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.

The symbols are defined as:

• ${\displaystyle \nu }$, ${\displaystyle \nu _{f}}$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