It is possible to calculate quantities of the form

‘(E)R(E) dE

where ‘(E) is the fluence and R(E) is an additive and/or multiplicative response function from the MCNP cross-section library. It is very useful to find the reaction rates, the averaged cross-sections (over the flux) and so

on. Such quantities are obtained with FM cards (related to F (tally) cards); an FM card is written as

FMn C m-L R1

or

FMn (C mL Rl)(C m2 R2)

where n is the tally number, C is the multiplicative constant, mt is the material and R is a reaction code (or a list of reactions); table 5.3 gives some useful reaction codes.

But let us see some useful examples of FM cards. Suppose that cell 1, of volume V, is made of material 2 with a given density.

F4:n 1 F14:n 1

FM14 (1 2 102) (-1 2 102) (-1 2 -6:102) (-1 2 -6 -7)

Tally 4 just gives the flux in cell 1. Tally 14 gives:

• (1/V) J ‘(E)an7f (E) dE for material 2.

• (1/ V) ‘(E)^n7(E) dE for material 2; the density of cell 1 is used because the muJltiplicative constant is negative to calculate the number of atoms.

• (1/V'(E)(Xn,7(E) + Xfis(E)) dE for material 2; the ‘:’ means that reactions -6 and 102 are summed.

• (1/V) J" ‘(E)v(E)Xfis(E) dE for material 2; the ‘ ’ means that reaction-6 is multiplied by ‘reaction’ -7.

To calculate

‘(EK,7(E) dE ‘(E) dE

one just has to take the ratio of the first value of tally 14 by the value of tally 4 (for complex cells, MCNP is not able to calculate the cell volume and one has to specify it with an SD or VOL card).

It is also possible to calculate these quantities for a material which is not directly present in the geometry; it is just necessary to define this material. This is particularly interesting if one wants to see the contribution of individual nuclei to a mixture (for example the role of hydrogen and of oxygen in the water). Note that, in this case, the multiplicative constant C must be positive, because the density of this perturbing material is not given.

As said in chapter 4, the CERN FEAT experiment [122] gave a value of G0k = 3, for incident proton energies larger than 1 GeV and for a uranium target. The experiment consisted of mapping out the number of fissions produced in a multiplying array surrounding a uranium target bombarded by the CERN PS proton beam. The multiplying array consisted of natural uranium bars immersed in a light-water swimming pool. The fission density within the uranium bars was obtained from the measurements, after careful corrections for the flux depression within the bars and influence of the surroundings of the detectors. The value of k was deduced from a measurement using a known 252Cf source. From the value of G0 it is possible to deduce a value of N0 = G0uEp/0.18 = 41.5 neutrons per GeV proton, to be compared with a value of 32 which can be read on figure 6.10. The ratio of neutron multiplicities for uranium and lead amount to 1.35 for the CERN experiments and 1.50 for the measurements of Hilscher et al., to be compared with the value of 1.4 corresponding to the multiplication in uranium. Another important result of the FEAT experiment was that the neutron multiplicity per GeV saturated for proton energies above 0.8 GeV. This behaviour is illustrated in figure 6.15. For lead and 1 GeV protons, the value of G0 should be between 2.5 and 1.8 according to the value retained for the neutron multiplicity. The value of G0 = 2 was retained by the CERN group for its calculations of the energy amplifier [76].

While processes taking place in galvanic cells are spontaneous and follow the principle of free energy minimization, electrolysis involves the separation of the constituents of a molecule with a consequent free energy increase obtained from electric power.

Figure 9.9 is a schematic drawing of an electrolysis cell. It describes the electrolysis of an MA salt. The reactions taking place at the electrodes are

M+ + e— ) M — AGM at the cathode (9.57)

A ) A + e — AGa at the anode (9.58)

MA ) M + A — (AGm + AGa). (9.59)

The voltage applied between the electrodes has to exceed a threshold VTh for the separation between A and M to take place. This threshold is obtained when the work done in transporting an electron mole from the cathode to the anode equals the molar free energy of the separation reaction:

96 500 Vt = AGm + AGa = AG. (9.60)

For multiply charged n+ negative ions one gets:

96 500nVt = AGm + AGa = AG. (9.61)

In the US [174], the Avanced Accelerator Applications programme is under way to develop a technology base for transmutation, to demonstrate this as an approach to long-term nuclear materials management, to build an accelerator driven test facility, and to strengthen the domestic nuclear infra­structure. An accelerator driven transmutation facility with a power rating in excess of 20 MWth, driven by a high-power proton linear accelerator with a beam power of approximately 8 MW, is planned to start operation in 2010.

