For thermal reactors a temperature increase leads to a hardening of the neutron spectrum. In turn this leads, in general, to a decrease of the capture and fission cross-sections. The effects of these reductions depend on the specific properties of the fissile and fertile nuclei in the thermal region. For example, a standard enriched uranium fuel has a negative spectrum temperature coefficient which becomes positive with increase of the plutonium content of the fuel.

A temperature increase tends to decrease the density of the materials. This effect is especially important for liquid coolants whose dilatation transfers matter into the expansion vessels. Thus, a temperature increase decreases the relative concentration of a liquid coolant. This may have very different effects for different systems. In PWR reactors the slowing down of the neutrons tends be less efficient because of the drop in water density. This leads to a drop of the fission probability while the capture rate in water is also decreased. These two effects are opposite but their net result is a decrease of the reactivity. In RBMK reactors the neutrons are slowed down by the graphite, while water ensures the cooling and captures a fraction of the neutrons. A temperature increase leads to a decrease of the number of captures in water which is not counterbalanced by a decrease in the fission rates; thus, the temperature dilatation effect tends to increase the reactivity. In liquid sodium cooled fast reactors the decrease of the sodium density leads to a hardening of the spectrum and, therefore, increases the fission rate, at least for large reactors.[19] For lead cooled reactors, because of the smaller slowing down power of lead, this effect is very small.

Void effect

In the case of a large temperature increase, in an accidental configuration, vapour bubbles may appear in the coolant. For PWR reactors this has a negative effect on the reactivity, because of the dominant influence of the spectrum hardening. For RBMK reactors the effect is strongly positive because of the dominant effect of the decrease of the capture rate in the coolant. In sodium cooled reactors the effect on reactivity is positive, but it occurs at a much higher temperature than for PWR reactors. In lead cooled reactors the void coefficient is negative [45].

If the reaction is a fission, Np neutrons are emitted according to the value of v(En), the mean number of neutrons per fission (given in cross-section files). The Np neutrons are chosen as

Np = I + 1 if C < v(En )-I

Np = I if C>v(En)-I

where I is the largest integer smaller than v(En) . The energies of outgoing neutrons are chosen using a Maxwell fission spectrum, or an energy — dependent Watt spectrum, or an evaporation spectrum. However, such treatment is not totally correct, because the NP obey a Poissonian distribu­tion; for example, if v = 2.5, MCNP gives two or three neutrons (in order to have 2.5 on average), but it never gives bigger and smaller neutron number (with the same average).

The INC model lacks justification for nucleon energies (inside the nucleus) below around 100-150 MeV. Pre-equilibrium models [85, 97-99] have been used for some time in nuclear physics in this energy domain. These models follow a population of quasi-particle excitations of the nuclear Fermi gas by means of a master equation. Quasi-particle states are characterized by their particle escape and damping widths. Angular distributions are asso­ciated with the escaping particles. In a sense, pre-equilibrium models allow an easier phenomenological adjustment of angular distributions than does the intranuclear cascade. There are many versions of pre-equilibrium models, but unhappily no clear criteria to choose among them, except their ability to reproduce experimental data.

In these processes the spent fuels are first decladded by shearing and sawing. A dissolution of the oxide fuel in hot nitric acid follows. At present, on an industrial scale, uranium and plutonium are extracted by the Purex process.

9.2.1 The Purex process

Solvent properties

The Purex process uses an organic phase consisting of tributyl phosphate (TBP) soluted in a hydrocarbon diluent as extractant. The formula of TBP is (C4H9)3PO4. A possible structure for TBP is:

H

C

C

C

C

O

H—C—C—C—C—O—P—C—C—C—C—H

O

It is thought that TBP forms bonds via the electron of unsaturated oxygen.

This molecule forms a complex with uranyl nitrate UO2(NO3)2 which is soluble in the hydrocarbon diluent but not in water. Similarly it can form a complex with plutonium nitrate Pu(NO3)4. The main complex forming equilibrium reactions for U and Pu read [138]

UO2+(aq) + 2NOf(aq) + 2TBP(o)^ UO2(NO3)2-2TBP(o) (9.11)

Pu4+(aq)+4NOf(aq)+2TBP(o)^ Pu(NO3)4.2TBP(o) (9.12)

which occur at the interface between the aqueous phase (aq) and the organic one (o).

The equilibrium concentrations are correlated by the law of mass action:

Distribution coefficients measure the ratio between the molar concentra­tions in the organic phase and those in the aqueous phase. For example, for uranium in the TBP-aqueous system:

which, using equation (9.13), reads

Du = Ku[NOf (aq)]2.[TBP(o)]2.

