Physics:Six factor formula

Short description: Formula used to calculate nuclear chain reaction growth rate

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: [math]\displaystyle{ k = \eta f p \varepsilon P_{FNL} P_{TNL} = k_{\infty} P_{FNL} P_{TNL} }[/math]

Symbol	Name	Meaning	Formula	Typical thermal reactor value
[math]\displaystyle{ \eta }[/math]	Thermal fission factor (eta)	neutrons produced from fission/absorption in fuel isotope	[math]\displaystyle{ \eta = \frac{\nu \sigma_f^F}{\sigma_a^F} }[/math]	1.65
[math]\displaystyle{ f }[/math]	Thermal utilization factor	neutrons absorbed by the fuel isotope/neutrons absorbed anywhere	[math]\displaystyle{ f = \frac{\Sigma_a^F}{\Sigma_a} }[/math]	0.71
[math]\displaystyle{ p }[/math]	Resonance escape probability	fission neutrons slowed to thermal energies without absorption/total fission neutrons	[math]\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) }[/math]	0.87
[math]\displaystyle{ \varepsilon }[/math]	Fast fission factor (epsilon)	total number of fission neutrons/number of fission neutrons from just thermal fissions	[math]\displaystyle{ \varepsilon \approx 1 + \frac{1-p}{p}\frac{u_f \nu_f P_{FAF}}{f \nu_t P_{TAF} P_{TNL}} }[/math]	1.02
[math]\displaystyle{ P_{FNL} }[/math]	Fast non-leakage probability	number of fast neutrons that do not leak from reactor/number of fast neutrons produced by all fissions	[math]\displaystyle{ P_{FNL} \approx \mathrm{exp} \left( -{B_g}^2 \tau_{th} \right) }[/math]	0.97
[math]\displaystyle{ P_{TNL} }[/math]	Thermal non-leakage probability	number of thermal neutrons that do not leak from reactor/number of thermal neutrons produced by all fissions	[math]\displaystyle{ P_{TNL} \approx \frac{1}{1+{L_{th}}^2 {B_g}^2} }[/math]	0.99

The symbols are defined as:

• [math]\displaystyle{ \nu }[/math], [math]\displaystyle{ \nu_f }[/math] and [math]\displaystyle{ \nu_t }[/math] are the average number of neutrons produced per fission in the medium (2.43 for uranium-235).
• [math]\displaystyle{ \sigma_f^F }[/math] and [math]\displaystyle{ \sigma_a^F }[/math] are the microscopic fission and absorption cross sections for fuel, respectively.
• [math]\displaystyle{ \Sigma_a^F }[/math] and [math]\displaystyle{ \Sigma_a }[/math] are the macroscopic absorption cross sections in fuel and in total, respectively.
• [math]\displaystyle{ N_i }[/math] is the number density of atoms of a specific nuclide.
• [math]\displaystyle{ I_{r,A,i} }[/math] is the resonance integral for absorption of a specific nuclide.
  • [math]\displaystyle{ I_{r,A,i} = \int_{E_{th}}^{E_0} dE' \frac{\Sigma_p^{mod}}{\Sigma_t(E')} \frac{\sigma_a^i(E')}{E'} }[/math].
• [math]\displaystyle{ \overline{\xi} }[/math] is the average lethargy gain per scattering event.
  • Lethargy is defined as decrease in neutron energy.
• [math]\displaystyle{ u_f }[/math] (fast utilization) is the probability that a fast neutron is absorbed in fuel.
• [math]\displaystyle{ P_{FAF} }[/math] is the probability that a fast neutron absorption in fuel causes fission.
• [math]\displaystyle{ P_{TAF} }[/math] is the probability that a thermal neutron absorption in fuel causes fission.
• [math]\displaystyle{ {B_g}^2 }[/math] is the geometric buckling.
• [math]\displaystyle{ {L_{th}}^2 }[/math] is the diffusion length of thermal neutrons.
  • [math]\displaystyle{ {L_{th}}^2 = \frac{D}{\Sigma_{a,th}} }[/math].
• [math]\displaystyle{ \tau_{th} }[/math] is the age to thermal.
  • [math]\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]} }[/math].
  • [math]\displaystyle{ \tau_{th} }[/math] is the evaluation of [math]\displaystyle{ \tau }[/math] where [math]\displaystyle{ E' }[/math] 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