# 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. The formula is[1]

${\displaystyle k_{\infty }=\eta fp\varepsilon }$
Symbol Name Meaning Formula
${\displaystyle \eta }$ Reproduction Factor (Eta)
${\displaystyle {\frac {\mbox{neutrons produced from fission}}{\mbox{absorption in fuel isotope}}}}$
${\displaystyle \eta ={\frac {\nu \sigma _{f}^{F}}{\sigma _{a}^{F}}}}$
${\displaystyle f}$ The thermal utilization factor
${\displaystyle {\frac {\mbox{neutrons absorbed by the fuel isotope}}{\mbox{neutrons absorbed anywhere}}}}$
${\displaystyle f={\frac {\Sigma _{a}^{F}}{\Sigma _{a}}}}$
${\displaystyle p}$ The resonance escape probability
${\displaystyle {\frac {\mbox{fission neutrons slowed to thermal energies without absorption}}{\mbox{total fission neutrons}}}}$
${\displaystyle p\approx \mathrm {exp} \left(-{\frac {\sum \limits _{i=1}^{N}N_{i}I_{r,A,i}}{\left({\overline {\xi }}\Sigma _{p}\right)_{mod}}}\right)}$
${\displaystyle \epsilon }$ The fast fission factor
${\displaystyle {\frac {\mbox{total number of fission neutrons}}{\mbox{number of fission neutrons from just thermal fissions}}}}$
${\displaystyle \varepsilon \approx 1+{\frac {1-p}{p}}{\frac {u_{f}\nu _{f}P_{FAF}}{f\nu _{t}P_{TAF}P_{TNL}}}}$

The six factor formula defines each of these terms in much more detail.

## Multiplication

The multiplication factor, k, is defined as (see Nuclear chain reaction):

${\displaystyle k={\frac {\mbox{number of neutrons in one generation}}{\mbox{number of neutrons in preceding generation}}}}$

If k is greater than 1, the chain reaction is supercritical, and the neutron population will grow exponentially.
If k is less than 1, the chain reaction is subcritical, and the neutron population will exponentially decay.
If k = 1, the chain reaction is critical and the neutron population will remain constant.

In an infinite medium, neutrons cannot leak out of the system and the multiplication factor becomes the infinite multiplication factor, ${\displaystyle k=k_{\infty }}$, which is approximated by the four-factor formula.