Method of quantum characteristics

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

$k=\eta fpϵP_{FNL}{P}_{TNL}$

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.	$f=\frac{{\mathrm{\Sigma }}_a^F}{{\mathrm{\Sigma }}_a}$	0.71
$p$	The resonance escape probability	Fraction of fission neutrons that manage to slow down from fission to thermal energies without being absorbed.	$p\approx \mathrm{e}\mathrm{x}\mathrm{p}(-\frac{\sum _{i=1}^N{N}_i{I}_{r,A,i}}{{\left(\bar{\xi }{\mathrm{\Sigma }}_p\right)}_{mod}})$	0.87
$ϵ$	The fast fission factor (Epsilon)	$\frac{\text{total number of fission neutrons}}{\text{number of fission neutrons from just thermal fissions}}$	$ϵ\approx 1+\frac{1-p}{p}\frac{u_f{\nu }_f{P}_{FAF}}{f{\nu }_t{P}_{TAF}{P}_{TNL}}$	1.02
$P_{FNL}$	The fast non-leakage probability	The probability that a fast neutron will not leak out of the system.	$P_{FNL}\approx \mathrm{e}\mathrm{x}\mathrm{p}(-{B_g}^2{\tau }_{th})$	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:

• $\nu$, ${\nu }_f$ 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.
• $N_i$ is the number density of atoms of a specific nuclide.
• $I_{r,A,i}$ is the resonance integral for absorption of a specific nuclide.
  • $I_{r,A,i}=\underset{E_{th}}{\overset{E_0}{\int }}d{E}^{\prime }\frac{{\mathrm{\Sigma }}_p^{mod}}{{\mathrm{\Sigma }}_t(E^{\prime })}\frac{{\sigma }_a^i(E^{\prime })}{E^{\prime }}$.
• $\bar{\xi }$ (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.
  • ${L_{th}}^2=\frac{D}{{\mathrm{\Sigma }}_{a,th}}$.
• ${\tau }_{th}$ 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):

$k=\frac{\text{number of neutrons in one generation}}{\text{number of neutrons in preceding generation}}$

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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.