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: $k=\eta fp\epsilon P_{FNL}{P}_{TNL}=k_{\mathrm{\infty }}{P}_{FNL}{P}_{TNL}$

Symbol	Name	Meaning	Formula	Typical thermal reactor value
$\eta$	Thermal fission factor (eta)	⁠neutrons produced from fission/absorption in fuel isotope⁠	$\eta =\frac{\nu {\sigma }_f^F}{{\sigma }_a^F}=\frac{\nu {\mathrm{\Sigma }}_f^F}{{\mathrm{\Sigma }}_a^F}$	1.65
$f$	Thermal utilization factor	⁠neutrons absorbed by the fuel isotope/neutrons absorbed anywhere⁠	$f=\frac{{\mathrm{\Sigma }}_a^F}{{\mathrm{\Sigma }}_a}$	0.71
$p$	Resonance escape probability	⁠fission neutrons slowed to thermal energies without absorption/total fission neutrons⁠	$p\approx \mathrm{e}\mathrm{x}\mathrm{p}(-\frac{\sum _{i=1}^N{N}_i{I}_{r,A,i}}{{\left(\bar{\xi }{\mathrm{\Sigma }}_p\right)}_{mod}})$	0.87
$\epsilon$	Fast fission factor (epsilon)	⁠total number of fission neutrons/number of fission neutrons from just thermal fissions⁠	$\epsilon \approx 1+\frac{1-p}{p}\frac{u_f{\nu }_f{P}_{FAF}}{f{\nu }_t{P}_{TAF}{P}_{TNL}}$	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	$P_{FNL}\approx \mathrm{e}\mathrm{x}\mathrm{p}(-{B_g}^2{\tau }_{th})$	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.
• ${\mathrm{\Sigma }}_a^F$ and ${\mathrm{\Sigma }}_a$ are the macroscopic absorption cross sections in fuel and in total, respectively.
• ${\mathrm{\Sigma }}_f^F$ 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.
  • $I_{r,A,i}=\underset{E_{th}}{\overset{E_0}{\int }}d{E}^{\prime }\frac{{\mathrm{\Sigma }}_p^{mod}}{{\mathrm{\Sigma }}_t(E^{\prime })}\frac{{\sigma }_a^i(E^{\prime })}{E^{\prime }}$
• $\bar{\xi }$ 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.
  • ${L_{th}}^2=\frac{D}{{\mathrm{\Sigma }}_{a,th}}$
• ${\tau }_{th}$ is the age to thermal.
  • $\tau =\underset{E_{th}}{\overset{E^{\prime }}{\int }}d{E}^{″}\frac{1}{E^{″}}\frac{D(E^{″})}{\bar{\xi }[D(E^{″}){B_g}^2+{\mathrm{\Sigma }}_t(E^{\prime })]}$
  • ${\tau }_{th}$ is the evaluation of $\tau$ where $E^{\prime }$ 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.

See also

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