Using this formalism, we have determined the fuel volume necessary to obtain ks = 0.98 as a function of R1, for a reactor using a U-Pu fuel with the following characteristics.

• The relative volumic fractions were 0.5 for lead, 0.08 for iron, 0.39 for the fuel and 0.03 for vacuum.

• The fuel was 88% uranium 238 and 12% plutonium in molar fractions. Both uranium and plutonium were in the dioxide form.

• The relative amounts of plutonium isotopes were 62.7% 239Pu, 24.3% 240Pu, and 13% 241Pu, corresponding to the concentration of spent PWR reactor fuel.

• The cross-sections were one-group cross-sections extracted from the MCNP Monte Carlo calculation which is described in the next section.

The results are shown in figure 10.1. The values shown in the figure correspond to a 1 MW proton beam, each proton being assumed to produce 30 neutrons. For an internal radius R1 = 0.15 m, the maximum neutron flux reached is 3.6 x 1015n/cm2/s, corresponding to a maximum specific power of 280W/cm3. The total thermal power was 120 MW. The

volume of the fuel zone is around 0.7 m3 for a fuel weight of 3.5 tons. This configuration can be considered as the smallest possible demonstration design able to:

• reach a multiplication factor of 0.98

• approach the maximum acceptable specific power

• reach representative neutron fluxes, so that fuel evolution can be studied in realistic conditions.

The 1 m internal radius could be representative of an energy producing reactor. The maximum specific power is 50W/cm3 for the 1mA beam. A 10 mA beam would lead to an acceptable 500 W/cm3 specific power and a 1200 MWth reactor, for a fuel zone volume of 2.7 m3, and a fuel weight of 14 tons. In this configuration, the ratio of maximum to minimum flux is only 1.25, a very reasonable value. The thickness of the fuel zone is less than 20 cm.

A realistic reactor could be neither homogeneous nor spherical. The fact that the proton beam has to penetrate inside the reactor leads to truncated cylindrical shapes, rather than spherical, which are not apt for a simple analytical treatment, even in the one-group diffusion approach.

More realistic calculations are necessary, and we give an example of one of these.

This appendix is based upon an article written by H Nifenecker and G Ouzounian (ANDRA, * France).

Deep disposal of nuclear wastes can be considered a reference strategy against which transmutation-incineration options can be evaluated. Here we give a simple account of the kind of performance which can be expected from a well conceived deep underground disposal site.

First, we give a sketchy description of what an underground disposal facility could be.’f Then, we will give the bases on which computation of the diffusion of radioactive elements through geological layers rest. Finally, we will give results in the case of clay. Of course, such theoretical cal­culations must be validated in the most realistic manner possible. Experimen­tal studies in underground laboratories have been described by one of the authors in the July 1999 issue of the SFP* journal [175].

The most commonly used focusing elements are magnetic quadrupoles. At low energies electric quadrupoles are also very widely used. Figure III.6 shows the geometric configurations of electric and magnetic quadrupoles.

The general features of quadrupolar focusing elements can be sum­marized by the pattern of the forces acting on the beam perpendicular to

its direction. If Oz is the direction of the beam, a quadrupolar pattern is obtained with the force components:

Fx = — f0x (III.49)

Fy = fy. (III. 50)

With such a pattern, particles which are off axis in the Ox direction receive an acceleration towards the beam axis and are, therefore, focused. In contrast, particles which are off axis in the Oy direction receive an acceleration away from the beam axis and are, therefore, defocused. More quantitatively the equations of motion of the particle, assuming no force in the Oz direction, read

equations (III.52) are separable. The solution of equations (III.52) follows, for an initial value of z, z0 = 0, with x = dx/dz:

r, sin^/Kz)

x(z) = X0 cos( KZ)+ X0———— —=

K

x’(z) = —x0yrK sin f/Kz) + x0 cos ([Kz)

