilization factor, p = resonance escape probability, ε = fast fission factor, P_FNL = fast non‑leakage probability, and P_TNL = thermal non‑leakage probability. The product k∞ = η f p ε P_FNL P_TNL determines whether a chain reaction can be sustained. For criticality, k_eff = k∞ P_NL must equal 1, where P_NL accounts for leakage. The four‑factor formula (without P_FNL and P_TNL) applies to infinite media. Understanding this formula is essential for reactor physicists to design fuel and moderator configurations, to predict reactivity changes, and to optimise core performance.
Worked example
Six‑Factor Formula – Two Examples
Real‑World| Parameter | Value |
|---|---|
| η | 2.0 |
| f | 0.71 |
| p | 0.87 |
| ε | 1.02 |
| P_FNL | 0.997 |
| P_TNL | 0.98 |
| Parameter | Value |
|---|---|
| η | 1.95 |
| f | 0.75 |
| p | 0.85 |
| ε | 1.03 |
| P_FNL | 0.998 |
| P_TNL | 0.985 |
Common mistakes
- Six‑factor formula: k∞ = η·f·p·ε·P_FNL·P_TNL – the product of six factors.
- η: Eta (neutrons produced per absorption in fuel).
- f: Thermal utilization factor.
- p: Resonance escape probability.
- ε: Fast fission factor.
- P_FNL: Fast neutron non‑leakage probability.
- P_TNL: Thermal neutron non‑leakage probability.
- Interpretation: k∞ is the infinite multiplication factor (no leakage) – includes all factors.
- k_eff: k_eff = k∞ × P_FNL × P_TNL (two‑factor formula) – for finite reactors.
Applications
The six‑factor formula, k∞ = η·f·p·ε·P_FNL·P_TNL, calculates the infinite multiplication factor for a reactor core, accounting for the neutron lifecycle. It includes factors for neutron production (η), thermal utilisation (f), resonance escape (p), fast fission (ε), and non‑leakage probabilities (P_FNL, P_TNL). This formula is the basis for reactor design and neutron economy analysis. Reactor physicists use it to evaluate the impact of different materials, geometries, and fuel enrichments on the chain reaction. It guides decisions on fuel type, moderator choice, and core layout. By understanding each factor, engineers can identify the most effective means to increase k∞, thereby improving reactor performance and safety.
- Reactor core design and neutronic optimisation
- Assessment of neutron economy and reactivity balance
- Evaluation of fuel types, moderators, and structural materials
- Comparison of different reactor concepts (PWR, BWR, CANDU)
- Education and understanding of the neutron life cycle
Frequently Asked Questions
The six‑factor formula expresses the effective multiplication factor (k_eff) as the product of six factors: k_eff = η · f · p · ε · P_FNL · P_TNL. The first four are the four‑factor formula (for an infinite medium), and the last two account for neutron leakage. It is used to estimate the reactivity of a reactor core.
- η – neutron yield per absorption in fuel (number of neutrons produced per fission).
- f – thermal utilisation factor (fraction absorbed in fuel).
- p – resonance escape probability (probability that neutrons avoid being absorbed in resonances).
- ε – fast fission factor (neutrons produced by fast fission).
- P_FNL – fast non‑leakage probability (probability a fast neutron does not leak out).
- P_TNL – thermal non‑leakage probability (probability a thermal neutron does not leak out).
The four‑factor formula (k∞ = η·f·p·ε) applies to an infinite reactor (no leakage). The six‑factor formula includes the leakage probabilities to give k_eff for a finite reactor.
Confusing the four‑factor formula (k∞) with the six‑factor formula (k_eff). Mixing them up leads to incorrect criticality assessment for finite reactors.
It is used for preliminary scoping calculations to estimate the multiplication factor. More detailed calculations use nuclear codes, but the six‑factor formula provides a useful conceptual framework.
- η depends on the fuel isotope and neutron energy.
- f depends on the fuel volume fraction and parasitic absorptions.
- p depends on the moderator and fuel arrangement (heterogeneity).
- ε depends on the fast neutron spectrum.
- P_FNL and P_TNL depend on the core size and reflectors.
p is the probability that a neutron escapes resonance absorption as it slows down from fast to thermal energies. This is crucial in water‑moderated reactors where uranium resonances can absorb many neutrons.
P_FNL ≈ exp(–B² τ), where τ is the Fermi age (slowing‑down area). P_TNL ≈ exp(–B² L²), where L is the diffusion length. B is the geometric buckling, determined by the core shape.
Approximate values for a PWR: η ~1.8, f ~0.7, p ~0.8, ε ~1.0, P_FNL ~0.95, P_TNL ~0.95, giving k_eff ~1.0.
As burnup increases, η decreases (due to depletion of U‑235), f changes (due to fission product absorption), p may change, and leakage probabilities may change if the core composition changes. These factors are used to predict the reactivity loss over life.