Let us see now the case of a subcritical ADSR. Suppose it is filled with

ooo Odd

Th/ U and cooled by lead; the target is a lead spallation target. In this example, the core is filled with a hexagonal lattice. As already said in the previous example, the lattice that we use is a nice tool but some fuel

(a) (b)

rods are cut. In order to have a better description, we present here the way to overcome that problem; the resulting MCNP file is much longer; we will just illustrate the method on our design with non-realistic sizes (large hexa­gons, thick fuel rods, small reactor). The spallation source cannot be described with MCNP alone (one can use MCNPX of course); we will take a cylindrical source as described in section 5.5.3 to ‘represent’ the spallation source.

The following MCNP input file (named ads) describes the reactor:

Heterogeneous core c

c Exterior c

1 0 1:-2:3 imp:n=0 c

c Iron tank c

2 1 -7.87 -1 2 -3 (4:-5:6) (50:-6) imp:n=1 $cyl.-(reflector

hole(middle))-(beam hole(top)) c

c Lead reflector c

3 2 -10.34 -4 5 -6 (10:-11:12) (50:-12) imp:n=1 $cyl -(core

hole(middle))-(beam hole(top)) c

c Core c

4 0 -10 11 -12 40 imp:n=1 fill=2

5 0 21 -22 23 -24 25 -26 imp:n=1 u=2 lat=2 fill=-10:10 -10:10 0:0 &

liiiiiiiiiiiiiiiiiiii 4 1111111111111111111114 1111111111111111111114 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 1 1 1 1 4 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 1 1 1 4 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 1 1 1 4 1111111333333333331114 1111113333333333331114 1111133333333333331114 1111333333113333331114 1111333331113333311114 1113333331133333311114 1113333333333333111114 1113333333333331111114 1113333333333311111114 1113333333333111111114 1113333333331111111114 1111333333111111111114 1111111111111111111114 1111111111111111111114 111111111111111111111

6 3 -10.34 30 imp:n=1 u=3 $

7 4 -10. -30 imp:n=1 u=3 $ fuel rod surround by lead

8 2 -10.34 31 -32 imp:n=1 u=1 $ c

$ lead from core to the target vessel surface

(51:-46) imp:n=1 $ iron vessel for the target

$ lead target $ lead under the target vessel $ lead above target vessel to pipe

$ hole for the beam in the target vessel $ pipe in the core $ vacuum in pipe $ pipe in the lead reflector $ vacuum in pipe $ pipe in the tank $ vacuum in pipe

c Tank/Reflector surfaces

1 cz 155

2 pz -155

3 pz 155

4 cz 150

5 pz -150

6 pz 150

c Reflector/Core surfaces

10 cz 100

11 pz -100

12 pz 100

c Hexagonal Mesh surfaces

c Beam Pipe surfaces

50 cz 2

51 cz 1.5

c Material

m1 26000.55c 1 $ Iron of the tank

m2 82000.50c 1 $ Pb of the reflector

m3 82000.50c 1 $ Pb of target and core

m4 92233.60c 0.15 90232.60c 0.85 & $ 233U(15%) and 232Th

8016.60c 4. $ 16O of ThO2 and UO2

c Neutron Source

SDEF POS 0 0 0 ERG=D1 RAD=D2 EXT=D3 AXS 0 0 -1 SP1 -5 1.3 $ p(E)/E exp(—E/1.3)

512 0. 9. $ Radial extension (disk of radius 9cm)

513 -9 29 $ Z extension (from source position)

TOTNU

PRDMP 2J -1

As can be seen, the core is a cylinder (cell 4) filled with universe 2, the hexagonal lattice (lat = 2). But this time, the position of each mesh of the lattice is specified: the lattice is made of 10 — (—10) + 1 = 21 hexagons in x, the same in y and there is no z lattice. Each mesh is filled with a specific universe: 1 for lead hexagons and 3 for fuel + lead hexagons (we have[33] organized the line in order to visualize a schematic view of the hexagonal

(a) (b)

lattice: the ‘fuel’ part of the core is more or less cylindrical and the ‘hole’ in its centre is for the target). The geometry is displayed in figures 5.6 and 5.7.

Intensities

Intensities may be limited by several factors:

• The intensity which can be provided by the ion source. ECR plasma sources are able to provide very high intensities with good emittance and very high reliability.

• The pre-injection accelerator. Until recently pre-acceleration was achieved with electrostatic accelerators like Cockroft-Waltons or SAMES. The need to have the source at a potential close to 1 MV leads to complexity and is a cause of breakdowns. RFQs are nowadays able to accelerate reliably several tens ofmA at several MeV [126, 127].