. p, ’ sinh(—Kz)

y(z) = У0cosh( Kz)+ У0 —

K

y (z) = y^/Ksinh^/Kz)+y0 cosh^/Kz)

so that, after a field region of length L, the initial values of x and x are modified as:

yL = y0 cosh( VkL) + y0

K

yL = y^/K sinh^/KL)+y0 cosh^/KL). Defining the vectors

X0 =

one sees that X0 transforms into XL via the matrix multiplication

A focusing device should be such that a particle with initial velocity parallel to Oz (x0 = y0 = 0) exhibits a velocity change towards the beam. This happens when x’Lx0 < 0. Thus equations (III.60) describe a system which is focusing along x and defocusing along y. This is a general feature of quadrupolar configurations: they are focusing in one direction and defocus­ing in the other orthogonal one. Note that when к = 0 one gets the free motion solution.

The small-angle approximation of the focusing element reads

rn -1 kl2) l

F — kL (1 — 1 kL2)

while that of the defocusing element reads

Q Г (1 +1 kL2 ) l

D kL (1 +1 kL2)

The focusing or defocusing character of the element is therefore given by the sign of matrix element Q21. The association of a focusing device with a defocusing one allows one to get focusing in all cases. Qualitatively, this results from the fact that, after a first defocusing, particles reach out from the beam, leading to a stronger focusing field in the next optical element, while the reverse is true when focusing happens first. At the lowest approxima­tion, the transfer matrix for the defocusing-focusing (DF) arrangement reads

+ kL2 2L

k2 L3 1 — kL2

which is focusing. Similarly the FD arrangement leads to the transfer matrix

which is also focusing. Other quadrupole arrangements have interesting properties such as the triplet which allows stigmatic focusing. In the triplet case the transfer matrix reads

A focusing device can be characterized by its focal length f. It can be shown that a quadrupole can be assimilated to a thin lens in the centre of a drift

space of length l. The focal distance of the lens is related to the focusing parameter of the quadrupole by

The magnitude of the nuclear waste problem can be deduced from table 2.17 which shows the amount of spent fuels discharged in the OECD countries in 1992 [36].

We recall that nuclear power only accounts for 5.8% of the total world energy production. This small percentage will, however, lead to a spent fuel inventory of about 200 000 tons by the year 2020. The annual production of spent fuels amounts to about 8000 tons. This figure is to be compared with

the present spent fuel recycling capabilities of around 2000 tons per year, mostly by the COGEMA La Hague facility.

At present, two different strategic approaches are proposed for high — activity nuclear waste disposal:

1. Direct spent fuel element disposal, without any reprocessing. Such an approach is favoured by, among others, the US, Sweden and Swizerland.

2. Spent fuel reprocessing with the aim of optimized extraction of transuranics and fission products and, possibly, their transmutation by nuclear reactions into less radiotoxic or short-lived species. This approach is followed, notably, by the UK, Japan, France and Belgium.

In both cases, some sort of storage of radioactive wastes is needed. Two options are considered:

1. Deep underground storage with or without possible retrieval.

2. Surface or sub-surface storage.

It is clear that these two last solutions can only be temporary, since the half­life of many of the wastes exceeds, by far, the life span of civilizations. The proponents of such solutions argue that technical progress may allow a better evaluation of the safety or feasibility of alternative solutions. One should note, however, that such progress requires experimenting with deep underground storages in dedicated laboratories on the one hand, and separation and transmutation studies on the other hand. It may seem, there­fore, paradoxical that the most vocal advocates of temporary storage oppose both underground laboratories and reprocessing. The paradox can be understood as an aspect of a strategy aimed at pulling out of nuclear power altogether. Only after a withdrawal is obtained will the question of existing wastes be seriously examined. In that case the only possible solution will be deep underground disposal, but it is untimely to acknowledge that fact while fighting the anti-nuclear struggle!

In the future, the relevance of the two basic choices, direct storage or reprocessing, will depend on the development of nuclear power.

1. In the case of withdrawal from nuclear power in the near future, direct storage is the most natural choice. Reprocessing policies are consequences of investments made in the frame of the deployment of fast breeders. The phasing out or standing still of the breeder programmes raised the question of the future of the large reprocessing facilities like those of BNFL and COGEMA. It was found that using the separated plutonium as fuel in thermal neutron reactors had some advantages (decreased need for enriched uranium and reduced volume of the most active wastes) at a very modest cost [39]. An a posteriori policy of waste separation and transmutation followed.