From equation (9.16) it appears that the concentration of uranium in the organic phase increases with the concentration of TBP in the organic phase as well as with the concentration of NOf in the aqueous phase. It is important to note that here we are dealing with free TBP concentration. In particular, a larger uranium concentration in the aqueous phase leads to a smaller free TBP concentration in the organic phase (for fixed total TBP concentration) since more TBP is used in the formation of the complex.

The concentration of NOf in the aqueous phase can be adjusted by adding more or less nitric acid, which reacts with TBP according to

H+(aq)+NOf(aq)+TBP(o)! HNO3TBP(o)

which leads to the equilibrium equation

Finally DU can be expressed as a function of the aqueous concentrations [H+(aq)], [NO^(aq)], UO2+(aq) and the total concentration of TBP in the organic phase. Typical values of KU and KH are 5.5 and 0.145 respectively. Table 9.1 from reference [138] illustrates these considerations. It shows that the concentration of the uranyl ions strongly influences the value of DU. The influence of the nitric acid concentration is less dramatic and becomes small for a concentration larger than 1.5 moles/l.

Figure 9.2 shows that TBP has much larger distribution coefficients for actinides than for most fission products. The separation properties can be controlled by playing on the concentration of nitric acid in the aqueous phase and on the uranium concentration in the organic phase.

From the preceding, we conclude that fast spectra are more efficient than thermal spectra for MA incineration, because of their high fissility to fast neutrons. We also get the result that, for the same fission densities, the incineration rates are similar for fast and thermal spectra. This is surprising since, for plutonium incineration, for example, it is usually claimed that the much higher fission cross-sections with low-energy neutrons lead to higher incineration rates in thermal reactors [133]. Indeed, while the fission density is w = nff jo-f у (here nf j is the density of fissile nuclei, afj their microscopic average fission cross-section and ‘ the neutron flux), the lifetime with respect to fission is Tfj = nfj/w = 1/afj’. It follows that small life­times can be obtained for large neutron fluxes and/or large fission cross­sections, even if the fission rates are kept small by limiting the fissile nucleus density. Thus, associating the large fission cross-sections for low — energy neutrons to small fissile density, one expects to obtain much smaller fission lifetimes for thermal than for fast neutrons with the same fission densities.

We want to understand the apparent contradiction between such con­siderations, which seem to hold in the case of plutonium, and the behaviour of minor actinide fuels. We first discuss the maximization of incineration rates in critical reactors.

The maximization of incineration rates

We consider a sketchy homogeneous infinite reactor with only two com­ponents:

1. The fuel, characterized by its atomic density nfuel, absorption cross-section a[fuelj, and its neutron multiplication coefficient kfuel > 1.

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

,• (cool)

cross-section ^a.

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

_(reac) nfuel _(fuel) , ncool _(cool)

oa — oa + oa

nreac nreac

and an effective multiplication coefficient

Note that, because of the condensed nature of the components of all practical reactors’ designs, it is not possible to make the value of nreac vary very much. For example, for water, the atomic density is 1023/cm3, while for lead it is 0.3 x 10 /cm and for uranium 0.6 x 10 /cm. In contrast, nfuel can be varied within large limits provided the reactor remains critical. We define the atomic fraction of the fuel x — nfuel/nreac. The criticality condition kreac — 1 allows us to express x as

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

In the case of plutonium or other fissile mixtures, it appears that the main difference between fast and thermal reactors lies in the ratio oifuel)/oicool). There is a clear advantage to using coolants with small absorption cross­sections. As examples, for heavy water ncoolo(cool) — 4 x 10-5 for thermal reactors and ncooloicool) — 3 x 10-4 for lead coolant and fast spectrum. Thermal neutron fuel absorption cross-sections exceed 500 barns while they range 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. For thermal neutrons the extremely high incineration rates (lifetimes of a few hours) could only be reached with liquid fuels, allowing very fast purification and replenishment. Note that these high incineration rates are made possible
by the high dilution of the fuel in the coolant, so that the small value of «fuel has to be compensated by a high value of the neutron flux

The above considerations are not valid for minor actinide mixtures. In this case the major difference between thermal — and fast-neutron incineration is that of the corresponding fuel multiplication factors, as seen in figure 11.12. The subcritical nature of the MA fuel with thermal neutrons makes fuel dilution 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 optimum situation (which might be unrealistic because of insufficient cooling power) would then be nreac = nfuel. The incineration rate reduces to the first term of the left-hand side of equation (11.16), i. e.

w

Ainc =—— (11.17)

nreac

which means that it depends, essentially, on the fission density, and only weakly on the fissile nucleus density.

