Six factor formula#Multiplication

{{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.

class="wikitable" style="text-align: center;" |+ Six-factor formula: $k\; =\; \backslash eta\; f\; p\; \backslash varepsilon\; P\_\{FNL\}\; P\_\{TNL\}\; =\; k\_\{\backslash infty\}\; P\_\{FNL\}\; P\_\{TNL\}${{cite book |last=Duderstadt |first=James |author2=Hamilton, Louis |title=Nuclear Reactor Analysis |year=1976 |publisher=John Wiley & Sons, Inc |isbn=0-471-22363-8 }} ! Symbol ! Name ! Meaning ! Formula ! Typical thermal reactor value

$\backslash eta$ | Thermal fission factor (eta) | {{sfrac|neutrons produced from fission|absorption in fuel isotope}} | $\backslash eta\; =\; \backslash frac\{\backslash nu\; \backslash sigma\_f^F\}\{\backslash sigma\_a^F\}\; =\; \backslash frac\{\backslash nu\; \backslash Sigma\_f^F\}\{\backslash Sigma\_a^F\}$ | 1.65

$f$ | Thermal utilization factor | {{sfrac|neutrons absorbed by the fuel isotope|neutrons absorbed anywhere}} | $f\; =\; \backslash frac\{\backslash Sigma\_a^F\}\{\backslash Sigma\_a\}$ | 0.71

$p$ | Resonance escape probability | {{sfrac|fission neutrons slowed to thermal energies without absorption|total fission neutrons}} | $p\; \backslash approx\; \backslash mathrm\{exp\}\; \backslash left(\; -\backslash frac\{\backslash sum\backslash limits\_\{i=1\}^\{N\}\; N\_i\; I\_\{r,A,i\}\}\{\backslash left(\; \backslash overline\{\backslash xi\}\; \backslash Sigma\_p\; \backslash right)\_\{mod\}\}\; \backslash right)$ | 0.87

$\backslash varepsilon$ | Fast fission factor (epsilon) | {{sfrac|total number of fission neutrons|number of fission neutrons from just thermal fissions}} | $\backslash varepsilon\; \backslash approx\; 1\; +\; \backslash frac\{1-p\}\{p\}\backslash frac\{u\_f\; \backslash nu\_f\; P\_\{FAF\}\}\{f\; \backslash nu\_t\; P\_\{TAF\}\; P\_\{TNL\}\}$ | 1.02

$P\_\{FNL\}$ | Fast non-leakage probability | {{sfrac|number of fast neutrons that do not leak from reactor|number of fast neutrons produced by all fissions}} | $P\_\{FNL\}\; \backslash approx\; \backslash mathrm\{exp\}\; \backslash left(\; -\{B\_g\}^2\; \backslash tau\_\{th\}\; \backslash right)$ | 0.97

$P\_\{TNL\}$ | Thermal non-leakage probability | {{sfrac|number of thermal neutrons that do not leak from reactor|number of thermal neutrons produced by all fissions}} | $P\_\{TNL\}\; \backslash approx\; \backslash frac\{1\}\{1+\{L\_\{th\}\}^2\; \{B\_g\}^2\}$ | 0.99

The symbols are defined as:{{cite book |last=Adams |first=Marvin L. |title=Introduction to Nuclear Reactor Theory |year=2009 |publisher=Texas A&M University}}

• $\backslash nu$, $\backslash nu\_f$ and $\backslash nu\_t$ are the average number of neutrons produced per fission in the medium (2.43 for uranium-235).
• $\backslash sigma\_f^F$ and $\backslash sigma\_a^F$ are the microscopic fission and absorption cross sections for fuel, respectively.
• $\backslash Sigma\_a^F$ and $\backslash Sigma\_a$ are the macroscopic absorption cross sections in fuel and in total, respectively.
• $\backslash Sigma\_f^F$ 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.
• $I\_\{r,A,i\}\; =\; \backslash int\_\{E\_\{th\}\}^\{E\_0\}\; dE\text{'}\; \backslash frac\{\backslash Sigma\_p^\{mod\}\}\{\backslash Sigma\_t(E\text{'})\}\; \backslash frac\{\backslash sigma\_a^i(E\text{'})\}\{E\text{'}\}$
• $\backslash overline\{\backslash xi\}$ 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.
• $\{L\_\{th\}\}^2\; =\; \backslash frac\{D\}\{\backslash Sigma\_\{a,th\}\}$
• $\backslash tau\_\{th\}$ is the age to thermal.
• $\backslash tau\; =\; \backslash int\_\{E\_\{th\}\}^\{E\text{'}\}\; dE$ \frac{1}{E} \frac{D(E)}{\overline{\xi} \left[ D(E) {B_g}^2 + \Sigma_t(E') \right]}
• $\backslash tau\_\{th\}$ is the evaluation of $\backslash tau$ where $E\text{'}$ is the energy of the neutron at birth.