Nuclear energy production is accompanied by the production of various radioactive wastes:

• fission products

• activation products due to neutron captures by nuclei belonging to the structure of the reactor, such as, for example, cobalt 60

• transuranic nuclei due to neutron captures by the nuclear fuel.

Nuclear wastes are characterized by their radiotoxicity and their half­life. Only wastes with half-lives exceeding ten years are associated with significant storage problems. These are essentially some fission products (LLFP) and transuranic elements. Their noxiousness is, traditionally, measured by their ingestion radiotoxicity.

The previous examples show integrated flux over a cell, independent of energy and time. It is, however, useful to know the energy and/or time dependence of such quantities. Below are three examples related to the three previous tallies:

E4 0.01 0.1 1. 10.

T14 0.1 0.2 0.3 0.4 E24 0.1 1. 10.

T24 10. 99I 1000.

The first line defines, for tally 4, an energy binning where the upper bounds of each bin are 0.01, 0.1, 1 and 10MeV; tally 14 give the flux in cell 1 and in cell 2 as a function of time (from 0.1 to 0.4 shakes). Tally 24 will give the mean flux of cells 1 and 2 as a function of energy (three bins) and of time (101 linear bins from 10 to 1000 shakes).

• The first bin is always from 0 to the first specified upper bound.

• MCNP gives also the sum of all bins (1D binning), or for 2D binning the sum of all energy bins as a function of time and the sum of all time bins as a function of energy.[32]

Electron beams have been used extensively to produce neutron pulses. It has been proposed, for example by Abalin et al. [124], to use an electron accelera­tor to generate the neutron source of hybrid reactors.

Electrons with energies above a few tens of MeV are slowed down essentially by Bremsstrahlung photon emission. Above 10MeV the domi­nant process for energy loss of the photons is pair creation. In that manner an electromagnetic shower develops, consisting of a population of electrons, positrons and photons. Independently of their electromagnetic interactions these particles may also interact with nuclei. The photonuclear cross-section is dominated by the giant dipole resonance, with an energy which depends weakly on the nuclear mass and is around 10 MeV. Thus, if one wants to optimize the photonuclear rate of interactions within the electromagnetic shower, one should maximize the number of photons (real or virtual) with energies around 10 MeV. The optimum initial electron energy is then found to be between 100 and 200 MeV. For example, the Euratom Geel linear accelerator provides electrons with a maximum energy of 140 MeV, and operates, in practice, at 100 MeV. At this energy only 0.1 neutron is produced per incident electron. This corresponds to a neutron production yield of 1/GeV, to be compared with the typical 30 neutrons per GeV which can be obtained with protons. To overcome this small neutron production efficiency, Abalin et al. [124] suggest using multi­plying media with ks very close to one. We shall examine their proposal in more detail below.

Let us begin with the consideration of an electric cell.[51] Figure 9.8 is a schematic representation of a cell.

A half cell is composed of a metallic electrode immersed in a salt solu­tion. Here, for simplicity, we have assumed that the salt cation is the singly ionized metal of the electrode. Thus, on the electrodes, electron capture and loss reactions can take place such as:

M^ + e_^ M1 [AG1] (9.41)

M2 <=; MJ+ e_ [-AG2] (9.42)

We give a positive value to the free energy release AG12 of the electron capture by an ion, since this a rather natural choice corresponding to the elementary atomic reactions.

If a conductor connects the two electrodes, an electron transfer between the two half cells can occur which results in the possible reactions

M+ + M2 ) M+ + M1 [AG = AG1 — AG2] (9.43)

M+ + M2 ^ M+ + M1 [AG = AG2 — AG1]. (9.44)

The direction of the electron transfer depends upon the sign of AG = AG1 — ag2. If AG > 0, electrons will flow from right to left, and

therefore a positive current will flow from left to right. A positive potential

V will be established between the left and the right electrodes. Assume that a mole of M1 is produced.[52] The number of electrons transferred is thus equal to the Avogadro’s number, and the amount of charge transferred is a Faraday, i. e.

1 Faraday =(1.6 x 10-19)x(6 x 1023) = 96 000 Coulomb. (9.45)

The work done by this transfer is therefore

W = 96 000 x V. (9.46)

Neglecting resistive losses, this work has to be provided by the molecular free energy change AG, so that

AG = 96 000 V (9.47)

V = 1.04 x 10-5AG. (9.48)

The generalization of these equations to the case when the M1 ions are the n1 charged M?+ species is straightforward and yields

M?+ + n2M2 ) Mn22+ + n1M1 [AG] (9.49)

AG = 96000n1 V (9.50)

a g

V = 1.04 x 10-5——- . (9.51)

n1

In practice, electro-chemists use a standard reference for each element which is the cell of the element associated with the hydrogen cell H2/H+ with the reaction 2H+ + 2e- ) H2 with one molar electrolyte concentration and normal temperature and pressure conditions. By convention the value of AG for the hydrogen cell under these conditions is assumed to be 0. Each ionic state of elements is, therefore, characterized by the standard potential
of the cell it forms with the standard hydrogen half-cell, as well as by its free energy relative to that of H+:

AG0 = 96 500nV0 (9.52)

AG

V0 = 1.04 x 10—5——— — . (9.53)

n

Metals easily lose their electrons to hydrogen, and are therefore characterized by a negative value of both V0 and AG0. When the standard conditions are not fulfilled, equation (9.4) allows us to get

