Expression of $k_{\infty }$

We derive an expression for $k_{\infty}.$ For simplicity we assume that the only possible reactions are scattering, capture and fission, neglecting such reactions as (n,xnyp). Since the number $k_{\infty }$ is the number of secondary neutrons produced, on the average, following absorption of the primary neutron one can write:

$$k_{\infty}=<\nu>\frac{\mbox{probability for fission after absorption} }{\mbox{probability for absorption}}$$ (4.14)

where $<\nu>$ is the average number of neutrons emitted per fission. One should note that this expression is only of interest if $k_{\infty }$ remains constant with time during the multiplication process, i.e. if the neutron spectrum itself remains time invariant. In particular this requires that the neutron of the first generation have a spectrum similar to that of fission neutrons. If this is not the case, a correction to the treatment has to be made. An equivalent form allows to obtain a quantitative expression for $k_{\infty}.$ One considers that, at a given time, the medium is immersed in a neutron flux $\varphi(E,\overrightarrow{r} ),$ where we indicate a spatial dependance of the flux to take into account any possible inhomogeneities of the medium. Equivalent to equation 3.14 we can write:

$$k_{\infty}=<\nu>\frac{\mbox{number of fissions after absorption}}{\mbox{number of absorptions}}$$ (4.15)

In this form we can obtain the expression in terms of cross-sections:

$$k_{\infty}=<\nu>\frac{\iiiint\Sigma_{f}(E,\overrightarrow{r} ... ...(E,\overrightarrow{r} )\varphi(E,\overrightarrow{r} )dEd^{3}r}$$ (4.16)

If we consider a medium involving n nuclei, and use cross sections averaged over $\overrightarrow{r}$ and E, like in 3.16, we can write:

$$k_{\infty}=\frac{\underset{i}{\sum}\nu_{i}\Sigma_{f}^{(i)}}{\underset{i}{\sum }\Sigma_{a}^{(i)}}$$ (4.17)

Consider the simple case where the medium involves only three types of nuclei, one fissile, one fertile and one capturing. Then,

$$k_{\infty}\ =\ \nu\ \frac{\Sigma_{f}^{(fis)}}{\Sigma_{a}^{(fi... ...}}{\Sigma _{a}^{(fis)}+\Sigma_{a}^{(fert)}+\Sigma_{a}^{(abs)}}$$ (4.18)

where we have used the relation $\eta=\nu\frac{\sigma_{f}}{\sigma_{a}} =\nu\frac{\Sigma_{f}}{\Sigma_{a}}$, since it is clearly valid when there is only one fissile species. It follows that $k_{\infty}<\eta\frac{\Sigma _{a}^{(fiss)}}{\Sigma_{a}^{(fis)}+\Sigma_{a}^{(fert)}}$. The number of fissile nuclei per unit volume disappearing per unit time is $\Sigma_{a}^{(fis)}$ while the number of such nuclei created following neutron capture by fertile nuclei is $\Sigma_{a}^{(fert)}.$ Thus the breeding condition is that $\Sigma_{a}^{(fert)}>\Sigma_{a}^{(fis)}$. It follows that breeding is only possible if $\eta>2k_{\infty}$, and in particular, for critical systems, $\eta>2$.

It is often useful and quite usual to write $k_{\infty }$ as a product of four factors

$$k_{\infty}=\varepsilon pf\eta$$ (4.19)

where $\varepsilon$ is the enhancement factor due to fissions of fertile nuclei occuring by fast neutrons, f the probability that the neutron capture occurs in the fuel, p the probability for a neutron captured in the fuel to be specifically captured by a fissile nucleus, and $\eta$ the mean number of neutrons emitted following a capture in a fissile nucleus. While these definitions are valid for fast reactors, they are modified for thermal reactors: $\varepsilon$ becomes the enhancement factor due to fissions of fertile and fissile nuclei by fast neutrons, p the probability that the neutron escapes capture during the slowing down process(especially in the large resonances of the fertile nuclei), f the fraction of thermal neutrons captured in the fuel, and $\eta$ the number of neutrons emitted after capture in one of the fuel nuclei (both fertile and fissile).