Even when a beam emittance is constant, its spatial extension can and usually does increase. This is easily seen by considering an ensemble of particles having initial positions between +d and —d and transversal velocities distributed uni­formly between +v and —v. As shown in figure III.5, at time t, particles with velocity +v will span positions between —d + vt and d + vt; those with velocity —v will span positions between —d — vt and d — vt. In phase space the initial rectangular volume ((—d, —v), (d, —v), (d, v), (—d, v)) transforms into a parallelepiped ((—d — vt, —v), (d — vt, —v), (d + vt, v), (—d + vt, v)). Thus, the beam spatial extension increases from 2d to 2(d + vt) while the phase space volume remains constant. Clearly, such an uncontrolled increase of the spatial extension of the beam is unacceptable in any accelerator, hence the need for spatial focusing.

The use of breeding or converter[6] reactors would change the picture consid­erably. Converters and breeders allow full use not only of the fissile 235U isotope, but also of the fertile U and Th isotopes. Thus, in principle, a 1 GWe reactor requires only 1 ton of natural uranium or an equivalent amount of thorium. This means that, at the current market cost, assuming a production capacity of 9000 GWe, corresponding to the most nuclear­intensive scenario of table 2.13, the reserves would amount to 2000 years for natural uranium and about four times more for thorium. In fact, the very efficient use of the uranium and thorium would allow the use of very low grade ores, including sea water uranium, which means that the resources would be practically unlimited. The mill tailings would also be considerably reduced by a factor more than 100.

While the plutonium present in spent fuels has to be considered a waste, it is the fissile material for breeders and converters. Only long-lived fission products (LLFPs) and minor actinides (MAs)^ can thus be considered as nuclear wastes. In the absence of specific transmutation of these wastes, their radiotoxicity, after a cool-down period of 300 years,* would be at least one order of magnitude smaller than that of the PWR spent fuels, for an equivalent energy production. Since fuel reprocessing is a prerequisite for any breeding or converting cycle, it is quite logical to consider the trans — mutating of LLFPs and MAs. We shall discuss this possibility in some detail below. It has been shown that the incineration of MAs and the transmutation of some of the most significant LLFPs are feasible. Nuclear wastes would then be reduced to the reprocessing losses. Modern reprocessing is claimed to have 99.9% efficiency in the recovery of plutonium and 99% in the recovery of MAs [38]. It would then be possible to reduce the total radiotoxi­city of the wastes by several orders of magnitude after a few hundred years of cooling. With such a reduction, long-term disposal might not be necessary, or at least will be considerably reduced.

While the reliability and safety of PWR reactors has been widely demon­strated in industrialized western countries,§ the experience with breeder or converter reactors is limited and ambiguous. By far the best known breeders are of the liquid metal fast reactor type. Practically all of these liquid metal reactors have used sodium as their coolant,[7] with the exception of several recent Russian submarine propulsion reactors which are cooled with a liquid lead-bismuth eutectic. While it seems that the records of the Russian sodium cooled reactors like BOR60, BN350 and BN600 are very good, the records of such western reactors are much more questionable. The small American reactor EBR2 worked satisfactorily until its final shutdown in 1995. Another American reactor, Enrico Fermi, could never run. The small French reactor RAPSODIE ran very nicely until its final shutdown. The 250 MWe Phenix reactor ran satisfactorily for ten years until unexplained reactivity fluctuations led to its shutdown. It has been started again recently at reduced power, pending safety improvements. The large, 1200 MWe Superphenix reactor was plagued by sodium leaks and administrative imbroglio until the decision was made to stop it indefinitely. The Japanese Monju reactor is, also, suffering from sodium leaks. It seems that the combination of increasingly stringent safety constraints and the use of the very reactive liquid sodium led to difficult running conditions. Further­more, the investment costs for a reactor like Superphenix are about twice as large as those necessary for a standard PWR reactor with the same power. The cost of electricity which would have been produced by Superphenix in normal running conditions would have been twice that produced by a PWR. Of course, it can be argued that, for an industrial-scale series of reactors, the investment cost as well as the fuel cost would have decreased.

