Four factor formula

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 fission/absorption in fuel isotope	${\displaystyle \eta ={\frac {\nu \sigma _{f}^{F}}{\sigma _{a}^{F}}}}$	1.65
${\displaystyle f}$	Thermal utilization factor	neutrons absorbed by the fuel isotope/neutrons absorbed anywhere	${\displaystyle f={\frac {\Sigma _{a}^{F}}{\Sigma _{a}}}}$	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
${\displaystyle \epsilon }$	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:[2]

• ${\displaystyle \nu }$, ${\displaystyle \nu _{f}}$ 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 cross sections for fuel, respectively.
• ${\displaystyle \Sigma _{a}^{F}}$ and ${\displaystyle \Sigma _{a}}$ are the macroscopic absorption cross sections in fuel and in total, respectively.
• ${\displaystyle N_{i}}$ 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