Six factor formula

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.
  • ${L_{th}}^{2}={\frac {D}{\Sigma _{a,th}}},$ 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):

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.

See also

• Critical mass
• Nuclear chain reaction
• Nuclear reactor
• Four factor formula