The six-factor formula is used in nuclear engineering to determine the multiplication of a nuclear chain reaction in a non-infinite medium.

Six-factor formula: ${\displaystyle k=\eta fp\varepsilon P_{FNL}P_{TNL}=k_{\infty }P_{FNL}P_{TNL))$^[1]

Symbol	Name	Meaning	Formula	Typical thermal reactor value
${\displaystyle \eta }$	Thermal fission 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	${\displaystyle p\approx \mathrm {exp} \left(-{\frac {\sum \limits _{i=1}^{N}N_{i}I_{r,A,i)){\left({\overline {\xi ))\Sigma _{p}\right)_{mod))}\right)}$	0.87
${\displaystyle \varepsilon }$	Fast fission factor (epsilon)	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
${\displaystyle P_{FNL))$	Fast non-leakage probability	number of fast neutrons that do not leak from reactor/number of fast neutrons produced by all fissions	${\displaystyle P_{FNL}\approx \mathrm {exp} \left(-{B_{g))^{2}\tau _{th}\right)}$	0.97
${\displaystyle P_{TNL))$	Thermal non-leakage probability	number of thermal neutrons that do not leak from reactor/number of thermal neutrons produced by all fissions	${\displaystyle P_{TNL}\approx {\frac {1}{1+{L_{th))^{2}{B_{g))^{2))))$	0.99

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.
• ${\displaystyle {B_{g))^{2))$ is the geometric buckling.
• ${\displaystyle {L_{th))^{2))$ is the diffusion length of thermal neutrons.
  • ${\displaystyle {L_{th))^{2}={\frac {D}{\Sigma _{a,th))))$.
• ${\displaystyle \tau _{th))$ is the age to thermal.
  • ${\displaystyle \tau =\int _{E_{th))^{E'}dE''{\frac {1}{E'')){\frac {D(E'')}((\overline {\xi ))\left[D(E''){B_{g))^{2}+\Sigma _{t}(E')\right]))}$.
  • ${\displaystyle \tau _{th))$ is the evaluation of ${\displaystyle \tau }$ where ${\displaystyle E'}$ is the energy of the neutron at birth.

Multiplication

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

k = number of neutrons in one generation/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.