• Space charge increases with the maximum accelerated current. The critical region is at low energy. As shown in Appendix III, a way to decrease space charge effects is to reduce the distance between focusing devices. Here again RFQs are helpful since their wavelengths are of the order of a few mm. It seems that Linacs are more promising than cyclotrons as far as space charge limitations are concerned. In cyclotrons, space charge limitation decreases when the separation between turns increases [125] and thus when the energy gain per turn increases.^ However, increasing the energy gain per turn also increases the high voltage on the cavity and, as discussed below, the risks of trips. In practice, cyclotrons’ intensities seem to be limited to approximately 10 mA. Intensities as high as 100mA should be feasible with Linacs and have been demonstrated for the low-energy injec­tion part [126, 127].

• High transmission of the accelerators is required in order to minimize beam losses. In principle, Linacs seem to have an advantage here. Indeed extraction of the beam from cyclotrons is a delicate point. However, transmissions as high as 99.98% have been reported for PSI [125]. The [46] operational experience from PSI shows that high-intensity accelerators can be run with very limited irradiation risks for operators.

• Beam availability already reaches 85 to 90% today both at LAMPF and PSI. Such figures could be improved for industrial accelerators by using components far enough from their design values and through redundancy of the critical pieces of equipment. Regular and more frequent maintenance would also be efficient. With these improvements, it is estimated that availabilities exceeding 95% should be possible.

• Short-duration trips cause sharp decreases of the beam intensity. These might induce fatigue in the structure elements of the ADSR so that they have to be reduced in number as much as possible. Many of them are due to the failure of individual elements such as RF power units, cavity windows and focusing devices. The number of trips at LAMPF and PSI amount to over 10000 per year [127]. This number should be drastically reduced by several orders of magnitude. Several dispositions seem able to improve the situation considerably. (i) Minimize the number of single items whose failure leads to a beam loss, e. g. RF cavities and power units. These should be such that when one of them breaks down the energy decrease is small enough that the beam is not defocused away from the target. An example is provided by the ATW project [127] which plans to have an individual power unit for each SC cavity. The failure of one unit leads to a decrease of the beam energy by 5.5 MeV to be compared with a total energy of 1000 MeV. This 0.55% decrease is not sufficient for a beam loss at the target position. (ii) Use devices which rapidly compensate the energy loss consecutive to the breakdown of a unit [127] by increasing the gain of the nearby units.

• It is shown, on general grounds, in Appendix III, that energy efficiency improves for higher intensities. Table 6.4 is an example of this gain. It com­pares the energy efficiencies of the present PSI cyclotrons and of a possible 10 MW accelerator based on an extrapolation of the PSI concept [128].

In the process of designing a possible demonstration facility, a realistic study was made, using an MCNP calculation [130]. The fuel composition was the same as that described in the preceding section, except that the total concen­tration of the industrial plutonium could be varied, and the lead fraction was reduced to 30%. In addition to the constraints given above, ks was required to vary by not more than 1% in the first year. Such a value is certainly more than would be acceptable for an industrial reactor, but allows us to start with a higher plutonium concentration (no breeding), and thus to decrease the size of the reactor.

The geometry of the simulated reactor is shown in figure 10.2. The inner radius of the fuel zone was 15 cm throughout the calculations, while the external radius and height were such as to minimize the surface of the fuel zone for a given volume. Figure 10.3 shows how ks depends on the total initial plutonium concentration and on the volume of the fuel. It was found that the evolution of ks with time (dks/dt) was essentially independent of the fuel volume, as expected from the fact that neutron leakage should not vary much with time. A choice of an initial plutonium concentration of 12% led to an acceptable value of (dks/dt) = —0.007/year. In this case a fuel zone volume of 1.5 m3 gave the required initial value of ks = 0.98, together with an average specific power of 150 W/cm3 for a 1 MW beam power.

It appears that the realistic simulations led to about twice as large fuel volumes as the analytic calculations. The main reason for such a

discrepancy is the existence of large neutron losses through the open ends of the fuel zone cylinder.

To quantify the risks associated with underground disposal, we use here the geological data of the future evaluation laboratory in the east of France. This implies nothing concerning the location of the site that would finally be chosen in the event of a decision for definitive storage.

The reference scenario deals with the direct disposal of irradiated fuel, i. e. without fuel reprocessing. The irradiated fuel packages are themselves protected by a system of man-made barriers. The disposal site proper lies at a depth of about 500 m in the middle of a 130 m thick Callovo-Oxfordian

* Agence Nationale pour la gestion des Dechets RAdioactifs: French National Agency for Radioactive Waste management.

t Although schematic, our treatment is rather detailed, since we have not found such a descrip­tion in the general nuclear literature.

