Because of structural constraints, the flux in critical reactors cannot be too arbitrarily high.

We first discuss the maximum neutron flux which could be achieved in either a fast or a thermal reactor. This is an important quantity since the lifetime of a nucleus in a neutron flux is ‘o-a, independent of the nucleus concentration. The maximum heat density which can be extracted provides the maximum value of the product Xf’. At present, values of 500 W/cm3 are design values for liquid metal cooled reactors. This leads to a value of £f’ = 500/ef = 1.5 x 1013 fissions/cm3/s. Very high values of the flux can be obtained if Xf is very small. However, Xf cannot be arbitrarily small since the reactor has to be critical, thus kx = ^{Xf/[Xf(1 + a) + Sc]} > 1 where Xc is the macroscopic capture cross-section of components other than the fissile part. Thus one should have Xf/Xc > 1/(v — 1 — a)~ 1. As an example of a thermal system, we consider pipes containing a molten salt immersed in a heavy water tank. The lower limit of Xc is given by that of heavy water, Xc = 0.000 044. With a fission cross-section of 500 barns this corresponds to a fissile nucleus density of 0.8 x 1017/cm3. The maximum maximorum of the neutron flux in a thermal reactor is thus 3.4 x 1017. Of course, due to the components of the salt and of the pipes, such a value will probably never be reached. However, fluxes ten times smaller have been considered, for example, by Bowman [2]. For such high flux the lifetime of fissile nuclei would be extremely short: 5 hours for a 1017 n/cm2/s flux. The inventory in fissile material of the reactor would also be extremely small: for a density of fissile nuclei of 3.0 x 1017/cm3, corresponding to a flux of 1017 n/cm2/s, the total fissile mass necessary to produce 3 GW would be only 700 g! The total volume of the reactor would be 6 m3.

For fast reactors we consider a fissile species diluted in molten lead with Xc = 3 x 10 4 cm—1 and thus a maximum maximorum flux of 5.0 x 1016 n/cm2/s. For such a flux the lifetime of the fissile species would be around 3 hours. The minimum inventory for a 3 GW reactor would be 350 kg. This shows that thermal reactors have a higher potential for small inventories and fast burn-up. Of course the actual implementation of these potentialities could be very difficult. In fact, the more or less fissile nature of the fuel for thermal neutrons has a deep influence on the incineration rate achievable. This can be seen in the following more quantitative, although schematic, analysis.

We consider a homogeneous infinite reactor with only two components:

1. the fuel, characterized by its atomic density nfuel, absorption cross-section ^, and its neutron multiplication coefficient kfuel > 1, and

2. the coolant, characterized by its atomic density ncool, and its absorption

cool)

cross-section aa.

The aggregate reactor is characterized by its atomic density nreac = nfuel + ncool, its absorption cross-section

nfuel (fuel) . ncool (cool)

—- aa 4——— aa 7

nreac nreac

and an effective multiplication coefficient

We define the atomic fraction of the fuel x = n^l/n^c. The criticality condition kreac = 1 allows us to write x as

Aside from the criticality condition it seems reasonable to assume that the fission density is limited to a specific value w. Thus

with

(fuel) a(fuel)

a — af

(fuel)

In the case of fissile mixtures (kfuel > 1), it appears that the main difference between fast and thermal reactors lies in the ratio aafuel)/aacool). There is a clear advantage to using coolants with small absorption cross-sections. As examples, for heavy water ncoolaicool) = 4 x 10-5 for thermal reactors, and ncoolaicool) = 3 x 10-4 for lead and fast spectra. Fuel absorption cross­sections for thermal neutrons exceed 500 barns while they lie around 2 barns only for fast neutrons. It follows that, for fissile mixtures, incineration rates with thermal neutrons could, in principle, be three orders of magnitude larger than those with fast neutrons.

The situation is different for non-fissile (minor actinides, for example) mixtures. In this case the major difference between incineration of thermal

and fast neutrons is that of the corresponding fuel multiplication factors. The subcritical nature of the MA fuel with thermal neutrons implies the use of an ADSR to perform the incineration. Dilution of the fuel would thus be counterproductive, since it would decrease the reactor multiplication coefficient kreac below kfuel, and thus require higher accelerator current to keep the neutron flux constant. The incineration rate reduces to the first term of the right-hand side of equation (3.99), i. e.

w

Ainc =—— (3.100)

nreac

which means that it depends, essentially, on the fission density. Indeed, because of the condensed nature of the components of all practical reactor designs, it is not possible to play very much on the value of nreac. For example, for water, the atomic density is 1023/cm3, while for lead it is 0.3 x 1023/cm3 and for uranium 0.6 x 1023/cm3.