In conclusion, while the interest of breeder reactors is clear in the hypoth­esis of an extension of nuclear power, one cannot consider the present type of sodium cooled reactor to be the only solution. In this context it is possible that hybrid reactors may help develop alternatives to sodium cooled reactors. They may also facilitate a switch to the thorium breeding cycle which would lead to a much reduced production of minor actinides.

In order to evaluate some of the possible advantages of hybrid reactors over critical reactors, we think it useful to review how these are controlled. It is essentially through the motion of neutron-absorbing control rods that the value of the criticality coefficient keff is modified. In discussions of the evolution of a reactor it is usual to use the reactivity p = (keff — 1)/keff. The time constant associated with the motion of control rods is, typically, measured in seconds. The time delay rnf between two neutron generations is much smaller, typically 10—7 s for fast reactors and 10 4s for thermal reactors [48]. Such numbers would imply a very fast evolution of the reactor, even for very small positive reactivities. As we saw in section 3.2.4, the reactor power increases exponentially with time t:

W (t) = W»“p( (1—pf (3Л01)

For p = 0:01 the power is multiplied by 2 after 70 neutron generations, i. e. less than 10 ps for fast reactors and less than 10 ms for thermal reactors! With such a fast rise in the reactor power, one might think that reactor control by control rods would be hopeless. In fact, the presence of a small fraction of delayed neutrons makes things tractable.

A collision between a neutron and an atom is affected by the thermal motion of the atom, and in most cases the collision is also affected by the presence of other atoms nearby.

Depending on the compound (essentially, moderators) and on the neutron energy, two treatments are done. For slow neutrons (En < 4eV) and for carbon, water, etc., the crystalline structure as well as effects of chemical binding have to be taken into account via an S(a,0) treatment[31] (these S(a, 0) tables are special sets of ENDF files). For higher energies or other compounds, a free gas thermal treatment is applied: the medium is assumed to be a free gas, and, in the range of atomic weight and neutron energy where thermal effects are significant, the elastic scattering cross­section at zero temperature is assumed to be independent of neutron energy and the reaction cross-sections are independent of temperature; the elastic scattering cross-section at temperature T is then given by

where A is the atomic weight and En the neutron energy.^

Then the collision is sampled in the target-at-rest frame and the target laboratory velocity is chosen according to the Maxwell law P(V )/V exp(—V2/kT). However, if the elastic scattering cross-section has been processed (say, by NJOY) at the desired temperature, this treatment is not applied. The temperature entered in MCNP is used only to modify (if

Figure 5.2. Selection of nucleus k from N.

needed) elastic scattering, and thus total, cross-sections; no other Doppler effect is taken into account via this temperature.

The energy loss due to electronic collisions is given by Bethe’s formula which reads, to a very good approximation,

with A, Z and p respectively the mass number, the charge and the mass density of the target nucleus and Zp, ft and E respectively the charge, velocity and energy of the projectile. The constant D = 0.3071 MeV cm2/g; I is the average ionization potential of target atoms, with approximate value I(Z) = 16Z0 9 eV. Also mec2 = 0.511 MeV.

Bethe’s formula does not let us obtain an analytic expression of the projectile range. A common approximation, which allows reasonable proton range estimates, is

Note that the stopping power per atom is proportional to Z, and indepen­dent of A and p. For a 1 GeV proton one gets an electronic range

A

Rel(E = 1 GeV) = 205 . (6.6)

PZ

Taking the examples of beryllium and lead we get, for a 1 GeV proton,

• for beryllium: Rel(E = 1 GeV) = 250 cm

• for lead Rel(E = 1 GeV) = 45 cm.

From the above considerations it is apparent that fast hybrid reactors have more favourable neutronic characteristics than do thermal reactors. Practi­cally it seems difficult to design a hybrid thermal reactor with keff larger than 0.95 for the U-Pu cycle and 0.92 for the Th-U cycle. The corresponding values for fast reactors would be 0.99 and 0.98 respectively.

