Fick’s law was introduced in the frame of the kinetic theory of gases. It relates the particle flux or current to the gradient of the particle density,

j =-D grad (p). (3.20)

In neutron physics the density is replaced by the neutron flux and the particle flux by the neutron current. For those used to the theory of gases this change of definitions may be the source of some confusion.

Fick’s law relates the current J(r, v, t) to the flux ‘(r, v, t). It reads:

J(r, v, t) = — Dgrad (‘(r, v, t)). (3.21)

We give a simple derivation of this relation, following Lamarsh [55]. We assume that ‘(r, v, t) over a neutron mean free path can be considered as linear.[13] For simplicity we assume that we compute the current at the

origin. Thus,

We compute the neutron current through a surface dAz, placed at the origin, and perpendicular to the z axis. We use the spherical coordinates x = r sin(e) cos(^), y = r sin(e) sin(^), z = rcos(d). It is then clear that any integral over x and y of a linear functional of ‘ will only involve ‘0 + rcos(d)(d’/dz)0. Furthermore, the neutron currents originating from z > 0 and z < 0 have opposite directions. This means that only odd terms will be involved in the total neutron current,* i. e. the term rcos(0)(d’/dz)0. Terms from z < 0 and z > 0 are equal, so that it suffices to evaluate the con­tribution of the z < 0 region and double it. In the absence of external sources, neutrons crossing surface dAz come either from a scattering or a fission event. In a first approach we neglect the contribution of fission neutrons, assuming that Xs ^ Xf. We also assume that the time delay between neutron scattering and the arrival of the neutron at the origin is negligible.

The number of neutrons scattered in an elementary volume dV is Xs'(r, v, t), and those heading towards the surface dAz, assuming isotropic laboratory scattering, is

Xs'(r, v, t) cos(6) dAz d V
4кг2

On their way towards surface dAz, these neutrons may undergo a reaction
which takes them out, thus the number of neutrons reaching the surface is

e_SrrXs'(r, v, t) cos(0) dAz d V
4кг2

Taking into account the remarks of the preceding paragraph we obtain the neutron current,

and

r = M

z 3XT dz J0

Similar relations hold for the other components of J, so that we get Fick’s law, equation (3.21), with

D=X • (323)

Note that, for Xa ^ Xs, D = 1/3Xs.

* This circumstance is responsible for the validity of Fick’s law up to second order.

Although our derivation of Fick’s law assumed isotropic scattering in the laboratory frame, it is in fact possible to extend its validity to the case of mod­erately anisotropic scattering. In particular if Xa ^ Xs, D = 1/3Es(1 — p) where p is the average value of the cosine of the scattering angle.

The derivation also assumed an infinite, homogeneous medium. It is in fact valid when applied in regions several mean free paths away from the medium’s boundary. It is even valid at the boundary between two media, provided the absorption cross-section is small.

Similarly Fick’s law is valid in the presence of external sources, in regions sufficiently far from the sources (several mean free paths).

One of the most serious limitations of Fick’s law, in its simple, mono­group, form, is that it assumes no velocity modification after scattering. This is true in thermal reactors where neutron spectra can be considered to have reached an equilibrium in which up-scattering is as probable as down-scattering. It may also be true for low lethargy[14] fast flux reactors. In both cases Fick’s law can be used to simplify the Boltzmann equation to a diffusion equation.

The only practical way to dispose of actinides is to induce their fission. Fission is accompanied both by energy and neutron production. However, several neutron captures may be necessary before fission occurs, so that the net neutron number necessary for actinide incineration will be Ncap + (1 — v) where Ncap is the number of captures before fission and v the number of fission neutrons. The number of neutrons required depends on the neutron flux magnitude as well as on how hard it is. Let us consider one nucleus of species j(Zj, Aj). It can suffer fission with average cross­section o(f, capture a neutron with average cross-section (here nucleus k is Zk = Zj, Ak = Aj + 1) or decay to several possible other nuclei k, with partial decay rates Xjk (here nucleus k is (Zk = Zj + 1, Ak = Aj), (Zk = Zj — 1, Ak = Aj), (Zk = Zj — 2, Ak = Aj — 4) depending on the type

of radioactivity involved). The fission probability reads

The production of nucleus k from nucleus j can be defined as

Starting with one nucleus i, the number of nuclei j which are ultimately produced is given by the system

yj = X Akjyk + Sij. ( 3.144)

k

The Kronecker symbol Sy expresses the fact that, initially, there was one nucleus i. Knowing the yj, it is possible to compute the number of neutrons necessary to incinerate nucleus i:

Di = X RaPja)yj ( 3.145)

j, a

where the set f yjg is the solution of system (3.144), Ra is the neutron balance for reaction a (fission, capture or decay) and Р(a’) is the reduced transition

rate for reaction a and nucleus j. The values of Ra are given in table 3.9. The expression of D was first given in a slightly different form by Salvatores and Zaetta [38], and generalized to mixtures of nuclei. Table 3.10 gives values of D for important nuclei, as well as for spent fuel mixtures.