As already said, particle transport looks like a theoretical experiment: a particle is followed from its birth (the source), throughout its life, to its death (absorption, escape). Probability distributions are randomly sampled using transport data to determine the outcome at each step of its life (figure 5.1). Specific techniques have to be implemented for critical problems, where the neutron chain length can reach infinity. In MCNP, critical calcula­tions are known as KCODE calculations. In all cases one has to know:

• if the neutron interacts or not in a medium

• if yes, on which nuclei of the medium

• what kind of reaction occurs

• what are the ‘secondary’ particles emitted

Let us see how these steps are handled in MCNP.

Interaction: yes or no?

If one considers a neutron in a material, this neutron can escape or interact in the material. The probability for a collision to occur between / and / + d/ is

p(/) d/ = exp(—XT/)XT d/

where XT is the macroscopic total cross-section. One has to sample / accord­ing to this exponential probability law. Let £ be a random number in [0,1[ uniformly distributed. One can write

p/

p(/) d/ = 1 — e—StZ

0

that is to say / = — (1/XT) ln(1 — £) which can be replaced by / = — (1/XT) ln £ because 1 — £ and £ have the same distribution. The probability distribution of / is obtained by estimating the length of the interval A£ corresponding to the interval d/,

d — = d = exp(—St/)St (5.12)

which verifies that p(/) obeys the distribution given in equation (5.10).

• If / is greater than the distance to the edge of the material, the neutron escapes; the neutron is then placed on the surface separating the medium being exited and the test for the medium being entered is done again.

• Otherwise, an interaction occurs at distance /.

What is the interaction? Depending on the interaction, MCNP answers the following:

1. What is the velocity of the target nucleus?

2. On which nucleus does the collision happen?

3. How many photons are emitted? (This is optionally done if MCNP follows neutrons and photons; but here, we will not discuss that process.)

4. Is the neutron still alive or is it captured?

5. Is it an elastic scattering or an inelastic reaction?

6. What are the energies and directions of the new outgoing particles (if any)?

In the following, we show the main ways to answer these questions. £ will denote a random number in [0,1[ uniformly distributed.

In most hybrid reactor concepts, the external neutrons are provided by the interaction of accelerated charged particles with matter. The most widely proposed systems use high-energy protons. A few other propositions resort to electrons or deuterons as well as muons as originators of neutron — producing reactions. Because of the importance of high-energy protons we discuss this case more thoroughly.

6.1 Interaction of protons with matter

Energetic protons and nuclei interact with matter mostly by collisions with electrons. These lead to progressive energy loss.

The reactivity of any reactor is generally temperature dependent. Critical reactors have, for obvious safety reasons which we have discussed in chapter 3, a negative reactivity temperature coefficient. For example, PWRs have a coefficient between 5 x 10~5 and 10~4 per °C [48]. This means that a PWR reactor has a reactivity at zero power between 0.03 and

0. 015 higher than at nominal power. The temperature coefficient of fast reactors is, usually, smaller than that of thermal rectors. For sodium cooled fast reactors it is around 10~5 per °C [48]. A similar value has been calculated by Rubbia et al. [76] for their Fast Energy Amplifier.

As already stressed, the neutron balance of 239Pu is deteriorated in a thermal spectrum, due to a large parasitic capture cross-section. Moreover, the isotopes of Pu, Am and Cm with an even number of neutrons will reach large inventories because of their small absorption cross-section. Even if the asymptotic Th/U cycle has large advantages in an epithermal neutron spectrum, the Th/Pu —- Th/U transition will be more difficult to optimize in such a spectrum. Moreover, the disappearance of 241Pu by decay plays an important role in the plutonium inventory needed to start a Th/Pu reactor. The isotopic composition is also an important parameter, and a plutonium coming from a MOx fuel, where the 239Pu proportion is smaller than for a plutonium coming from a UOx fuel, will be less advantageous. Figure 11.10 emphasizes this phenomenon.