13.2.2 Europe

In Europe, the situation is characterized by a number of projects but an uncertain state of financing. In 1998, some European countries established a Technical Working Group (TWG) which has to identify the technical issues for which an important R&D effort is needed. These experts have recommended to design and operate an experimental ADSR (XADS) at a significant thermal power, so as to come close to what an industrial ADSR dedicated to minor actinide incineration could be. The whole time scale (preliminary design studies, R&D, construction) is approximately 10 or 12 years.

As a pedagogical exercise, it is interesting to describe the various projects related to this programme in some detail. A complete overview of the European approach can be found in reference [134].

Figure III.3 is a schematic representation of the Alvarez Linac. As compared with the Wideroe Linac, it is characterized by the polarization of the drift tubes, due to the TM010 mode. The voltage drop is concentrated along the gaps between the isopotential drift tubes. On average, the voltage gradient is constant along the axial direction. Since the length of the drift tubes, as

in the Wideroe case, has to increase as the particle is being accelerated, in order to keep the synchronism, it follows that the voltage drop between two adjacent drift tubes increases with the particle energy, in contrast to the case of the Wideroe accelerator where this voltage drop is independent of the particle energy.

In the case shown on figure III.3, the particle travels one drift tube length during one period of the RF. This means that

I = j3X. (III.39)

As compared to the Wideroe I = (f3/2)X relation, this implies a length of the accelerator twice as long. For this reason, side-coupled cavity Linacs are used at the highest energies, where it is very important to reduce the length of the Linac as much as possible. In the side-coupled Linac, cavities work on the I = (f3/2)X mode but particles cross only one cavity out of two, the odd one for example. This way the particles still see cavities with the same polarization but in the I = (f3/2)X mode, which allows division by two of the length of the accelerator.

III.1.2 Phase stability

Up to now we have assumed that particles were exactly in phase with the RF and that they were accelerated at the maximum field. However, in this case, any small deviation of the particle energy from the resonance energy would lead to a loss of synchronism. For example, a particle with less than the resonance energy would take more time than the synchronous particle to reach the following gap and thus be accelerated by a reduced electric field and, therefore, become even more distant in energy from the synchronous particle. These particles would eventually be lost. Due to this mechanism, beam intensities would be extremely reduced. It has been realized that, if the particles cross the inter-drift tube gap at a time when the electric field is rising, particles with energies larger than that of the resonant one cross the next gap at an earlier phase and, thus, for a smaller field value and, there­fore, their energy comes closer to that of the resonant particles. On the contrary, particles with lower energy than that of the resonant one will cross the next gap at a later phase, and, thus, for a larger field value; therefore their energy again comes closer to that of the resonant particles. This principle, which allows capture in the accelerated beam of particles with non­resonant energies, is called the phase stability principle. Although it makes a low loss acceleration of intense beams possible, it has some drawbacks:

• It decreases the effective accelerating field, and thus requires longer accelerators.

• It leads to a weaker focusing, or even a defocusing of the beam. This can be seen in figure III.4. Indeed, the figure shows that, in static fields, particles

Figure III.4. Partial representation of the electric field line in an Alvarez Linac. The tank and the drift tubes are in black. Electric field lines are shown with dashed lines. A non-axial particle trajectory is shown with a dotted line. Note the modification of the field lines due to the presence of the drift tubes. While the drift tubes are equipotential the full potential difference appears along the gaps.

exiting from the drift tube at an angle to the beam are first deflected towards the beam axis. In the second half of the gap they are deflected off the beam axis, but less strongly than they are towards it in the first half, since the deflecting field becomes smaller the closer it is to the beam axis. The net result of these deflections is a focusing effect. However, if the electric field is increasing while the particle crosses the gap, the deflection off the beam axis in the second half of the gap may exceed that in the first half. The net result may be a defocusing of the beam, or at least a weaker focusing. It is usually necessary to correct this effect by the interposition of magnetic focusing devices such as quadrupoles along the beam axis. Note that focusing and defocusing by the electric field are only significant at low energies where the increment of the particle energy within the gap is noticeable with respect to the particle energy itself.

