Four factor formula

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The four-factor formula, also known as Fermi's four factor formula is used in nuclear engineering to determine the multiplication of a nuclear chain reaction in an infinite medium.

Four-factor formula: ${\displaystyle k_{\infty }=\eta fp\varepsilon }$^[1]

Symbol	Name	Meaning	Formula	Typical thermal reactor value
${\displaystyle \eta }$	Reproduction factor (eta)	⁠neutrons produced from thermal fissions/thermal absorption in fuel isotope⁠[2]	${\displaystyle \eta ={\frac {\nu \sigma _{f}^{F}}{\sigma _{a}^{F}}}={\frac {\nu \Sigma _{f}^{F}}{\Sigma _{a}^{F}}}}$	1.65
${\displaystyle f}$	Thermal utilization factor	⁠thermal neutrons absorbed by the fuel isotope/thermal neutrons absorbed anywhere⁠[2]	${\displaystyle f={\frac {\Sigma _{a}^{F}}{\Sigma _{a}}}}$	0.71
${\displaystyle p}$	Resonance escape probability	⁠fission neutrons slowed to thermal energies without absorption/total fission neutrons⁠	${\displaystyle p\approx \exp \left(-{\frac {\sum \limits _{i=1}^{N}N_{i}I_{r,A,i}}{\left({\overline {\xi }}\Sigma _{p}\right)_{mod}}}\right)}$	0.87
${\displaystyle \varepsilon }$	Fast fission factor	⁠total number of fission neutrons/number of fission neutrons from just thermal fissions⁠	${\displaystyle \varepsilon \approx 1+{\frac {1-p}{p}}{\frac {u_{f}\nu _{f}P_{FAF}}{f\nu _{t}P_{TAF}P_{TNL}}}}$	1.02

The symbols are defined as:^[3]

• ${\displaystyle \nu }$, ${\displaystyle \nu _{f}}$ and ${\displaystyle \nu _{t}}$ are the average number of neutrons produced per fission in the medium (2.43 for uranium-235).
• ${\displaystyle \sigma _{f}^{F}}$ and ${\displaystyle \sigma _{a}^{F}}$ are the microscopic fission and absorption thermal cross sections for fuel, respectively.
• ${\displaystyle \Sigma _{a}^{F}}$ and ${\displaystyle \Sigma _{a}}$ are the macroscopic absorption thermal cross sections in fuel and in total, respectively.
• ${\displaystyle \Sigma _{f}^{F}}$ is the macroscopic fission cross-section.
• ${\displaystyle N_{i}}$ is the number density of atoms of a specific nuclide.
• ${\displaystyle I_{r,A,i}}$ is the resonance integral for absorption of a specific nuclide.
  • ${\displaystyle I_{r,A,i}=\int _{E_{th}}^{E_{0}}dE'{\frac {\Sigma _{p}^{mod}}{\Sigma _{t}(E')}}{\frac {\sigma _{a}^{i}(E')}{E'}}}$
• ${\displaystyle {\overline {\xi }}}$ is the average lethargy gain per scattering event.
  • Lethargy is defined as decrease in neutron energy.
• ${\displaystyle u_{f}}$ (fast utilization) is the probability that a fast neutron is absorbed in fuel.
• ${\displaystyle P_{FAF}}$ is the probability that a fast neutron absorption in fuel causes fission.
• ${\displaystyle P_{TAF}}$ is the probability that a thermal neutron absorption in fuel causes fission.
• ${\displaystyle P_{TNL}}$ is the thermal non-leakage probability