2. At the present world level, nuclear power has only a marginal role in alleviating the waning of reserves and the environmental degradation problems associated with energy production. Long-term continuation of nuclear power would only be justified at a much higher level than at present. In that case, we have previously noted that breeding will be mandatory. Reprocessing will be necessary and the nuclear waste issue will be completely different. For example, using the values given above for scenario A2N in 2050, nuclear power would reach as much as 9000 GWe, more than 20 times more than at present. With the PWR technology, the annual discharge of spent fuel would rise to 260000 tons. Should this be disposed of underground, four sites equivalent to the US Yucca Mountain would be needed each year. Using fast reactor technology, both plutonium and uranium should be recovered from the spent fuel and, aside from tech­nological losses, the highly active wastes would be limited to fission products and minor actinides, i. e. about 9000 tons per year. Furthermore, given the existence of reprocessing facilities, it might be feasible to transmute minor actinides as well as some of the long-lived fission products.

Underground disposal

From the preceding, it seems probable that deep underground disposal will be necessary in all cases. It is, therefore, important to understand the

nature and amplitude of the hazards which might be associated with such a site. An order of magnitude of the dangers associated with deep underground storage can be obtained by comparing the activity of the stored wastes with the radioactivity of the earth’s crust. Assuming that the storage site is 500 m deep, the comparison can be made with that of the first kilometre of crust. The mean activity of the crust is around 1500 Bq per kg. Considered sites have areas of order 1 km2 corresponding to a crust activity of the order of 3.5 x 1015 Bq, a little less than half being due to 40K and more than half to thorium and uranium decay. The crust activity for the area of a country like France amounts to 1.7 x 1021 Bq. This activity has to be compared with that of the materials stored. We take the example of a storage of 100000 tons of irradiated fuel, corresponding to 50 years of operation of the nuclear reactors of France, a very highly nuclearized country. The activity of the storage is shown as a function of time in table 2.18 and figure 2.6.

10[8]‘

1011

Although this is a very rudimentary approach, the comparison of the activities of table 2.18 with the total activity of the crust of France shows that the average increase of radioactivity over France due to nuclear waste storage will remain very small at all times. Assuming that at least 100000 years are needed for a complete diffusion of the wastes one sees that, even locally, the dose which might be delivered to the most exposed population will not exceed a few times that due to natural radioactivity.

More precise diffusion calculations, such as those displayed in Appendix I, give the following results.

• At no time in the future, in a normal situation, will the dose delivered to the most exposed population exceed 0.25 mSv/year, i. e. a factor of ten below natural irradiation

• The main contributor to the dose is I. Almost all stored I will be released within a time span of approximately 1 million years. Due to their small solubility and mobility, actinides have a very small contri­bution.

• In case of an accidental situation, such as drilling a well through the repository and drinking the extracted contaminated water, the maximum dose to the most exposed population should not exceed a few mSv/year. In this case 9I remains an important contributor but Ra takes the lead for longer times. It is a descendent of 238U. In the case of reprocessing, its influence will decrease considerably.

• The amount of heating by the radioactive wastes at the storage site will have an essential role in determining the surface, and thus the cost of the storage facility. After 100 years plutonium and minor actinides will play the dominant role in heat production and the cost of their incineration will have to be evaluated in comparison with the ensuing cost saving of storage.

In order to have an estimate of the cost of deep underground disposal, we take the example of the US Yucca Mountain site which has been accepted as a site for deep underground storage of nuclear wastes in the US. The site would cover about 6 km2 honeycombed with about 100 km of tunnels [37], while the maximum storage capacity should be 70 000 tons. The cost of the site would be more than 15 billion dollars, corresponding to an additional cost of nuclear electricity of about 1 mil/kWh.