Table 3.10 shows [38] that incineration by fast neutrons is always a net neutron producer. This is due to the fact that fission cross-sections of fertile nuclei, which are very small or vanishing for thermal neutrons, are large for fast neutrons. The table also shows under which conditions breeding can be obtained from 232Th and 238U. The protoactinium effect is clearly visible for high thermal fluxes where its extraction is, clearly, mandatory. While, with a moderate flux, breeding can be obtained for 232Th for both a thermal and a fast flux, only a fast flux allows breeding for 238U.

Neutron balance is not the only parameter that should be considered for the choice of a particular system with the aim of waste incineration. The half-life of the waste in the neutron flux is also very important since it determines the inventory needed to reach a specified transmutation rate and the time it takes to get rid of the waste.

Emax is the maximum neutron energy and Elim is the boundary separating implicit and analogue capture: below Elim, capture is analogue (i. e. a true capture) and above it is implicit (the weight of neutrons is reduced). In the latter case, the CUT card is useful to specify a minimum neutron weight before killing them.

PRDMP card

When an MCNP calculation is run, a lot of information and the tally results are written in an ‘o’ file, which is easy to read for a human being, but less so for a computer (lots of text). The PRDMP card is useful to save tally results in an ‘m’ file, which has a pretty standard format and is easier to read with a

computer; in addition, MCNP can read the ‘m’ file and plot some results. To produce such a file, the following example is pretty good:

PRDMP 2j -1

(2j means jump the first two entries, and -1 is to write the ‘m’ file).

In chapter 4 we have seen that, without elaborate source enhancement as described in section 6.4, an energy gain of the subcritical array on the order of 100 can be anticipated. With a desired total thermal power of the ADSR around 1 GWth, one sees that beam powers around 10 MW are needed. This is an order of magnitude larger than beam powers delivered by present accelerators. It is, therefore, important to examine the feasibility of such high-intensity accelerators. Since this discussion requires some knowledge of accelerator physics we give an introduction to this subject in Appendix III. It is shown there that, in practice, linear accelerators and cyclotrons are the best choices available for obtaining high average intensities that are almost continuous. It is thus instructive to review the characteristics of the existing high-intensity Linacs and cyclotrons.

While the size of critical reactors may be arbitrary, the presence of a localized primary neutron source constrains the size and total power of hybrid reac­tors. In this chapter we shall examine this issue, first using a simple, intuitive spherical reactor model, and then taking the example of the optimization of the size of a possible demonstration set-up.

10.1 The homogeneous spherical reactor

Because the neutron source is localized in character, one expects the size of hybrid reactors to be limited. Optimization of the reactor has to be done with respect to several key quantities:

1. The value of the source multiplication factor ks, which relates the beam power to the total power of the reactor, and thus to the energy gain. The possibility of innovative designs, in this respect, is discussed in section 6.4.

2. The value of the effective multiplication factor keff which determines the safety of the reactor. Because of the general positive correlation between keff and ks, the highest values of keff, compatible with safety, are sought. In section 3.6 we have seen that for fast systems a limit of keff = 0.98 seems reasonable.

3. A specific power maximum value is set by the heat removal system. Prac­tically, maximum specific powers of the order of 500 W/cm3 are possible with standard liquid metal cooling.

4. It is important to minimize the fuel volume, and at the same time to minimize the range of specific powers within the system.

In order to give the reader a feeling for the size of hybrid reactors we study a simple spherical reactor model. The reactor is made of three concentric zones:

• The central zone (1), where the spallation reaction takes place and where we neglect neutron absorptions. This spherical zone has radius R1.

• The fuel zone (2) between radius R1 and radius R2.

• A reflector zone (3) between R2 and infinity.

Our treatment is based on the solution of the one-group diffusion equa­tion, which appears to be a reasonable approximation for fast reactors.

The aim of this project is to couple a proton accelerator with a target and a subcritical system of sufficient size to produce a sizable power. This experi­ment could be carried out in the TRIGA reactor at the ENEA Casaccia Centre (Italy) operating as a subcritical assembly. TRIGA (Training Research Isotope General Atomic immersed test reactor) is an existing 1 MW thermal power swimming pool reactor cooled by natural convection
of water in the reactor pool. The fuel elements are cylinders of uranium (enriched to 20% in 235U) with a cylinder of metallic zirconium inside.

At the present stage of the feasibility study, the TRIGA project is based on the coupling of an upgraded commercial proton cyclotron with a tungsten solid target surrounded by the TRIGA reactor scrammed to undercriticality. The flexibility offered by the swimming pool reactor is well suited for conversion to subcritical configuration, which is achieved through:

• the replacement of the outermost fuel ring with a graphite reflector,

• the removal of the innermost ring of fuel core.

The target should be hosted in the central thimble, at present used for high-neutron-flux irradiation. A beam power of a few tens of kW appears adequate to run a subcritical system with an appropriate multiplication coefficient. With ks of about 0.97, the system may produce several hundred kWth power in the reactor and a few tens of kWth in the target.