11.2.1 Plutonium incineration

Plutonium incineration represents a special case. As we have seen, the stabilization of the plutonium inventory, and even its slow decrease,

should be possible within the existing nuclear energy producing system. However, if nuclear energy were to come to a stop and if the solution of underground disposal is not considered satisfactory, the reduction of the plutonium inventory would take a very long time, if standard solid fuel fast or thermal reactors are used. This is due to the need to mix fertile and fissile nuclei to limit the reactivity decrease during irradiation. For example, in the CAPRA concept [38], liquid metal fast reactors (LMFR) are used as plutonium burners. A CAPRA burner producing 8 TWhe burns 800 kg of plutonium but only consumes 160 kg, because of Pu regen­eration. Similar figures could be obtained with PWR reactors [41, 47]. Table 11.4 shows the production of transuranic elements for different schemes proposed for plutonium incineration in thermal reactors. Aside from the once-through UOx and well known single MOx schemes, the three additional schemes are:

• the MIX [169] scheme in which the fuel elements involve a homogeneous mixture of uranium oxide and plutonium oxide

• the CORAIL [40] scheme where both UOx and PuOx rods are placed within the same fuel assembly

• the APA scheme [41] which is a heterogeneous arrangement of UOx and Pu on inert matrix fuel elements.

These figures show that plutonium incineration would be slower than theoretically possible. In view of this situation, Bowman [133] has recently proposed using a molten salt hybrid reactor. The reactor would have the fol­lowing characteristics:

• thermal power: 750 MWth

• molten salt fuel with an NaF-ZrF4 carrier, fission products and plutonium fluorides

• thermal flux: 2 x 1014n/cm2/s

• moderator: graphite

• ks = 0.96.

The reactor is fed with a mixture of fission fragments, zirconium and plutonium fluorides obtained through fluorization of the spent fuels and extraction by sublimation of the uranium hexafluoride. The annual input would be 300 kg of plutonium and minor actinides, 1200 kg of fission products and zirconium cladding.[56] The output would be 65 kg of plutonium and minor actinides, 1435 kg of fission products and carrier salt.

Following Bowman [133], the advantages of such a system would be:

• no weapons plutonium or other weapons materials in repository

• possibility of underground criticality in repository eliminated

• 80% of fission energy recovered before waste emplacement

• instant irreversible elimination of weapons potential upon entry into transmuter.

The emphasis is clearly put on the prevention of uncontrolled military use of the plutonium in spent fuels. One TIER reactor would be associated with every 3000 MWt reactor, thereby eliminating the need for radioactive material transportation.

Further incineration of the remaining plutonium and minor actinides would require more elaborate chemical processing in order to separate fission fragments. Special reactors would be devoted to the second stage of incineration. In this case, however, transportation would again be necessary, but with no risk of weapons materials smuggling.

While, as has been seen above, in normal operation, deep disposal of nuclear wastes does not seem to represent significant potential problems for public health at any time in the future, the consequences of accidental situations must also, obviously, be examined. Let us assume that drilling occurs more than 500 years after the packages are installed, beyond a time during which memory and monitoring have been ensured. Drilling in such condi­tions could represent two hazards:

1. Radioactive elements brought to the surface along with the drilled samples; these elements do not, in principle, enter the food chain, but they can be responsible for external radiation.

2. If drilling is done to obtain drinking water, contamination can occur as the water percolates through the disposal site.

We show in section 4.1.2 that the additional number of neutrons provided by an ADSR is the number of initial spallation neutrons N0 whatever the value of the multiplication coefficient ks. It is then possible to estimate the addi­tional transmutation capabilities of an accelerator. Typically an accelerator can provide 1.5 x 1025 neutrons/year/mA. Thus if all neutrons are captured in the nuclei to be transmuted, one gets an annual transmutation rate of 25 moles/year/mA. For example, the production rate of 233U would be around 5kg/year/mA. One sees that a 20 mA accelerator, as typically considered for ADSRs, would produce 100 kg/year of 233U, in addition to the production of the subcritical system, which in the best case amounts to 50kg/GWe/year. Thus ADSRs can increase quite significantly the 233U breeding capabilities. For 239Pu the gain is smaller since a fast breeder can produce up to 150kg/GWe/year.

Similarly around 2.5 kg of long-lived fission products could be trans­muted each year per mA proton beam. Since a typical 1 GWe reactor produces each year 4 kg of 129I, for example, a 20 mA accelerator could transmute annually the production of more than ten reactors.