In the INC code of Bertini the incident proton collides with one or several nucleons of the target nucleus. These, in turn, collide with the unperturbed nucleons. A cascade develops. The INC calculation for a specific nucleon stops whenever its energy falls below a specified value, related to the depth of the nuclear potential well. Collisions between cascade nucleons are not allowed. This limitation is lifted in more recent calculations, like those of Yariv and Fraenkel [86, 87], Iljinov et al. [88] and Cugnon and co-workers [89] where cascade-cascade collisions are allowed, at the cost of calculation time. The initial code of Bertini, which is still the most widely used, was poorly documented and is difficult to improve. In particular, better nucleon-nucleon and pion-nucleon cross-sections are now available. Efforts are being made to use newer codes like the code ISABEL of Yariv et al. incorporated as an option in the Los Alamos code LAHET [90], the code of Cugnon incorporated in the CERN GEANT system [94], or a code written by the authors of the FLUKA system [95]. Classical intra­nuclear cascade calculations, like those just referred to, do not treat, in general, the emission of clusters of nucleons during the process. Work is being done to account for such emissions in the frame of the QMD model [96]. However, such calculations are very time consuming and will, probably, be restricted to specific calculations like those necessary for estimating the production of helium in structural materials. Note that both the evaporation and pre-equilibrium model account for the emission of light nuclei, in the low-energy regime.

In practice the conditions are more complex than for gases or very dilute solutions. Reaction rates depend upon the environment of the molecules which can form complexes with the solvent or other molecules. Such complex behaviour has led chemists to use the concept of activities at instead of concentrations in the mass action relation, which then reads

П a=к (p, t, {Ф ig). (9.6)

The activity coefficients are defined as the ratios of activities to concentra­tions:

By convention it is assumed that, for a pure solvent where only molecules of the solvent interact, 7 = 1. For very dilute solutions, since the dissolved molecules interact essentially with the solvent, it is expected that the activity coefficients are independent of the concentrations so that

In intermediate cases the activity coefficients vary with the concentrations. However, when these concentrations do not vary strongly during a reaction it is justified to consider that the activity coefficients are constant so that one can write a modified law of mass action in terms of concentrations rather than activities

An example of the dependence of the activity coefficients on the concentrations is given by the regular model which states that

7i(ci) = exp[а(1 — Ci)2]. (9.10)

As expected, equation (9.10) gives the limiting values 7i = 1 for ci = 1 and 7i = e“ for Ci « 0. With а > 0 the dilute phase is more active than the con­centrated one while the reverse is true for а < 0. The two types of behaviour are shown in figure 9.1 for а = 1 and а = — 1.

The reactivity decrease observed in figure 11.2 is due to progressive poisoning by fission fragments. This can be seen in figure 11.16, where a comparison is made of the reactivity behaviour with and without fission product removal. The figure shows that, in the fast-neutron case, the poisoning effect of fission products becomes significant only after approximately 10 years of irradia­tion. In practice, even when using solid fuels, periodic fuel reprocessing is needed for fission product removal. Even with metallic fuels, a period of 10 years between two reprocessing events would appear a maximum, due to irradiation damage to the fuel elements. Thus, the difference between solid and liquid fuels appears here as rather semantic, as can also be seen in the figure. The figure also shows that, with a small admixture of a fissile mix like industrial plutonium, an almost constant value of kx can be obtained for all fast-spectrum cases. After 10 years of irradiation quasi­equilibrium is achieved. Molten fuels do not seem advantageous for the minor actinide incinerator, at least as far as neutronic behaviour is con­cerned. However, other considerations may make molten salt fuels a good choice: the presence of large quantities of 244Cm in the fuel at equilibrium concentration will make the fuel extremely radioactive, with ample neutron

emission due to the high spontaneous fission rate of this isotope. The only practical chemistry available for such highly radioactive fuels is, probably, pyrochemistry through fluorization or chlorination. This pyrochemistry uses molten fluorides or chlorides as a mandatory step. It is, then, tempting to use these salts themselves as fuels.

