Six factor formula

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title: "Six factor formula" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["nuclear-technology", "radioactivity"] description: "Formula used to calculate nuclear chain reaction growth rate" topic_path: "general/nuclear-technology" source: "https://en.wikipedia.org/wiki/Six_factor_formula" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Formula used to calculate nuclear chain reaction growth rate ::

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

::data[format=table title="Six-factor formula: k = \eta f p \varepsilon P_{FNL} P_{TNL} = k_{\infty} P_{FNL} P_{TNL}{{cite book |last=Duderstadt |first=James |author2=Hamilton, Louis |title=Nuclear Reactor Analysis |year=1976 |publisher=John Wiley & Sons, Inc |isbn=0-471-22363-8 }}"]

Symbol	Name	Meaning	Formula	Typical thermal reactor value
\eta	Thermal fission factor (eta)		\eta = \frac{\nu \sigma_f^F}{\sigma_a^F} = \frac{\nu \Sigma_f^F}{\Sigma_a^F}	1.65
f	Thermal utilization factor		f = \frac{\Sigma_a^F}{\Sigma_a}	0.71
p	Resonance escape probability		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)		\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		P_{FNL} \approx \mathrm{exp} \left( -{B_g}^2 \tau_{th} \right)	0.97
P_{TNL}	Thermal non-leakage probability		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.
  • {L_{th}}^2 = \frac{D}{\Sigma_{a,th}}, where D is the diffusion coefficient.
• \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.

Multiplication

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

• 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 , the chain reaction is critical and the neutron population will remain constant.

References

References

1. Duderstadt, James. (1976). "Nuclear Reactor Analysis". John Wiley & Sons, Inc.
2. Adams, Marvin L.. (2009). "Introduction to Nuclear Reactor Theory". Texas A&M University.

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