From the preceding it is seen that the main health hazards in the distant future associated with deep underground storage come from I and U. The contribution of other actinides will be very weak, with the possible exception of 237Np. One might, therefore, conclude the futility of incinerating actinides, including plutonium, for waste management. That this might not be the case stems from the consideration of heat production within the storage. We give a simple derivation of the temperature which might be produced at the storage site due to the heat production of the stored wastes.

In reactor physics it is customary [48] to define the number of neutrons per unit volume, per velocity bin and solid angle unit n(r, v, fi, t) such that the number of neutrons in volume d3r, at a position r, with a velocity between v and v + dv pointing in direction fi (unit vector) within solid angle d2^ is n(r, v, fi, t) d3r dv d2^. The flux of neutrons is defined as

ф(г, v, fi, t)=vn(r, v, fi, t). (39)

The number of neutrons per time unit with velocity v and direction fi which cross a planar unit surface at position r and unit normal vector u is:

ф(г, v, fi, t)fi • u. The case where ф is isotropic is particularly interesting, since it is nearly satisfied in reactors. Then, if measuring angles with respect to the normal vector u, we obtain the total number of neutrons crossing the surface, regardless of their direction, by

■»::/2

cos(9) sin(0) d9 = 2:ф(г, v, fi, t).

0

We now consider a thin slab with unit surface and an atomic thickness of ns identical nuclei per unit surface. The nuclei have reaction cross-section a. The number of reactions in the slab per unit time reads

The total flux is the directional flux integrated over angle, thus

‘(r, v, t) = 4:ф(г, v, fi, t).

This quantity is usually called the neutron flux. In term of this quantity, the number of reactions per time unit is thus

while the number of neutrons crossing a unit surface plane per time unit is

1 Mr, V; t)).[12]

Instead of computing the number of reactions per unit time, it is instructive to compute the total length travelled by all neutrons traversing a thin unit surface slab, which we assume to have thickness l, that is very small compared with the transverse dimensions of the slab. Then the total length travelled by the neutrons is

^n/2

sin(h) dh = ‘(r, v, t)l

0

where the volume V of the slab is simply l, since it has unit surface. Thus we obtain an expression for the flux:

The formula was demonstrated for an infinitely thin slab. However, for iso­tropic neutron fluxes, it can be generalized to any arbitrary volume. Indeed, any volume can be subdivided, at will, into n small, thin, parallelograms. For each of the elementary volumes we have the flux value ‘ = Lj/V. The average flux over the volume is the average of the ‘ weighted by the volume,

и = V" = Li = L

(‘) E V’ v v

since the total length travelled by the neutrons is clearly the sum of the elementary lengths. This is an important formula since it is the one used in all Monte Carlo simulations. It can be shown that this formula also holds for anisotropic neutron fluxes.

Note that the definition of the neutron flux is different from the usual definition of flux in other fields of physics. For example, in thermodynamics, a finite heat flux through a surface requires a temperature gradient across this surface. The analogy of the heat flux in neutronics is the neutron current defined as

This current is only different from 0 for anisotropic neutron fluxes. One-sided currents are also frequently used and defined by

where N is the direction in which the flux is measured.

The number of interactions of type (a) per cm3 is S(a)p where ‘ is the neutron flux expressed in n/cm2/s. The most important macroscopic cross­sections are the scattering cross-section XS, the absorption cross-section Xa and the fission cross-section Xf = XaPf where Pf is the fission probability.

In systems using solid fuels as small a variation of kx as possible between two refuelling events is sought. From equations (3.133), (3.134) and (3.135) it is

(a) (b)

Figure 3.9. Model fuel evolution in a Th-U hybrid system. The fast neutron flux is 4 x 10 n/cm /s. The evolution of the concentrations of U and fission fragments (F. F.) with respect to 232Th are shown in (a), the evolution of in (b).