Figure 11.16 shows that kx is much greater than unity, at least when using metallic fuels. To bring it below unity, neutrons have to be absorbed or lost by escape. One particularly tempting way to use these extra neutrons would be to transmute long-lived fission products like 99Tc and 129I. The excess number of neutrons per fission is close to 0.6, while the yields of 99Tc and 129I are only of the order of 2% per fission. It follows that the minor actinide incinerator could, in addition, transmute 30 times more 99Tc and 129I than it produces.

The fuel assemblies are surrounded by a 0.6 m thick man-made barrier. Each module comprises about 12 fuel elements and one barrier contains four modules. Each element weighs 0.5 metric tons. The total weight of the fuel accessible for each cell is then 24 metric tons. Each cell is sealed with a 2m thick stopper. The galleries access approximately 80 cells, i. e. about 2000 metric tons per gallery. The case of a drilling through a gallery is considered, which corresponds to the worst case scenario. The clay thickness is then

1.6 m.

Among the heavy elements, radium is characterized by a short diffusion delay, as can be verified in table I.1. The diffusion delay through the stoppers is 4200 years. The decay half-life of 226Ra, 1600 years, is of the same order as this delay, so that a large fraction of the radium may be in the water extracted from the well. The situation here differs a great deal from the site’s normal operation. Ra is a descendant of U, which accounts for most of the mass of irradiated fuels, 2000 metric tons in the present case. At radioactive equilibrium, the mass of Ra compared with that of U is like the ratio of their half-lives, which means 700 g of radium for 2000 tons of uranium.

The computed impact is notably larger than in the normal evolution described earlier. However, it remains acceptable in so far as it is of the same order as natural radioactivity. It is dominated by iodine and radium. It is interesting to note that the latter would have been produced anyway since its origin is natural as it is a descendant of 238U. The problem, if any, would be due to the fact that uranium is concentrated in one place, much like a particularly concentrated and rich vein in a mine. If uranium were used as fuel in breeder reactors, the need for uranium storage in geological repositories would disappear.

It is clear that neither 238U nor iodine can be confined: uranium because of its long half-life, iodine because of its mobility. In both instances, the question is whether it would be preferable to dilute them as much as possible in the environment (in particular in the ocean) rather than concentrating them in one particular site.

On one hand, concentration as it is considered here has small con­sequences in the absence of any intrusion. The reason for such an intrusion could be the existence of a particularly well concentrated vein of fertile matter and, as we cannot anticipate future energy options, we cannot anticipate how attractive this disposal site would be. On the other hand,
careful dilution of the 100000 tons of uranium in the 3 billion tons of the ocean could not lead to any significant health damage. Similarly, dilution of I in the ocean, where it would mix with huge quantities of stable I, lead to very small hazard: it would require the production of 5 x 106 reactor years to reach an average irradiation of human thyroids of 1% of the natural rate.

From the point of view of safety, a debate based on scientific data weigh­ing the advantages of confinement versus dilution would be worthwhile.

Nuclear reactors are macroscopic media where neutrons are propagated. It is, therefore, worthwhile to define macroscopic entities characteristic of the neutronic properties of the medium. We consider a homogeneous mixture of different nuclei i in number N. Let n, be the number of nuclei i per unit volume (usually cm3). Let of’* be the cross-section of type (a) (for example fission, absorption, capture or scattering) of nucleus i. The

macroscopic cross-section is defined as:

The mean free path for reaction a is simply

In this case fission products are, generally, extracted from the fuel soon after they are produced. It is also possible to keep the concentration of fertile elements constant by continuous feeding. The relative proportion of fissile and fertile nuclei evolves towards an equilibrium:

In the case of simple fissile nuclei regeneration, equations (3.135) and (3.137) show that the maximum value of kM is equal to ^/2.

So far, proposed liquid fuels have been molten salts. A reactor using a mixture of uranium, thorium, beryllium and lithium fluorides has run successfully for several years in Oak Ridge National Laboratory [49]. This experience led to the molten salt breeding reactor project. This reactor was supposed to use the Th — U cycle described in section 3.5. The capture of neutrons by 233Pa tends to decrease the reactivity of the reactor.[24] This is why, in the MSBR project, online fuel processing was assumed. This pro­cessing aimed at extracting both the fission products and the protactinium. After protactinium decay, the resulting 233U was reinjected into the reactor. This procedure predicts breeding of the order of 5% per year.

