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Six factor formula

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.

Symbol	Name	Formula	Typical thermal reactor value
\eta	Thermal fission factor (eta)	\eta = \frac{\nu \sigma_f^F}{\sigma_a^F} = \frac{\nu \Sigma_f^F}{\Sigma_a^F}	1.65
f	Thermal utilization factor	f = \frac{\Sigma_a^F}{\Sigma_a}	0.71
p	Resonance escape probability	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	Fast fission factor (epsilon)	\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}	Fast non-leakage probability	P_{FNL} \approx \mathrm{exp} \left( -{B_g}^2 \tau_{th} \right)	0.97
P_{TNL}	Thermal non-leakage probability	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.
\Sigma_a^F and \Sigma_a are the macroscopic absorption cross sections in fuel and in total, respectively.
\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} = \int_{E_{th}}^{E_0} dE' \frac{\Sigma_p^{mod}}{\Sigma_t(E')} \frac{\sigma_a^i(E')}{E'}
\overline{\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}{\Sigma_{a,th}}, where D is the diffusion coefficient.
\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): :

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 , the chain reaction is critical and the neutron population will remain constant.

References

Duderstadt, James. (1976). "Nuclear Reactor Analysis". John Wiley & Sons, Inc.
Adams, Marvin L.. (2009). "Introduction to Nuclear Reactor Theory". Texas A&M University.

Info: Wikipedia Source

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