Delayed neutrons are associated with the beta decay of fission fragments. Indeed, after prompt fission neutron emission the residual fragments are still neutron rich. They undergo a beta decay chain. The more neutron rich

the fragment, the more energetic and faster the beta decay. In some cases the available energy in the beta decay is high enough to leave the residual nucleus in such a highly excited state that neutron emission instead of gamma emission occurs. This process is illustrated in figure 3.5.

The eventually emitted neutron is said to be delayed (with respect to the fission). The delay is determined by the beta decay time constant. Delays vary between fractions of a second and several tens of seconds. Probabilities for delayed neutron emission are of the order of or less than 1% per fission, or per prompt fission neutron.

Beta-delayed neutron emission is highest when the emitted neutron binding energy is small. This is true when the neutron emitter has an odd number of neutrons just above a neutron shell closure. In particular beta decaying nuclei with neutron numbers equal to 52 (N = 50 closed shell) and 84 (N = 82 closed shell) are good delayed neutron emitter precursors. Examples are 87Br and 137I.

Beta-delayed neutrons are characterized by their yields flb relative to the total neutron number per fission, and their decay constants rt. The total number of delayed neutrons yield per fission is fl = 22 fli. One may, also, define a mean decay time Td = fliTi)/fl. Thus, in first approximation,

the time which determines the time constant of the reactor is rnf(1 — fl) + flTd rather than rnf. Table 3.6 shows the values of fl, Td and rd = flTd for a number of nuclei. The data are for fast neutron fission. We have also given the values of N/A for these nuclear species, since the more neutron rich fissioning nuclei generally lead to higher values of fl, but often to smaller values of Td.

From table 3.6 and using equation (3.101), we see that the doubling time will range between 0.1 s and 1 s. The smaller the value of rd, the more difficult

will reactor control be. In particular reactors fuelled exclusively with minor actinides would have low values of rd.

Because of the important influence of the delayed neutron fraction ft on reactor safety it is customary to express reactivity in $ units: a positive reactivity of 1 $ is a reactivity equal to ft, corresponding to a multiplication coefficient keff ~ 1 + ft. Of course reactivities can also be expressed in fractions of unity.

The importance of the delayed neutrons in reactor safety can be seen better examining a modified neutron kinetic equation that includes the effect of delayed neutrons.

If the material is composed of N different nuclei, the interaction occurs on nucleus k if

k -1 n k

ST <£ ST < ST. (5.14)

i = 1 i = 1 i = 1

Figure 5.2 shows an illustration of this selection method.

(1) Capture

There are two ways to handle capture:

• Analogue capture, which is a ‘true’ capture: the neutron is absorbed with probability oa/o-T where absorption denotes all reactions like

((n, f), (n, a),…) except (n, n,…).

• Implicit capture: the neutron has a weight Wn which is decreased to W = (1 -(oa/oT))Wn by the capture. This way of handling captures is useful to obtain better statistics (the neutron is still alive after a capture, but its contribution is smaller).

While being slowed down, protons may undergo nuclear reactions. For proton energies larger than, typically, 100 MeV, the most violent reactions are called spallation. These account for most of the neutrons produced.

In a crude, black nucleus model, the reaction cross-section reads

°(е )=^^1 — — E (6.7)

with the geometrical cross-section

T = ^(1.3A1/3 + 1)2 barns.

Vc is the coulomb barrier:

1.44Z A + Ap

1.3A1/3 + 1 A

With these expressions it is possible to derive a nuclear range. For high — energy protons, the cross-section reduces to the black nucleus value, and thus the nuclear range reads approximately

Thus, for

• beryllium Rnuc = 35 cm

• lead Rnuc = 16 cm.

The probability that 1 GeV protons suffer nuclear reactions is very high both for beryllium and for lead. The nuclear range is smaller, relative to the electronic range, for light nuclei. On the other hand, the energy depos­ited in the target nucleus following a nuclear encounter is larger for heavy targets. In a simple forward scattering picture (a la Glauber) one expects that the number of target nucleons hit, and hence the energy deposited in the target nucleus, is proportional to the target thickness, i. e. to A1/3. It follows that the ratio of nuclear energy loss to electronic energy loss scales like (A/Z)E0:75.