In temperate regions the average annual production of dry ligneous material is 10 tons per hectare, with the maxima reaching 20 t/ha. This corresponds to a gross resource between 3.6 and 7.2 toe/ha, i. e. between 40 and 80 MWh. As compared with the average annual insolation of 1.5 x 104MWh/ha this corresponds to an efficiency between 0.2 and 0.5%. Taking into account the thermodynamical efficiency for electricity production, the maximum efficiency for electricity production is 0.2%. A facility producing 7 TWh per year would require a cultivated area of about 2500 km2, in the best cases. This surface could be halved in tropical conditions.

It is generally considered that the use of biomass is neutral as far as CO2 emissions are concerned. However, this is only true if it is ensured that all incinerated biomass is compensated by adequate replantating. This is, presently, not the case in most developing countries where deforestation contributes to greenhouse gas emission.

Today’s world biomass energy production amounts to approximately 70 GToe/year. Humans use about 4% of this production either for producing food (2GToe) or for energy production (1 Gtoe). In the most biomass intensive scenarios given at the 1992 UN Rio Conference, biomass energy production would be as high as 5 GToe, and the total human use would amount to 13 GToe, i. e. around 20% of the available resources. Because of its large volume, biomass has to be transformed into high energy content material close to its production location. Aside from local uses, the transfor­mation of biomass into gas (methane), alcohol (ethanol, ETBE) or vegetable oil ester is considered. At present, electricity production using biogas is only marginally competitive when the cost of the biomass is negligible. Otherwise the cost of biogas electricity is three times more than that which can be obtained with fossil fuels. The cost of biofuels is about three times that of fossil fuels.

Criticality can be obtained only if the probability of neutron escape is small enough. This condition leads to the concept of a critical minimum mass of
fuel needed to sustain the chain reaction, in the absence of an external neutron source.

Monte Carlo calculations follow the history of individual neutrons. The most used codes are MORSE [72] and MCNP [74]. The CERN group has written its own code, MC2 [76], which is, however, not in the public domain. The physics involved is basically the same in all these codes.

In the Monte Carlo scheme, there is no space (or time) discretization. One does not solve any differential equations and there is no need to write such an equation. MC methods supply information only about specific quantities, requested, a priori, by the user. In that sense, MC codes solve the integral transport equations. The principle is to follow individual particle histories (as many as possible). Then, the particle average behaviour is inferred, using the central limit theorem, from the simulated particles. Indi­vidual probabilistic events (such as interactions of particles with materials) are simulated sequentially. The probability distributions governing these events are statistically sampled to describe the total phenomenon. This is very similar to a real physics experiment: you plan to measure some quanti­ties (with specific detectors). By recording the result for many particles, the experiment supplies information on the physics of your system.

Fission product

Since not all the relevant cross-sections for the production and disappearance of fission products are included in the MCNP data base (ENDF-VI), one has to use a kind of ‘mean fission product’; the cross-sections for all fission fragments have to be analysed, together with the mass distribution of fission fragments for each of the fissile nuclei relevant to the study to evaluate the mean fission product and its cross-section (a linear combination of the data available in ENDF-VI is used to reproduce the mean values).[36] A time evolution study of these average cross-sections for the fissile nuclei has to be done to modify, if necessary, the linear combination after exposure to the neutron flux. In fast spectra, these cross-sections are generally more or less stable, but in a thermal case, the variation of these cross-sections being much faster, one has to modify the linear combination.

Short-term fuel evolution as well as temperature changes may lead to reactiv­ity changes. For thermal reactors the very large capture cross-section of 135Xe

and 149Sm lead to such effects. In the case of the Th-U cycle, a specific effect arises, both for thermal and fast reactors, due to the 27 day half-life of 233Pa. We first examine this effect.