From the preceding, we conclude that fast spectra are more efficient than thermal spectra for MA incineration, because of their high fissility to fast neutrons. We also get the result that, for the same fission densities, the incineration rates are similar for fast and thermal spectra. This is surprising since, for plutonium incineration, for example, it is usually claimed that the much higher fission cross-sections with low-energy neutrons lead to higher incineration rates in thermal reactors [133]. Indeed, while the fission density is w = nff jo-f у (here nf j is the density of fissile nuclei, afj their microscopic average fission cross-section and ‘ the neutron flux), the lifetime with respect to fission is Tfj = nfj/w = 1/afj’. It follows that small life­times can be obtained for large neutron fluxes and/or large fission cross­sections, even if the fission rates are kept small by limiting the fissile nucleus density. Thus, associating the large fission cross-sections for low — energy neutrons to small fissile density, one expects to obtain much smaller fission lifetimes for thermal than for fast neutrons with the same fission densities.

We want to understand the apparent contradiction between such con­siderations, which seem to hold in the case of plutonium, and the behaviour of minor actinide fuels. We first discuss the maximization of incineration rates in critical reactors.

The maximization of incineration rates

We consider a sketchy homogeneous infinite reactor with only two com­ponents:

1. The fuel, characterized by its atomic density nfuel, absorption cross-section a[fuelj, and its neutron multiplication coefficient kfuel > 1.

2. The coolant, characterized by its atomic density ncool, and its absorption

,• (cool)

cross-section ^a.

The aggregate reactor is characterized by its atomic density nreac = nfuel + ncool, its absorption cross-section

_(reac) nfuel _(fuel) , ncool _(cool)

oa — oa + oa

nreac nreac

and an effective multiplication coefficient

Note that, because of the condensed nature of the components of all practical reactors’ designs, it is not possible to make the value of nreac vary very much. For example, for water, the atomic density is 1023/cm3, while for lead it is 0.3 x 10 /cm and for uranium 0.6 x 10 /cm. In contrast, nfuel can be varied within large limits provided the reactor remains critical. We define the atomic fraction of the fuel x — nfuel/nreac. The criticality condition kreac — 1 allows us to express x as

Aside from the criticality condition, it seems reasonable to assume that the fission density is limited to a specific value w:

In the case of plutonium or other fissile mixtures, it appears that the main difference between fast and thermal reactors lies in the ratio oifuel)/oicool). There is a clear advantage to using coolants with small absorption cross­sections. As examples, for heavy water ncoolo(cool) — 4 x 10-5 for thermal reactors and ncooloicool) — 3 x 10-4 for lead coolant and fast spectrum. Thermal neutron fuel absorption cross-sections exceed 500 barns while they range around 2 barns only for fast neutrons. It follows that, for fissile mixtures, incineration rates with thermal neutrons could, in principle, be three orders of magnitude larger than those with fast neutrons. For thermal neutrons the extremely high incineration rates (lifetimes of a few hours) could only be reached with liquid fuels, allowing very fast purification and replenishment. Note that these high incineration rates are made possible
by the high dilution of the fuel in the coolant, so that the small value of «fuel has to be compensated by a high value of the neutron flux

The above considerations are not valid for minor actinide mixtures. In this case the major difference between thermal — and fast-neutron incineration is that of the corresponding fuel multiplication factors, as seen in figure 11.12. The subcritical nature of the MA fuel with thermal neutrons makes fuel dilution counterproductive since it would decrease the reactor multi­plication coefficient kreac below kfuel, and thus require higher accelerator current to keep the neutron flux constant. The optimum situation (which might be unrealistic because of insufficient cooling power) would then be nreac = nfuel. The incineration rate reduces to the first term of the left-hand side of equation (11.16), i. e.

w

Ainc =—— (11.17)

nreac

which means that it depends, essentially, on the fission density, and only weakly on the fissile nucleus density.