Liquid fuels have also been considered for fast reactors. In this case chlorides rather than fluorides have been proposed [64].

MCNP has a lot of possibilities for the definition of neutron sources and it is beyond the scope of this book to describe them. Source description is done in the Data cards section of the input file. Here we just want to give two simple source examples. The first one (the simpler) is a punctual isotropic and mono-energetic source,

SDEF POS 000 ERG=2.5

which defines a neutron source at position (0,0,0) of energy 2.5 MeV. This kind of source is very useful for starting a KCODE.

When dealing with an ADSR, the spallation source has to be defined; because this involves high-energy particles, MCNP alone cannot process that kind of source; nevertheless, if one can create a file accounting for neutrons below 20 MeV from such a source, MCNP can read that file (the position (x, y, z), direction (cosine), energy, time and weight of each neutron have to be specified). Such a source file is read with the FORTRAN subroutine ‘source. f’ included in the MCNP distribution. The source can be built by any high-energy transport code like FLUKA, HETC, or directly by MCNPX, which is a version of MCNP coupled to LAHET. In order to give a more realistic ADSR description in the following section, we give here an MCNP source to represent, more or less, the

spallation source. It is a cylindrical source:

SDEF POS 0 0 0 ERG=D1 RAD=D2 EXT=D3 AXS 0 0 -1 $ Dn=to be described later

SP1 -5 a $ (related to D1)

512 r1 r2 $ Radial extension (ring of radii r1 and r2) (related to D2)

513 zmin zmax $ Z extension (from source position) (related to D3)

Its energy is described by the Source Probability card (SP1); the first parameter (-5) means that the source energy probability is p{E) / Eexp(—E/a) (evaporation spectrum) and the second parameter is the value of a in MeV.

The axis of the cylinder passes through point POS (0,0,0) in the direc­tion AXS (0,0, —1) = z axis. Neutron positions are sampled uniformly (in volume) within a ring of inner radius r1 and outer radius r2 (Source Informa­tion, SI2 card). The ring lies in a plane perpendicular to AXS at a distance from POS sampled by EXT (here defined on the SI3 card from zmin to zmax).

Note

A source neutron, when it is born from a punctual source or an external source (like a spallation one) cannot be on a surface. If a neutron is born on a surface, it has to be pushed a little (let us say, by 1 pm).

While most proposals for the neutron source of hybrid reactors resort to spallation reactions with high-energy protons, some other possibilities have been proposed, which we discuss briefly here.

6.2.1 Deuteron-induced neutron production

Deuterons are weakly bound nuclei which easily undergo break-up reactions. This is well known at low energies, around 10 MeV, where deuteron beams

have been used to produce neutrons very efficiently. Light targets, like beryllium, are used in this context to obtain a peaked forward neutron beam, with a most probable energy close to half the deuteron beam energy. The mechanism at work is that, in the nuclear field of the target, the extended deuteron may break up into its proton and neutron constitu­ents. The neutron then escapes the target easily. On the other hand, proton beams are not expected to produce many forward peaked neutrons, since charge exchange reactions at low energies are not probable. At high energies one expects that, because of the high cross-section for charge exchange, neutrons and protons behave similarly. Thus, after break-up, the deuteron would become equivalent to two protons with half the deuteron energy. Therefore, one would not expect any significant gain by using a high — energy deuteron beam as a source of spallation neutrons. Ridikas and Mittig [120] have done a comparison between protons and deuterons. They simulated the interactions of deuterons and protons with thick targets, using the LAHET + MCNP system. Figure 6.12 shows the variations of the ratio of total neutron multiplicities with deuteron beams divided by those with proton beams as a function of the beam energy. Even at 1 GeV there seems to be an advantage for the deuteron beam, except for the uranium target.[42] This may be related to a larger cross-section of the