seen that the value of k1{t) depends on the initial concentration of the fissile element. An initially breeding value of this concentration induces an increase of k1(t) with time. This increasing trend may be more or less exactly com­pensated by the decrease of k1 caused by the increase of the concentration of fission products. Rubbia et al. [76] have shown that such a compensation was possible over long periods of time. To illustrate the mechanism of this compensation, we use our simple three-component model where we choose representative values of the cross-sections for a fast reactor using the thorium cycle. Thus, referring to table 3.2, the capture cross-section of the fertile nucleus is taken to be 0.45 barns and the fission cross-section of the fissile nucleus to be 2.75 barns. The average capture cross-section for fission products was taken to be 0.15 barns, according to recent calculation results.[25] Starting from a state without any fissile component, figure 3.9 shows the evolution of the fissile part, of the fission product part (a), and that of the multiplication factor k1 (b). The evolution of k1 shows a maximum after about 7 years, starting from zero concentration of 233U. After 3 years, the concentration of 233U is close to 0.135. Starting with this concentration the value of k1 is reasonably found to be constant for at least 5 years, as shown in figure 3.10(b). The maximum value of кж shows that the neutron economy for a critical reactor would be difficult since only 6% of the neu­trons are available for sterile captures and leakage. This point will be dis­cussed later, in more realistic terms. In figure 3.10(a) we show the evolution of k1 when the fissile component initial load noticeably exceeds the equilibrium value. Here there is a fast and continuous decrease of the reactivity. This means that solid fuel hybrid reactors would not be a good choice for incinerating without regenerating a highly fissionable nucleus like 239Pu, for example.

Figures 3.11(a) and (b) are equivalent to figures 3.9(b) and 3.10(b), but for a thermal reactor with the same specific power corresponding to a flux of 4 x 1014 n/cm2/s.[26] One sees that, if the neutron economy is slightly improved (higher value of kx at maximum), kx is stable only for a very short time, less than 1 year. This difference between fast and thermal systems was stressed by Rubbia et al. [76]. Figure 3.11(a) also shows that the electro-breeding^ of 233U is much faster for thermal reactors than for fast reactors. This is a reflection of the fact that the equilibrium concentration of 233U is seven times smaller for thermal reactors.

(a) (b)

Years Years

Figure 3.11. Variation of kx for a thermal system using the Th-U cycle. (a) Starting with no U present in the system at time 0. (b) Starting with an initial concentration of U slightly below the equilibrium value.

Tallies are quantities that are stored during an MCNP run. There are several tally types depending on what one wants to store (current, flux, …). Here, we will present only one type of tally and associated ‘modifier’: this tally is the one which calculates the neutron flux in a cell (or, when used with modifiers, counting rate, mean cross-section and so on). They are always normalized by the number of source neutrons and the volume (or surface for surface tallies) of the cell.

Tallies are defined in the Data cards block. The keyword for tally declarations is ‘F’ followed by a number (three digits maximum). The last digit corresponds to the tally basic type.*

For example, the neutron fluence in cell 1 is written

F4:n 1

* This last digit ranges from 1 (number of neutrons integrated over a surface) to 7 (fission energy deposition) for neutrons. For our purpose, tally basic type 4 (fluence over a cell) is the most useful; then, one can choose for this tally type F4, F14, …, F994, i. e. a total of 100 flux tallies in the same MCNP run.

where ‘:n’ indicates that neutrons are concerned. It is possible to calculate a given quantity in more than one cell; for example, the neutron flux in cell 1 and 2 will be

F14:n 1 2

The result of this tally will consist of four numbers, the flux value and its statistical error in each of the two cells, whereas the line

F24:n (1 2)

is the mean flux in cells 1 and 2 (thus two numbers, the flux and its error).

Muon catalysed d-t fusion has been suggested as a possible means for producing high yields of 14MeV neutrons. As an example we mention the proposal by Petitjean et al. [123]. In the muon catalysed fusion process, negative muons are captured on the lower Bohr orbit of deuterium or tritium atoms. The muon’s orbit radius is around 2.5 fermis, close to the nuclear radius, so that the Coulomb field of the deuteron (or triton) is almost cancelled. The probability of fusion of the muon accompanied deuteron or triton with another t or d nucleus becomes large. After fusion, most often, the muon is shaken out and becomes available for another cycle until it decays or is captured by a heavy nucleus. It has been found that up to 150 fusions per muon can be obtained. According to Petitjean et al. the optimum beam-target combination for negative pion, and thus negative muon, production is a beam of 1.5 GeV deuterons impinging on a carbon target. The HETC [92] simulation gives a maximum negative pion yield of 0.16. It follows that the maximum possible number of produced 14MeV neutrons per GeV of deuteron is around 15. These neutrons could be further multiplied by (n, 2n), and even (n, 3n), reactions. A multi­plication factor of 2 seems a maximum. Finally we see that no more than 30 neutrons per GeV-deuteron can be produced. Since not all muons will be captured by the heavy hydrogen atoms, a maximum number of 15 is more likely. This is a factor of 2 below the neutron yield from protons on uranium. Note the advantage that the pion production target can be completely disconnected from the neutron source. However, the (d, t) cell requires high pressures, and a high magnetic field is necessary to trap and focus the muons. It is doubtful that this technique could be used competitively.