AG = AG0 + RT ln(K) (9.54)

which reads, for reaction (II),

The above discussion implies that the radioelements are fully soluble in water. In truth, many elements are poorly soluble and that is liable to moderate their transfer. If the flow through the clay layer, or the man­made barrier, is insufficient to evacuate the radioelements as they are released by the fuel or the package, their concentration in the water that is in contact with them will increase until, possibly, a saturation limit Cmax is reached, beyond which they will precipitate in the vicinity of the source. To such a concentration limit corresponds a value of the flow entering the clay layer or the man-made barrier. This can be obtained from the stationary density equation:

On the earth’s surface the solar constant measures the power received from the sun by a 1 m2 surface perpendicular to the sun’s rays. It amounts to 1 kW. Practically, in order to estimate the energy available at a specified location, one has to take into account the latitude and the average daily and yearly insolation. Typical annual sunshine ranges from 1000kWh/m2 in northern Europe to 2500kWh/m2 in deserts like the Sahara. In Spain values reaching 1600kWh/m2 are obtained, while in California values as high as 2000 kWh/m2 are observed [32]. Taking, as an example, an insolation of 1800 kWh/m2 and an efficiency for transformation of the solar energy into electricity of 15%, one gets an annual electricity production of 270 kWh/m2, a number approximately valid for photovoltaic as well as thermodynamical systems. To obtain an annual production of 7 TWh, similar to that of a 1 GW nuclear plant, 26 km2 are needed. At current solar cell costs, such a facility would cost around $17 billion, more than ten times that for a nuclear plant of similar power. The corresponding electricity cost would reach about 500 per kWh, compared with the current costs of around 50 per kWh. Such high costs will restrict the use of photovoltaic systems to remote locations not connected to an electricity distribution network. Note, however, that the need to store the energy would increase its cost by at least a factor of two. Thermodynamic systems allow much lower costs down to 120 per kWh and may become competitive for significant energy production in such places as Africa, India and South America, provided the current output of the facility can be fed into a network able to cope with the essentially intermittent form of the solar energy. One should note, in this context, that day/night storage capacities are present in all thermal solar systems, in the form of oil, sodium or molten salt heat storage reservoirs.

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).

These methods consist of more or less elaborate approximations of the Boltzmann equation. The most widely used approximation is the multigroup diffusion theory which we outline here, as an example. The different groups correspond to energy bands Ei < E < Ei +1. The set of multigroup equations reads

DiA’i(r)- (r) + X! Sr, j!i’j(r) + Xi^2v^f, j’j(r) (5-8)

j j

where i( j) denotes the i( j)th group. Xr, j^ i is the cross-section for a jump from group j to group i. St i = Sa i + j Xr, j i is the cross-section for removing neutrons from group i. Xf, j is the fission cross-section in group j.

Xi is the fraction of the fission neutrons which have energies within group i. The cross-sections should be computed as averages over the group energy domain by

(E)'(E) dE

which means that equation (5.8) is, in fact, a set of complicated integro — differential equations. In particular, in the resonance regions, the flux has a complicated structure due to its depletion at energies in the vicinity of a resonance energy. Thus, approximations are made on the calculation of the group cross-sections. In particular, in heterogeneous reactors the cross­sections for the cells are first computed, with a large number of groups, with simplifying assumptions on the shape of the flux, and, possibly, correc­tion factors. In a second step the flux on the cell network is computed. In practice, experiments are needed to validate the group cross-sections for each type of reactor.

Since the neutron flux depends on the spatial position within the reactor, the core volume is divided into evolution cells (cell dimensions are typically Az « 5 cm and Ar « 5 cm). The evolution of each type of nucleus within the cell takes into account nucleus disappearance by neutron-induced reac­tions and by natural decay, and nucleus production by neutron reactions on a parent nucleus and by natural decay of a parent. The description of these four processes constitutes a system of Bateman’s coupled equations whose solution is a vector that contains the number of nuclei of each type.

Since hybrid reactors should not require control rods, it is of course very important to check that, in time, the reactor cannot become critical. We address this question in the present chapter, having in mind, especially, the possible evolution of the fuel.

8.1 Long-term evolutions

More complete calculations than those presented in section 3.6 are needed in order to characterize the behaviour of specific fuels which might be used in hybrid reactors. As examples we show the results of three such calculations [129] in figure 8.1 where we show the evolution of кю for a plutonium mixture originating from PWR spent fuel, for the Th — U system, and for a fuel made of a mixture of minor actinides. Figure 8.1 confirms that hybrid reactors with solid fuels are not fit for plutonium incineration. On the other hand, minor actinide fuels behave like a mixture of fissile and fertile nuclei. Figure 8.2 shows the evolution of the fission rates due to the various nuclei as a function of time. It appears that the stabilization of the variation of кж is chiefly due to the formation of 238Pu, which has a high fast-neutron fission probability. It is formed by neutron capture by 237Np, which behaves like a fertile species. To a lesser extent the rise of 244Cm fissions counteracts the decrease of the 241Am and 243Am fissions. Figure

8.2 also gives an idea of the long times required for a significant decrease of the fission rate of the fuel and, correspondingly, of the total number of transplutonium nuclei.