deuteron. The striking feature is the very large enhancement of neutron multiplicities for a deuteron beam impinging on a beryllium target, even at rather high energies. The calculations shows that the excess neutrons are strongly forward peaked. It is true that the neutron multiplicities of the d + Be reaction are less than those of the p + U reaction, for energies higher than 200 MeV. However, the authors suggest using a hybrid target where deuterons impinge on a thick beryllium target surrounded by a neutron multiplying medium. They compare this arrangement with a similar one where the beryllium is replaced by uranium. The result is shown in figure 6.13, in terms of energy gain (see below) of the set-up. It appears that, at 1 GeV, the advantage of the beryllium target is not striking. The neutron yield decreases less rapidly with energy with deuterons than with protons, which might ease the requirement on the accelerator. However, deuteron beams are known for activating the accelerator very strongly, so that the final advantage of employing deuteron beams is not obvious. One interesting result shown in figure 6.13 is that, even for protons, the light target is almost as prolific as the heavy one. This is true although, as can be seen in figure 6.14, the neutron multiplicity obtained from a beryllium thick target is nearly three times less than that obtained with a uranium target. This surprising apparent contradiction means that the neutrons pro­duced in the beryllium are more prolific than those born in the uranium: they are more energetic. This is a clear illustration of the difficulty of defining source neutrons.

A helium flow through the salt can be used to remove rare gases and, partially, noble metals which form small aggregates within the salt.

9.3.1 Liquid-liquid extraction

In principle, liquid-liquid extraction of components from the salt mixture resembles the solvent extraction of section 9.2.[50] A contact is established between the molten salt phase and a liquid metallic phase. The most abun­dant component of the metallic phase is a metal with very weak interaction with the fluoride salts and a low melting point. Cadmium and bismuth have the required chemical and physical properties. However, for a reactor, bismuth is preferred because of its low neutron cross-section. The extraction proper is done via reduction by a reductive component added to the liquid bismuth. In the MSBR project the reductive component was chosen to be metallic lithium. This choice is advantageous since lithium is automatically in equilibrium with its own salt:

LiF + Li Li + LiF. (9.32)

The basic reduction reaction thus reads:

MF^ + ^Li ^LiF + M (9.33)

where M is the metallic element to be extracted from the molten salt. The distribution coefficient of element M is defined as the ratio of the molar concentration of M in the metallic phase to that in the salt phase:

Using the expression (9.4) of K(P, T), we obtain for KM

vAGLiF — AGmf^

KM(T) = exp(——————- RT——— )■ (9-40)

Some examples of values of the formation enthalpies and entropies are shown in table 9.3. Values of log(KM(T = 600 °C)) obtained from equation (9.40) are also shown in the table. Large values of KM generally correspond to large values of the distribution coefficients in the metallic phase.

Although, in practice, KM(T) differs, sometimes significantly, from KM(T), assuming that the two values coincide (activity coefficients equal to unity) and using equation (9.38) allows us to obtain, at least, a qualitative understanding of the conditions for an efficient liquid separation. With this assumption, figure 9.7 shows the variations of the distribution coefficient for selected metals as a function of the concentration of lithium in the metal­lic phase. Positive values of log(KM(T)) correspond to possible extraction from the salt into the metal while the reverse is true for negative values. It is seen that noble metals (palladium) are easily extracted. Plutonium, uranium and protactinium are extracted with increasing difficulty, but still significantly less than rare earths (lanthanum) and thorium. Uranium and zirconium should be very difficult to separate.

In practice, the situation is much more complex than that displayed in figure 9.7. This complexity is not only due to various values of the activity coefficients which differ from unity, but also to the presence of several states of oxidation of the actinides and rare earths. While the selective liquid-liquid extraction of uranium and protactinium appears to be easy, by playing with the content of lithium in bismuth, the extraction of rare earths in the presence of thorium appears to be a difficult challenge. It has been proposed in the frame of the MSBR project [50] to strip rare earths

from the liquid bismuth by a counter-current of an LiCl salt. This salt appears to have very different properties for thorium and rare earths.

13.1.1 The FEAT experiment

The FEAT experiment carried out at CERN [122] was the first to couple a high-energy proton accelerator to a subcritical medium. The accelerator was the CERN PS and the subcritical medium a set of natural uranium rods immersed in water. This implied a rather large degree of subcriticality and thus could not give insight into the possible difficulties of controlling the subcriticality level of an ADSR. However, for the first time, a direct and very detailed comparison of neutronic measurements and simulations could be made. Some of the results obtained in this pioneering experiment were presented earlier in this book (see section 6.3).