The thermal spectrum, largely favourable to the asymptotic Th/U cycle compared with the fast spectrum, seems not to be the best way to start the transition towards the thorium cycle, starting from the plutonium coming from the PWR spent fuel. An optimized way to operate the transition could thus be a coupling between fast and epithermal reactors. The role of the fast-neutron reactors would be the management of plutonium and minor actinides produced by the present PWRs, and the production of 233U which could be used to start Th/U MSR very close to equilibrium. Different possibilities can be considered for the fast-neutron systems, depending on the development target. For example, fast-neutron reactors could have a U/Pu core which breeds the required quantity of plutonium to maintain the reactivity, surrounded by thorium blankets, in which 233U is produced. This coupling offers the possibility of fast deployments, as can be seen in figure 11.11.

An example of a complete computation involving all the radionuclides con­tained in the spent fuel is shown in figure I.1. It has been done by ANDRA as

Table I.2. Mass, radioactivity and half-life of the main radionuclides present in 20 000 metric tons of irradiated fuel at discharge.

Figure I.1. Doses at the outlet as a function of time for various nuclides. The source is composed of 21 600 metric tons of non-processed irradiated fuel. Computation made by ANDRA.

part of the verification of the characteristics and performance of the geo­logical barrier on the Meuse/Haute-Marne site. As a consequence, zero efficiency was ascribed to the man-made containment devices that are to surround the packages, whose purpose is, on one hand, to delay the establish­ment of water contact with the packages, and subsequently, on the other hand, to hold back and delay the migration of any radionuclides released. This migration delay is obtained either by inserting strongly absorbing modified natural materials (bentonite) or by inserting concrete, thus creating an alkaline environment to decrease the solubility of many compounds. The source term includes the following.

• A fraction of the radionuclides are assumed to be unstable, i. e. they are released as soon as water reaches the fuel; this concerns in particular 15% of the iodine and 20% of the niobium and the nickel in the inventory.

• The solubility of the elements is taken into account right at the beginning of the fuel dissolving step.

• The computation is done for 21 600 metric tons IF. This value takes into account the present IF French reprocessing status and fits into a larger framework of complete computation, including the contributions due to vitrified wastes, hulls and endcaps. It corresponds to that portion of the fuel that will not have been reprocessed.

The maximum flow of activity is due to iodine, with a peak of 107 Bq/ year at 2 x 106 years. The dose, too, is dominated by iodine. Its behaviour being governed by diffusion, the appearance of iodine at the outlet will be roughly proportional to the source term, that means to the inventory stored. It follows that, in order to compare results for a larger volume stored, the activity flows should be multiplied by the corresponding factor, a maximum of about 5 for 100000 metric tons of IF stored.

Niobium is limited both by its diffusion and by its precipitation. Tech­netium, like selenium, is limited by the precipitation of solid compounds; the dose associated with these two is thus not very sensitive to the source term but, rather, proportional to the area of penetration of the radioelements in the clay layer.

In principle, criticality accidents such as that of Chernobyl should be impos­sible for an ADSR. However, this is true only as long as one can monitor the effective value of the reactivity. As shown in chapter 7, this monitoring cannot be done solely by relating the beam energy to the reactor output energy: an increase of the reactivity of the subcritical part can be accom­panied by local poisoning of the spallation source in such a way that the output energy does not increase but may, on the contrary, decrease until a critical situation appears. It is thus necessary to devise elaborate ways to monitor the effective reactivity of the subcritical array.

One aspect, which is seldom stressed, of ADSRs is that it requires more technical skill and good maintenance to keep them running than for critical reactors. Indeed, high-intensity accelerators are and will remain rather diffi­cult to operate. Loss of expertise of the staff as well as poor maintenance will, inexorably, decrease the performance of the machine until it finally stops. In contrast, as the recent past shows, critical reactors are apt to run in rather bad shape and do not necessarily need the best team to be operated, with the dangers associated with such a situation. It can be argued, therefore, that ADSRs can offer safety against a societal disorder.

Aside from criticality accidents, ADSRs are subject to risks similar to those of critical reactors, such as solid fuel core melt-down, radioactive leaks into the environment, etc. In addition, the coupling between the accelerator and the subcritical medium may be the origin of weaknesses with respect to safety, such as window breaking or propagation of radio­activity through the accelerator.

For large subcriticality levels of more than a few per cent, the delayed neutron fraction has no influence on the safety of the reactor. This means that it becomes possible to use fuels with large minor actinide concentrations or plutonium without compensating for the small delayed fraction by the presence of 238U. Similarly, the sign of the temperature and void coefficients have a reduced influence. However, they should be limited so that sub­criticality should be guaranteed at all power levels of the reactor. In particu­lar, overly negative coefficients should be avoided to prevent criticality in the case when an accelerator trip leads to a sharp fall of the reactor power.

