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: $k=\eta fp\varepsilon P_{FNL}P_{TNL}=k_{\infty }P_{FNL}P_{TNL}$ ^[1]
Symbol	Name	Meaning	Formula	Typical thermal reactor value
$\eta$	Thermal fission factor (eta)	neutrons produced from fission/absorption in fuel isotope	$\eta ={\frac {\nu \sigma _{f}^{F}}{\sigma _{a}^{F}}}$	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:^[2]

$\nu$ , $\nu _{f}$ 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.
$N_{i}$ is the number density of atoms of a specific nuclide.
$I_{r,A,i}$ $I_{r,A,i}$ is the resonance integral for absorption of a specific nuclide.
- $I_{r,A,i}=\int _{E_{th}}^{E_{0}}dE'{\frac {\Sigma _{p}^{mod}}{\Sigma _{t}(E')}}{\frac {\sigma _{a}^{i}(E')}{E'}}$ .
${\overline {\xi }}$ ${\overline {\xi }}$ 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.
${L_{th}}^{2}$ ${L_{th}}^{2}$ is the diffusion length of thermal neutrons.
- ${L_{th}}^{2}={\frac {D}{\Sigma _{a,th}}}$ .
$\tau _{th}$ $\tau _{th}$ 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.

↑ Duderstadt, James; Hamilton, Louis (1976). Nuclear Reactor Analysis. John Wiley & Sons, Inc. ISBN 0-471-22363-8.
↑ Adams, Marvin L. (2009). Introduction to Nuclear Reactor Theory. Texas A&M University.

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