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Six factor formula

Formula used to calculate nuclear chain reaction growth rate

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated February 14, 2026

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: k=\eta fp\varepsilon P_{FNL}P_{TNL}=k_{\infty }P_{FNL}P_{TNL}
Symbol Name Meaning Formula Typical thermal reactor value
\eta Thermal fission factor (eta) ⁠neutrons produced from fission/absorption in fuel isotope⁠ \eta ={\frac {\nu \sigma _{f}^{F}}{\sigma _{a}^{F}}}={\frac {\nu \Sigma _{f}^{F}}{\Sigma _{a}^{F}}} 1.65
f Thermal utilization factor ⁠neutrons absorbed by the fuel isotope/neutrons absorbed anywhere⁠ f={\frac {\Sigma _{a}^{F}}{\Sigma _{a}}} 0.71
p Resonance escape probability ⁠fission neutrons slowed to thermal energies without absorption/total fission neutrons⁠ 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
\varepsilon Fast fission factor (epsilon) ⁠total number of fission neutrons/number of fission neutrons from just thermal fissions⁠ \varepsilon \approx 1+{\frac {1-p}{p}}{\frac {u_{f}\nu _{f}P_{FAF}}{f\nu _{t}P_{TAF}P_{TNL}}} 1.02
P_{FNL} Fast non-leakage probability ⁠number of fast neutrons that do not leak from reactor/number of fast neutrons produced by all fissions⁠ P_{FNL}\approx \mathrm {exp} \left(-{B_{g}}^{2}\tau _{th}\right) 0.97
P_{TNL} Thermal non-leakage probability ⁠number of thermal neutrons that do not leak from reactor/number of thermal neutrons produced by all fissions⁠ P_{TNL}\approx {\frac {1}{1+{L_{th}}^{2}{B_{g}}^{2}}} 0.99

The symbols are defined as:

  • \nu, \nu _{f} and \nu _{t} are the average number of neutrons produced per fission in the medium (2.43 for uranium-235).
  • \sigma _{f}^{F} and \sigma _{a}^{F} are the microscopic fission and absorption cross sections for fuel, respectively.
  • \Sigma _{a}^{F} and \Sigma _{a} are the macroscopic absorption cross sections in fuel and in total, respectively.
  • \Sigma _{f}^{F} is the macroscopic fission cross-section.
  • N_{i} is the number density of atoms of a specific nuclide.
  • I_{r,A,i} is the resonance integral for absorption of a specific nuclide.
    • I_{r,A,i}=\int _{E_{th}}^{E_{0}}dE'{\frac {\Sigma _{p}^{mod}}{\Sigma _{t}(E')}}{\frac {\sigma _{a}^{i}(E')}{E'}}
  • {\overline {\xi }} is the average lethargy gain per scattering event.
    • Lethargy is defined as decrease in neutron energy.
  • u_{f} (fast utilization) is the probability that a fast neutron is absorbed in fuel.
  • P_{FAF} is the probability that a fast neutron absorption in fuel causes fission.
  • P_{TAF} is the probability that a thermal neutron absorption in fuel causes fission.
  • {B_{g}}^{2} is the geometric buckling.
  • {L_{th}}^{2} is the diffusion length of thermal neutrons.
  • \tau _{th} is the age to thermal.
    • \tau =\int _{E_{th}}^{E'}dE''{\frac {1}{E''}}{\frac {D(E'')}{{\overline {\xi }}\left[D(E''){B_{g}}^{2}+\Sigma _{t}(E')\right]}}
    • \tau _{th} is the evaluation of \tau where E' is the energy of the neutron at birth.

01Multiplication

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.
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Sources and credits

This article is adapted from the Wikipedia article “Six factor formula”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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