The high tolerance level of ADSRs with respect to the fuel’s neutronic properties should make them excellent tools to study new reactor concepts by relaxing many safety conditions. For example, the same accelerator could feed different subcritical systems like molten salt, gas or lead cooled reactors. Such prototypes could allow studies of corrosion, radiation defects and fuel evolution in realistic conditions with less stringent criticality- control-related constraints.

The largest coolant temperatures are limited to 350 °C by pressure in the case of water cooled reactors and to 600 °C by corrosion in the case of liquid metal cooled reactors. Higher temperatures would allow higher efficiencies for electricity conversion, using combined cycles, as well as heat cogeneration. They might also have interesting chemical applications like thermal decomposition of water to produce hydrogen. High temperatures can be reached with a gas coolant, especially helium. These considerations were at the origin of the studies on high-temperature gas reactors (HTGRs). These reactors also have, potentially, interesting safety properties, although they use graphite as their neutron moderator like the British Wind — scale or the Chernobyl RBMK reactors. The high operating temperature would prevent the Wigner effect which led to the Windscale reactor accident. Using helium rather than water as coolant would ensure strong negative temperature coefficients, in contrast to the case of the water cooled RBMK reactors. The strong negative temperature coefficient ensures a breaking off of the chain reaction in the case of a loss of cooling. After reactor shut­down, the fuel element temperatures will rise until radiation cooling takes over. This is possible because fuel elements are designed to be able to sustain very high temperatures. The fuel is made of microspheres (TRISO spheres) of fissile and fertile nuclei surrounded by several layers of carbon, which ensure that no fission products can escape from the spheres. The microspheres are, themselves, embedded in carbonaceous materials which constitute the fuel rods. These are placed in graphite blocks, through which holes allow cooling gas circulation. Extensive tests were carried out in Germany, on the AVR reactor, to evaluate the behaviour of the fuel with temperature. The operating temperature is around 1000 °C. The fuel was tested at 1600 °C for several hundred hours and very small fission product release was observed. For moderate power reactors with around 150 MWe, calculations show that, in the absence of cooling, a maximum temperature of 1600 °C can be reached for a few tens of hours. The temperature is limited by radiation cooling. This is efficient because not only the total power but also the specific power of the reactor are kept small. The specific power is limited to 6kW/l, to be compared to the 100kW/l for PWRs.

The probability of significant radiation release has been estimated to be 10~8 per year, i. e. three orders of magnitude less than for PWRs.

The main safety concern for the HTGR is that air intrusion in the vessel would cause the graphite to burn.

The problem description is in one file*, the input file. This file contains geometry description (cells defined with surfaces), materials, neutron source, calculation * Note that the neutron source may be defined in a separate file.

type (critical or standard), etc. This file has a special format. It contains delimiters, data entries (or cards) and comments. Each line is limited to 80 char­acters; but a single data entry may lie on more than one line if each of these lines (except the last one) ends with an & (ampersand). MCNP is case insensitive.

The general structure of the input file is:

Title line

Cell cards

Blank line delimiter

Surface cards

Blank line delimiter

Data cards

Blank line terminator

Comments can be inserted on separate lines: the line must begin with a ‘c’ followed by at least one blank. They can also be placed at the end of a data entry: after a $ (dollar sign).

As explained above, for thick target measurements, the connection between the high-energy INC codes and the low energy is usually done at neutron energies of 20 MeV. However, at such low energies, the INC codes are not expected to perform well. Work is in progress to extend the low-energy codes to much higher energies, typically 150 MeV. To that end, the MCNPX code of Los Alamos [114] uses pre-equilibrium models to generate data files specific to the different nuclear species. The approach of the Bruyeres le Chatel group is, rather, to use an optical model calculation [115]. Figure 6.7, prepared by Koning, compares an experimental neutron spectrum, as observed at JAERI [116], to the traditional calculation and to a calculation incorporating the Bruyeres le Chatel approach. The improvement is striking. Calculations using the new library prepared by Los Alamos give an agreement with experiment similar to that obtained with the Bruyeres le Chatel method. In particular one can see that the strong discontinuity at 20 MeV displayed by the traditional calculation, which is due to the inaccuracy of the INC model, is completely removed in the new calculations.