# 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 formula is[1]

$k = \eta f p \varepsilon P_{FNL} P_{TNL}$
Symbol Name Meaning Formula Typical Thermal Reactor Value
$\eta$ Thermal Fission Factor (Eta) The number of fission neutrons produced per absorption in the fuel. $\eta = \frac{\nu \sigma_f^F}{\sigma_a^F}$ 1.65
$f$ The thermal utilization factor Probability that a neutron that gets absorbed does so in the fuel material. $f = \frac{\Sigma_a^F}{\Sigma_a}$ 0.71
$p$ The resonance escape probability Fraction of fission neutrons that manage to slow down from fission to thermal energies without being absorbed. $p \approx \mathrm{exp} \left( -\frac{\sum\limits_{i=1}^{N} N_i I_{r,A,i}}{\left( \overline{\xi} \Sigma_p \right)_{mod}} \right)$ 0.87
$\varepsilon$ The fast fission factor (Epsilon)
$\tfrac{\mbox{total number of fission neutrons}}{\mbox{number of fission neutrons from just thermal fissions}}$
$\varepsilon \approx 1 + \frac{1-p}{p}\frac{u_f \nu_f P_{FAF}}{f \nu_t P_{TAF} P_{TNL}}$ 1.02
$P_{FNL}$ The fast non-leakage probability The probability that a fast neutron will not leak out of the system. $P_{FNL} \approx \mathrm{exp} \left( -{B_g}^2 \tau_{th} \right)$ 0.97
$P_{TNL}$ The thermal non-leakage probability The probability that a thermal neutron will not leak out of the system. $P_{TNL} \approx \frac{1}{1+{L_{th}}^2 {B_g}^2}$ 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.
• $I_{r,A,i}$ is the resonance integral for absorption of a specific nuclide.
• $I_{r,A,i} = \int_{E_{th}}^{E_0} dE' \frac{\Sigma_p^{mod}}{\Sigma_t(E')} \frac{\sigma_a^i(E')}{E'}$.
• $\overline{\xi}$ (often referred to as worm-bar or squigma-bar) 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):

$k = \frac{\mbox{number of neutrons in one generation}}{\mbox{number of neutrons in preceding generation}}$

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.