Neutron multiplying assemblies.

In nuclear reactors the fission of a nucleus results from a neutron absorption. This fission is accompanied with the emission of $\nu$ neutrons, with $\nu$ between 2.2 and 3., depending on the fissioning species.

These neutrons, in turn, may induce additional fissions, and, thus, produce new neutrons. However, each neutron does not produce a fission. It may be absorbed either in a non-fissile or in a fissile nucleus without fission of the said(fission probability after neutron capture by a fissile nucleus is never 100%). A neutron created in a medium(which we first consider infinite) with fissile nucleus will, thus, give birth to $k_{\infty }$ second generation neutrons. The number of neutrons of the third generation will be $k_{\infty}^{2}$ and that of the n generation $k_{\infty}^{n-1}$ . Each neutron generation is the result of a neutron producing nuclear reaction which can be a fission or, more rarely, a (n,xn) reaction. The total number of neutrons following the apparition of a neutron in the multiplying medium will be^4.2:

$$n_{chain}=1+k_{\infty}+k_{\infty}^{2}+...+k_{\infty}^{n}+...=\frac {1}{1-k_{\infty}}$$ (4.13)

The total number of neutrons created in the medium per source neutron is simply $k_{\infty}n_{chain}$. One defines a neutronic ``gain'' as the ratio of the total number of neutrons (source +created) to the number of source neutrons. This gain is then $\frac{1}{1-k_{\infty}}$. Since all neutrons are, ultimately, absorbed, the number of absorption reactions is, thus, n_reac=n_chain. For finite media one has to replace $k_{\infty }$ by an effective value of k_eff ^4.3 which is less than $k_{\infty }$ due to neutrons escaping from the system. One should also consider local values k_s dependent on the specific location of the apparition of the initial neutron. If k_eff is larger than unity the reaction diverges, i.e. from one initial neutron one obtains a final number of neutrons going to infinity. A controlled divergence allows to start a reactor. When uncontrolled it leads to a criticality accident like at Tchernobyl. Of course, in the nuclear weapons case, the divergence is aimed at. When k_eff is kept equal to unity one obtains a critical reactor. The possibility to keep precisely the condition k_eff=1 is due to the presence of a small fraction of delayed neutrons ^4.4 which allow correction of deviation of the criticality coefficient k_eff from unity. If k_eff is less than unity an incident neutron gives birth to a finite number of secondary neutrons. The medium is said to be multiplying. The multiplication factor is $\frac{1}{(1-k_{eff})}$ .