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: ${\displaystyle k=\eta fp\varepsilon P_{FNL}P_{TNL}=k_{\infty }P_{FNL}P_{TNL}}$^[1]

Symbol	Name	Meaning	Formula	Typical thermal reactor value
${\displaystyle \eta }$	Thermal fission factor (eta)	⁠neutrons produced from fission/absorption in fuel isotope⁠	${\displaystyle \eta ={\frac {\nu \sigma _{f}^{F}}{\sigma _{a}^{F}}}={\frac {\nu \Sigma _{f}^{F}}{\Sigma _{a}^{F}}}}$	1.65
${\displaystyle f}$	Thermal utilization factor	⁠neutrons absorbed by the fuel isotope/neutrons absorbed anywhere⁠	${\displaystyle f={\frac {\Sigma _{a}^{F}}{\Sigma _{a}}}}$	0.71
${\displaystyle p}$	Resonance escape probability	⁠fission neutrons slowed to thermal energies without absorption/total fission neutrons⁠	${\displaystyle p\approx \mathrm {exp} \left(-{\frac {\sum \limits _{i=1}^{N}N_{i}I_{r,A,i}}{\left({\overline {\xi }}\Sigma _{p}\right)_{mod}}}\right)}$	0.87
${\displaystyle \varepsilon }$	Fast fission factor (epsilon)	⁠total number of fission neutrons/number of fission neutrons from just thermal fissions⁠	${\displaystyle \varepsilon \approx 1+{\frac {1-p}{p}}{\frac {u_{f}\nu _{f}P_{FAF}}{f\nu _{t}P_{TAF}P_{TNL}}}}$	1.02
${\displaystyle P_{FNL}}$	Fast non-leakage probability	⁠number of fast neutrons that do not leak from reactor/number of fast neutrons produced by all fissions⁠	${\displaystyle P_{FNL}\approx \mathrm {exp} \left(-{B_{g}}^{2}\tau _{th}\right)}$	0.97
${\displaystyle P_{TNL}}$	Thermal non-leakage probability	⁠number of thermal neutrons that do not leak from reactor/number of thermal neutrons produced by all fissions⁠	${\displaystyle P_{TNL}\approx {\frac {1}{1+{L_{th}}^{2}{B_{g}}^{2}}}}$	0.99

The symbols are defined as:^[2]

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

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.

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