The reference design for the accelerator is a 220MeV-H2+ Super­conducting Cyclotron based on the concepts developed in the Cyclotron Laboratory of CAL (Nice) for compact superconducting fixed-frequency cyclotrons for hadron therapy. Protons are produced at 110 MeV by strip­ping; the requested beam current is in the range 1-2 mA.

The experiments of relevance to ADSR development to be carried out in TRIGA are:

• subcriticality operation at significant power,

• the possibility of operating at some hundred kWth of power and at different subcriticality levels (0.95-0.99) will allow the designers to validate experimentally the dynamic system behaviour versus the neutron importance of the external source and to obtain important information on the optimal subcriticality level,

• correlation between reactor power and proton current,

• reactivity control by means of neutron source importance variation,

• start-up and shut-down procedures.

During acceleration, individual particles can be defined by their positions X and momenta P, each specified by their Cartesian projections x, y, z, Px, Py, Pz. Within a beam, one can define distribution probabilities for the individual particle coordinates. These distributions are characterized by their variances:

^x = <(x -<x})2} (HL40a)

<2 = <(y -<y})2} (Kb)

<2 =<(z -<z})2} (HL40c)

<4 = <(Px -<Px})2} (III.40d)

4y = ((Py -<Py})2} (m.40e)

<Pz = ((Pz -<Pz})2} (III.40f)

the beam direction is taken to be Oz. It is assumed in the following that Pz ^ Px, Py. Hence the angles of the particles with respect to the beam axis are defined by their two projections dx = Px/Pz and dy = Py/Pz.

The production potential of hydroelectricity is considerable. It amounts, theoretically, to 36 000 TWh, while the resources that can be harnessed in practice are estimated at 14000TWh, more than the present world electric energy production of 12000TWh. In 1990 hydroelectricity produced was only 2200 TWh. From these numbers, it would appear that hydroelectricity could be the main alternative to the use of fossil fuels for electricity produc­tion. However, several factors will limit this possibility:

• Most of the potential is located in Asia (27%), South America (24%) and the former USSR (24%). For the industrially developed countries of Europe, North America and Japan, the unused potential is small.

• The local environmental impact of large hydroelectricity is, usually, quite significant. Not only are the local climate and ecosystem disturbed, but large populations have to be displaced. This is, certainly, the main limiting factor to the establishment of large hydroelectric dams.

• The risks of catastrophic dam rupture. In the recent past, dam ruptures led to some of the most catastrophic technological events, certainly compar­able with Chernobyl. Some of the most dreadful events were:

Morvi (India 1979) 30000 dead Vaiont (Italy, 1963) 2118 dead L’Oros (Brazil, 1960) 1000 dead St Francis (USA, 1928) 700 dead Gleno (Italy, 1923) 600 dead Logan (USA, 1972) 450 dead Malpasset (France, 1959) 423 dead.

Each year, although with less catastrophic outcomes, several dam ruptures are observed in the world.

These limitations led the World Energy Council [27] to anticipate a maximum share of 7% for hydroelectricity in the fulfilment of world energy needs.

Hydroelectricity, after the large initial investment is repaid over a period of 15 to 30 years, is very cheap. In many cases the kWh cost is less than 20.

Finally, note that if hydroelectricity is easily modulated according to need, it is dependent upon the pluviometric regime of the region where the dams are implanted. Long droughts may significantly affect its availability, as has been experienced recently in California.

In the thermal reactor neutrons should be slowed down before they are captured. The survival probability was expressed in equation (3.61). The survival integral reads

(3.85)

with

This integral is only valid for small absorption. For E lying between fission neutron energy and thermal energy a parametric expression of the survival probability p can be used [55]

■]1Q ТПЛ

with the values of a and c for U and Th (which are the main resonance absorbers) shown in table 3.5. cabs is the atomic concentration of the absorber nuclei.

For heavy water £ = 0.509 while Ss = 0.452 cm-1. Taking the case of 232Th and requiring that p should be equal to 0.95, we get the concentration of 232Th atoms.

and, at equilibrium

and the value of kn

The diffusion length has to take into account the slowing down stage. This amounts to adding half the age of the neutron given by equation (3.56) to the diffusion length Lc

a2 (r : E) = —2 ln 0 = 2r.

V 2 3£S2 E

For thermal final energies the slowing down term reads, in the case studied,

with L = JL2 + a2(r : Eth) = 19 cm. The critical mass of fuel is around 1.2 tons. The lower critical mass for thermal systems is, obviously, a consequence of the higher capture and fission cross-sections.

The above calculations of critical masses neglected absorption in the structural elements, and, most important, by fission products.

The Monte Carlo code MCNP (general Monte Carlo N-particle transport code) is one of the best known and used Monte Carlo codes. It can be obtained from the NEA. This code can transport photons, electrons and neutrons. In the following, we will only consider neutron transport. The statistical sampling process is based on the selection of random numbers, analogous to throwing dice in the Monte Carlo gambling casino.