Four Factor Formula
Nuclear Engineering · Reactor Physics
k = η × ε × p × f
k =
η ×
ε ×
p ×
f
·
k = multiplication factor ·
η = reproduction factor ·
ε = fast fission factor ·
p = resonance escape probability ·
f = thermal utilization factor
Four Factor Formula is a fundamental equation in reactor physics that describes the
multiplication factor k of a thermal nuclear reactor in terms of four factors:
η (reproduction factor), ε (fast fission factor),
p (resonance escape probability), and f (thermal utilization factor).
This formula is used for estimating the criticality of a thermal reactor.
For a critical reactor, k = 1. Typical values for a light water reactor are
η ≈ 1.6–1.8, ε ≈ 1.0–1.1, p ≈ 0.7–0.9, f ≈ 0.7–0.9.
Presets:
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Solve for:
Calculation Steps
Ready| Step | Operation | Value |
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| Enter values and press Calculate | ||
Four Factor Reference
Typical values for different reactor types
| Reactor Type | η | ε | p | f | k |
|---|
k = η × ε × p × f · k < 1 (subcritical), k = 1 (critical), k > 1 (supercritical)
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| k | Effective Multiplication Factor | dimensionless |
| η | Reproduction Factor | dimensionless |
| ε | Fast Fission Factor | dimensionless |
| p | Resonance Escape Probability | dimensionless |
| f | Thermal Utilisation Factor | dimensionless |
What it means
The formula gives the ratio of neutrons in one generation to the previous. If k=1, the reactor is critical; k>1 supercritical; k<1 subcritical.
Worked example
Four Factor Formula (k = η × ε × p × f)
Reactor Physics
Scenario: A reactor physicist evaluates the neutron life cycle in a thermal reactor to determine the effective multiplication factor (k). The four factors represent neutron gains and losses: η (neutrons produced per absorption in fuel), ε (fast fission bonus), p (neutrons surviving resonance capture), and f (neutrons absorbed in fuel vs. all absorptions). The product of these factors gives the infinite multiplication factor, a key parameter in criticality analysis.
| Factor | Symbol | Description | Typical Value (PWR) |
|---|---|---|---|
| Reproduction Factor | η | Neutrons produced per absorption in fuel (fission + capture) | 1.8 – 2.0 |
| Fast Fission Factor | ε | Extra neutrons from fission by fast neutrons | 1.02 – 1.05 |
| Resonance Escape Probability | p | Neutrons surviving resonance absorption in 238U | 0.7 – 0.9 |
| Thermal Utilisation Factor | f | Thermal neutrons absorbed in fuel vs. all absorbers | 0.7 – 0.8 |
| Multiplication Factor (k = η × ε × p × f) | ~1.0 – 1.06 | ||
1Determine η from the fuel's nuclear properties: η = ν × σf / σa (neutrons per absorption).
2Estimate ε from the fast fission rate relative to thermal fission (depends on fuel enrichment and geometry).
3Calculate p using the resonance escape probability formula (depends on moderation and fuel lattice).
4Find f as the ratio of thermal neutron absorption in fuel to total absorption (fuel + coolant + poisons).
5Multiply the four factors to obtain the infinite multiplication factor (k∞), which can be adjusted for finite core (leakage) to get keff.
Multiplication Factor
k = η × ε × p × f
For a typical PWR: η=1.9, ε=1.03, p=0.75, f=0.72 → k ≈ 1.06. Control rods adjust this to maintain criticality (k=1).
Common mistakes
- Mixing up k‑effective and k‑infinity: This formula gives the effective multiplication factor; applying it as infinite multiplication ignores leakage.
- Ignoring temperature feedback: η, ε, p, and f all change with temperature; assuming they are constant leads to incorrect reactivity predictions.
- Using wrong cross‑section data: Using microscopic cross‑sections instead of macroscopic, or using 2200 m/s values for non‑thermal spectra.
Applications
- Reactor core design: Determines the required enrichment and moderator‑to‑fuel ratio for a given geometry.
- Safety analysis: Evaluates how reactivity changes with burnup, poison buildup, or temperature excursions.
- Fuel cycle economics: Helps compare the neutron economy of different fuel types (e.g., UO₂ vs. MOX).
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