We derive an expression for к1. For simplicity we assume that the only possible reactions are scattering, capture and fission, neglecting such reac­tions as (n, xnyp). Since the number к1 is the number of secondary neutrons produced, on the average, following absorption of the primary neutron one can write

where (v) is the average number of neutrons emitted per fission. One should note that this expression is of interest only if к1 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 has to be made. A quantitative expression for к1 can be obtained as follows. One considers that, at a given time, the medium is immersed in a neutron flux ‘(E, r), where we indicate a spatial dependence of the flux to take into account any possible inhomogeneities of the medium. Equivalent to equation (3.72) we can write

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

If we consider a medium involving n nuclei, and use cross-sections averaged over r and E, as in equation (3.74), we can write

E

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

where we have used the relation v = v(af/aa) = v(Xf/Xa), since it is clearly valid when there is only one fissile species. It follows that

+ Eafert)

The number of fissile nuclei per unit volume disappearing per unit time is f while the number of such nuclei created following neutron capture by fertile nuclei is sifert). Thus the breeding condition is that s[fert) > xifis). It follows that breeding is only possible if v > 2k1, and in particular, for critical systems, v > 2.

It is often useful and quite common to write kx as a product of four factors,

ki= epf v (3.77)

where є is the enhancement factor due to fertile nuclei fissions occurring by fast neutrons, f the probability that the neutron absorption occurs in the fuel, p the probability for a neutron absorbed in the fuel to be specifically absorbed by a fissile nucleus, and v the mean number of neutrons emitted following an absorption in a fissile nucleus. While these definitions are valid for fast reac­tors, they are different for thermal reactors: є 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 absorbed in the fuel, and v the number of neutrons emitted after absorption in one of the fuel nuclei (both fertile and fissile).