• Source: 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:




      ν


      {\displaystyle \nu }

      ,




      ν

      f




      {\displaystyle \nu _{f}}

      and




      ν

      t




      {\displaystyle \nu _{t}}

      are the average number of neutrons produced per fission in the medium (2.43 for uranium-235).





      σ

      f


      F




      {\displaystyle \sigma _{f}^{F}}

      and




      σ

      a


      F




      {\displaystyle \sigma _{a}^{F}}

      are the microscopic fission and absorption cross sections for fuel, respectively.





      Σ

      a


      F




      {\displaystyle \Sigma _{a}^{F}}

      and




      Σ

      a




      {\displaystyle \Sigma _{a}}

      are the macroscopic absorption cross sections in fuel and in total, respectively.





      Σ

      f


      F




      {\displaystyle \Sigma _{f}^{F}}

      is the macroscopic fission cross-section.





      N

      i




      {\displaystyle N_{i}}

      is the number density of atoms of a specific nuclide.





      I

      r
      ,
      A
      ,
      i




      {\displaystyle I_{r,A,i}}

      is the resonance integral for absorption of a specific nuclide.





      I

      r
      ,
      A
      ,
      i


      =

      ∫


      E

      t
      h





      E

      0




      d

      E
      ′




      Σ

      p


      m
      o
      d




      Σ

      t


      (

      E
      ′

      )







      σ

      a


      i


      (

      E
      ′

      )


      E
      ′





      {\displaystyle I_{r,A,i}=\int _{E_{th}}^{E_{0}}dE'{\frac {\Sigma _{p}^{mod}}{\Sigma _{t}(E')}}{\frac {\sigma _{a}^{i}(E')}{E'}}}







      ξ
      ¯




      {\displaystyle {\overline {\xi }}}

      is the average lethargy gain per scattering event.
      Lethargy is defined as decrease in neutron energy.





      u

      f




      {\displaystyle u_{f}}

      (fast utilization) is the probability that a fast neutron is absorbed in fuel.





      P

      F
      A
      F




      {\displaystyle P_{FAF}}

      is the probability that a fast neutron absorption in fuel causes fission.





      P

      T
      A
      F




      {\displaystyle P_{TAF}}

      is the probability that a thermal neutron absorption in fuel causes fission.







      B

      g




      2




      {\displaystyle {B_{g}}^{2}}

      is the geometric buckling.







      L

      t
      h




      2




      {\displaystyle {L_{th}}^{2}}

      is the diffusion length of thermal neutrons.







      L

      t
      h




      2


      =


      D

      Σ

      a
      ,
      t
      h






      {\displaystyle {L_{th}}^{2}={\frac {D}{\Sigma _{a,th}}}}






      τ

      t
      h




      {\displaystyle \tau _{th}}

      is the age to thermal.




      τ
      =

      ∫


      E

      t
      h





      E
      ′



      d

      E
      ″



      1

      E
      ″






      D
      (

      E
      ″

      )




      ξ
      ¯



      [

      D
      (

      E
      ″

      )



      B

      g




      2


      +

      Σ

      t


      (

      E
      ′

      )

      ]






      {\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]}}}






      τ

      t
      h




      {\displaystyle \tau _{th}}

      is the evaluation of



      τ


      {\displaystyle \tau }

      where




      E
      ′



      {\displaystyle E'}

      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


      References

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