* French Physical Society.

clay layer. The clay layer is very poorly permeable, and constitutes the main containment barrier for the very long-lived radioelements.

Water has two actions: first, it is the main ageing agent for the radio­active waste packages, thus helping the release of the radioelements they contain; second, it is the vector that can convey the radioelements across the geological barrier.

In the course of the first few thousand years, water will slowly seep through the man-made barriers and eventually reach the packages. From then on, corrosion will begin, causing alterations that will lead to the pro­gressive release of radioelements to the water. Typically, safety evaluation calculations, taking a conservative slant, estimate that the dissolution process for irradiated fuels will last 106 years. Once they have dissolved in the water, the radioelements will be conveyed through the clay layer via diffusion. Once outside the clay, it is assumed that they will disperse quickly in the limestone layers and that they will reach the ground water used by a critical group of humans, the one most exposed to these losses. The fundamental safety rule (FSR) requires that the committed dose for this population remain constantly below a threshold of 0.25 mSv/year, i. e. one tenth of the dose due to natural radioactivity. We might note that the risk of cancer associated with such a dose is comparable with that caused by smoking one pack of cigarettes per year[62] [176].

Before presenting the computation results from ANDRA, it seems useful to summarize the physics that comes into play in the diffusion of radioelements in a layer of clay, so as to identify the significant parameters and get a feel for the scale of the phenomena.

A particle travelling through a series of electric quadrupoles sees a time oscillating quadupolar field. Conversely, a single long quadrupole with time oscillating fields has focusing properties along both principal directions. Such RF quadrupole structures are often used as guides for low-velocity ions. If the poles have a simple rectilinear and parallel configuration only focusing takes place. It has been realized that when the poles have a spatial periodic structure, acceleration becomes possible for resonant particles. Such very compact accelerating structures are used in modern accelerators to accelerate low-energy protons or other heavier ions. They replace the cumber­some electrostatic accelerators used previously which required the ion source to be put at a very high voltage. RFQs require very precise machining.

The nuclear reactions available for nuclear waste processing are of two types:

1. Transmutation, which by neutron capture transforms a radioactive nucleus into a stable one.* This method is suitable for fission products.

* This may involve intermediary steps through short-lived isotopes.

As stable nuclei could be, simultaneously, transformed into radioactive ones, the method may require an initial separation of the isotopes to be transmuted. However 99Tc and 129I do not require such separation.

2. Incineration, which amounts to nuclear fission following neutron capture.* This method is suitable for transuranic elements. It is always associated with energy and neutron production. It is already applied, on an industrial scale, to plutonium.

The plutonium case

From the preceding, plutonium can be considered according to two different viewpoints. In the breeding strategy it is a nuclear fuel. In standard PWR reactors it appears to be a nuclear waste which is apt to be incinerated. Incineration is possible with thermal reactors like PWRs, but complete incineration will be difficult in this case. Indeed, it is associated with the production of transplutonic elements (americium and curium) which are difficult to incinerate in a PWR.

For the thorium-uranium cycle, 233U would have a role similar to plutonium in the uranium-plutonium cycle. However, in this case, the production of transuranic elements is greatly reduced.

Different nuclear waste reprocessing strategies are possible, depending on the availability of existing reprocessing plants, on the experience of incineration in thermal reactors and on the prospect to use fast reactors.

The variation of the neutron population can be written as

dn{t) —absorptions — escapes + prompt neutrons

dt +delayed neutrons + source ’

where n(t) is the total number of neutrons present in the reactor. The number of neutrons with velocity u is n(t)ft(u), with J0°° ft(u) du = 1. The absorption rate reads

where ra = 1 / (Sa)(u) is the partial lifetime of neutrons for absorption. Similarly, the escape term can be written as

n() = n(t)(P)(u) rP

where rP is the is the partial lifetime of neutrons for escape. (P) is defined in such a way that (P)/((P) + (Xa)) is the relative probability that a neutron disappears from the reactor by escape rather than by absorption.

The prompt neutron production rate is

(1 — ft)vnf(t) (3.105)

where ft is the delayed neutron fraction.

Let Xi (t) be the population of delayed neutron precursor nuclei, A,- their decay constant, and ei their relative production of neutrons at decay time. Then the delayed neutron production rate reads

X Ai "X (t).

We define the total neutron lifetime

11 1 r rP ra

so that equation (3.102) takes the form:

dt) = Г) +(1 — ft)vnf(t) + X Ai"iXi(t) + S(t).

i

The evolution equations for the Xi(t) are

dX (t)

dt

= —A iXi (t)+ cft)

where ci is the yield of fission fragment i.

