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

These neutrons, in turn, may induce 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 absorption by a fissile nucleus is never 100%). A neutron created in a medium (which we first consider infinite) containing fissile nuclei will give birth to ko second-generation neutrons. The number of neutrons of the third generation will be k1 and that of generation n, kO—1. Each neutron generation is the result of a neutron-producing nuclear reaction which can be a fission or, more rarely, an (n, xn) reaction. The total number of neutrons following the apparition of a neutron in the multiplying medium will be[16]

1

nchain = 1 + ki + koo + ‘"+k1o + ‘" = 1 _ k. (3.71)

1 ko

The total number of neutrons created in the medium per source neutron is simply konchain. 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 1/(1 — ko). Since all neutrons are ultimately absorbed, the number of absorption reactions is thus nreac = nchain. For finite media one has to replace ko by an effective value of kef which is less than ko due to neutrons escaping from the system. One should also consider local values, ks, dependent on the specific location of the apparition of the initial neutron. If keff is larger than unity the reaction diverges, i. e. from

one initial neutron the final number of neutrons goes to infinity. A controlled divergence allows one to start a reactor. When uncontrolled it leads to a criticality accident as in Chernobyl. Of course, in the case of nuclear weapons, the divergence is sought. When keff is kept equal to unity one obtains a critical reactor. The possibility to keep precisely the condition keff = 1 is due to the presence of a small fraction of delayed neutrons[17] which allow time to compensate for deviation of the criticality coefficient keff from unity. If keff 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 1/(1 — keff).