Six factor formula

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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[edit]

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.

See also[edit]

References[edit]

  1. ^ Duderstadt, James; Hamilton, Louis (1976). Nuclear Reactor Analysis. John Wiley & Sons, Inc. ISBN 0-471-22363-8. 
  2. ^ Adams, Marvin L. (2009). Introduction to Nuclear Reactor Theory. Texas A&M University.