# 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.
** ${L_{th}}^{2}={\frac {D}{\Sigma _{a,th}}}$
* $\tau _{th}$ 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.