The only practical way to dispose of actinides is to induce their fission. Fission is accompanied both by energy and neutron production. However, several neutron captures may be necessary before fission occurs, so that the net neutron number necessary for actinide incineration will be Ncap + (1 — v) where Ncap is the number of captures before fission and v the number of fission neutrons. The number of neutrons required depends on the neutron flux magnitude as well as on how hard it is. Let us consider one nucleus of species j(Zj, Aj). It can suffer fission with average cross­section o(f, capture a neutron with average cross-section (here nucleus k is Zk = Zj, Ak = Aj + 1) or decay to several possible other nuclei k, with partial decay rates Xjk (here nucleus k is (Zk = Zj + 1, Ak = Aj), (Zk = Zj — 1, Ak = Aj), (Zk = Zj — 2, Ak = Aj — 4) depending on the type

of radioactivity involved). The fission probability reads

The production of nucleus k from nucleus j can be defined as

Starting with one nucleus i, the number of nuclei j which are ultimately produced is given by the system

yj = X Akjyk + Sij. ( 3.144)

k

The Kronecker symbol Sy expresses the fact that, initially, there was one nucleus i. Knowing the yj, it is possible to compute the number of neutrons necessary to incinerate nucleus i:

Di = X RaPja)yj ( 3.145)

j, a

where the set f yjg is the solution of system (3.144), Ra is the neutron balance for reaction a (fission, capture or decay) and Р(a’) is the reduced transition

rate for reaction a and nucleus j. The values of Ra are given in table 3.9. The expression of D was first given in a slightly different form by Salvatores and Zaetta [38], and generalized to mixtures of nuclei. Table 3.10 gives values of D for important nuclei, as well as for spent fuel mixtures.

Table 3.10 shows [38] that incineration by fast neutrons is always a net neutron producer. This is due to the fact that fission cross-sections of fertile nuclei, which are very small or vanishing for thermal neutrons, are large for fast neutrons. The table also shows under which conditions breeding can be obtained from 232Th and 238U. The protoactinium effect is clearly visible for high thermal fluxes where its extraction is, clearly, mandatory. While, with a moderate flux, breeding can be obtained for 232Th for both a thermal and a fast flux, only a fast flux allows breeding for 238U.

Neutron balance is not the only parameter that should be considered for the choice of a particular system with the aim of waste incineration. The half-life of the waste in the neutron flux is also very important since it determines the inventory needed to reach a specified transmutation rate and the time it takes to get rid of the waste.