Given a fission event, the partial delayed neutrons yield is clearly

fti V "ici

so that equation (3.106) becomes

Ш=-’Р1+(1 — ft)v„,(t) + + S(t)

which can be further simplified by writing

dt) = Г)+( 1 — ft)vnf (t) + v X Ai ftiYi (t) + s(t)

i

Xi t)

c

= -AiYi (t) + nf (t).

Equation (3.119) accepts exponential solutions. Suppose that

n{t) = n(0) e! t Ci(t) = Ci (0) e! t

it follows that, for S(t) = 0, exponential solutions exist if

and

The p(u) function is always positive for u > 0. The trivial solution p = u = 0 corresponds to the constant neutron flux of a critical reactor. The limiting value p = 1 (keff ^ i) corresponds to infinitely fast exponential increase with u! i. Figure 3.6 shows how p depends upon the exponential diver­gence slope. It shows that the approximate calculation using average values of the delayed neutron yield and lifetime is very close to the exact calculation, except for very small values of p. Both converge with the prompt curve for p larger than a few $.

Figure 3.7 compares the critical divergence with and without delayed neutrons for a PWR reactor. In the first case, it is seen that, for a reactivity below 1$, the evolution of the reactor is slow, with characteristic times above 1 s. For p > 1$, in contrast, the evolution of the reactor becomes very fast and is comparable with that of the case without delayed neutrons. For p = 2$, for example, the characteristic time is less than 0.02 s.

Inelastic reactions ((n, n’), (n, f), (n, np),…) are assumed to be independent of the temperature, whereas the elastic scattering cross-section (and thus the total cross-sections) is adjusted according to the free gas model previously described.

The probability that the reaction is elastic scattering is

oel _ oel

oinel + oel oT — oa

The probability that the reaction is an inelastic one is

Vmel

VT — Va

If the inelastic reaction ‘wins’, the jth reaction is chosen among M according to

j — 1 m j

Vi <C Vi < at. (5 Л7)

І = 1 І = 1 І = 1

Most existing codes used for high-energy proton-nucleus reactions are based on the intranuclear cascade (INC) model of Bertini [82] for the first stage of the reaction, the final stages being described by an evaporation (EVAP) model like those by Dostrovski et al. [83] or Dresner [84]. The philosophies of the INC and EVAP models are very different: the INC calculations follow the history of individual nucleons in a classical or semi-classical manner, while the EVAP calculations follow the de-excitation of the whole nucleus while it decays from one nuclear level to a lower one. The connection between the two approaches is one of the delicate points of high (or inter­mediate) energy simulations of proton-nucleus reactions. In principle the single-particle approach of INC is justified as long as the wavelength of the incident nucleon is smaller[37] than the nucleon radius, i. e. A < кг/2 Fermi and E > 160 MeV. On the other hand, the evaporation approach is valid as long as the energy of the nucleon does not too much exceed the nuclear potential depth, i. e. about 40 MeV. Thus, the transition energy between the INC and EVAP calculations cannot be specified rigorously. For that reason several codes have added an intermediate step whose domain of validity is expected to overlap the INC and EVAP domains. This step is the pre-equilibrium (PE) step [85].

In general the behaviour of reactants in a solution or in a diphasic system, as often met in reprocessing, is governed by the law of mass action which we find useful to summarize.

Consider a general binary chemical reaction

vaA T vbB —— ncC T UfiD. (9-1)

This can also be written as, vA, = 0. By tradition the values of the left-

hand side of the reaction equation, such as equation (9.1), are taken to be positive. At chemical equilibrium, the thermodynamical potential should be minimum, which leads to the condition щ = 0, where щ is the chemical potential of molecule A,. Defining the concentrations of the molecules cA = nA/N where nA is the number of molecules A, and N the total number of molecules, including those of the solvent(s), the condition of chemical equilibrium leads to the law of mass action, which is exact for perfect gases and dilute solutions:

П cV = K(P; T; {f,}) (9.2)

where f, is related to the chemical potential of molecule i and P and T are the pressure and temperature respectively. For example in a dilute solution f, = щ — T ln c,. For dilute solutions the equilibrium constant reads simply

K (P; T ; {ф,})= expr —Щ^- — (9.3)

Here the P dependence disappears. K(P; T) is, more frequently, defined in terms of the free formation energies of the molecules. For reaction (9.1), for example,

vaA. Ga(T) + vbAGb (T) — vc AGc (T)— vdAGd(T)
RT

(9.4)

where the temperature dependence of AG(T) can be expressed as

AG(T) =AH — TAS (9.5)

where AH and AS are the formation enthalpy and entropy of the molecules.

In the following, since we deal with condensed matter, we shall drop the dependence of K(P, T) upon P.