The over-simplistic considerations we have just made are only intended to give a feeling of the physics of the interaction of high-energy protons with nuclei. It showed that the proton energy should be chosen high enough that nuclear energy losses exceed electronic energy losses. A more detailed treat­ment requires nuclear cascade simulations.

9.1 Basics of reprocessing

Fuel reprocessing was born at the same time as nuclear energy. It was developed within the Manhattan project in order to recover the plutonium needed for the fabrication of the Nagasaki atomic bomb. An excellent account of the pioneering techniques used by Seaborg and his collaborators can be found in reference [138]. Here, a wealth of different methods which have been used for recovering uranium and plutonium can also be found. Aside from the military needs to recover plutonium, with the associated proliferation aspects, the attractive potential offered by fast neutron breeders led to the building of industrial plants to recover both plutonium and uranium from spent nuclear fuels. The largest of these plants that are opera­tional today are La Hague and Sellafield. Because the breeding programmes have been stopped, these large plants have been converted to MOx fuel production for commercial light water reactors, a high recovery efficiency of plutonium and uranium being required. These plants use organic solvents for highly efficient recovery of these two elements. The reference molecule which has become a standard is tributyl phosphate (TBP) from which the so-called Purex (plutonium extraction) process was developed. The high efficiency allows quasi-complete plutonium extraction (99.9%), so that this element could be practically absent from the wastes. Of course, this requires complete incineration of plutonium in standard or dedicated reactors. Logically the minimization of the waste radiotoxicity requires that ameri­cium and curium should also be separated and incinerated. This led to the development of enhanced separation methods, based on the same principle as Purex, such as Diamex and Truex.

Although the wet process, Purex (which implies dissolving the fuel elements in an aqueous acid solution), is by far the most used and the only one which has reached industrial status, other processes which do not require aqueous dissolution have been explored. This has been done in two main instances:

1. The molten salt reactor programme initiated at Oak Ridge National Laboratory [49, 50]. Here the problem was, essentially, to recover thorium and uranium from a mixture of lithium, beryllium and fission product fluorides.

2. The fast breeder programme. The interest in the anhydric recovery of plutonium and uranium stems from the high plutonium enrichment of fast breeder fuels which leads to increased risks of criticality, especially with hydric processes, and from the interest of metallic fuels which lead to more energetic neutron spectra, with higher breeding capability. Indeed, the metallic fuels lend themselves very easily to fluorization.

While it is generally admitted that incineration of plutonium leading to a stabilization of the plutonium inventory could take place in thermal reactors, using specially designed MOx fuels, or in specialized fast reactors, the incineration of minor actinides has not led to such a consensus. As could be seen in table 11.5, which we repeat here for clarity, MAs are strong neutron poisons in thermal reactors.

The most efficient plutonium thermal incinerators are, at the same time, efficient producers of minor actinides. This can be seen from table 11.4. It seems, therefore, if plutonium incineration is to be carried out in thermal reactors, that these must be supplemented with minor actinide spe­cific incinerators. Due to the bad neutronic properties of MAs (especially small delayed neutron fractions) it seems that ADSRs would be a good choice, if not the only choice. Below we discuss at some length the properties of ADSRs using minor actinide fuels.

In the case of radioactive elements brought to the surface along with the core samples, the first effect to be considered would be the irradiation of the workers through direct contact with the ‘waste’ samples brought up. Other scenarios impacting a larger group of population can be devised. Direct ingestion of the samples is obviously not to be considered. Direct ingestion through water contamination, however, is conceivable. The drinking water would then have to wash through the sample. Such a scenario is hard to imagine, the more so as the probability that drilling would run precisely through a fuel component is very small. In contrast, the external irradiation hazard must be evaluated, especially for the workers. Safety analyses must include a set of hypotheses concerning the society, in particular the fact that such a scenario implies that knowledge and technologies equivalent to those we have today (deep drilling capability) are available. The discovery of an atypical object should then trigger measurements whose results would naturally lead to limiting the risk solely to the workers on that parti­cular site. In any case the number of potential victims of a serious irradiation will remain small as well as the volume of extracted radioactive material (at most 10~7 of the total deposit).