A linear accelerator consists basically of a linear succession of metallic drift tubes separated by gaps. The simplest structure was invented by Wideroe [179].

III.1.1 The WiderOe linear accelerator

In this accelerator the drift tubes are subjected to alternative potentials such that two adjacent tubes have opposite parity. Within the tubes, particles are shielded from electric fields and drift with a constant velocity. The length of the tubes increases from the accelerator entrance to its end.

Figure III.1 is a schematic drawing of a fraction of a Wideroe Linac at two successive times differing by a half RF period. / is the distance between two adjacent accelerating gaps. Synchronism of the beam pulse with the RF implies that the time spent by the particles between two accelerating gaps is equal to a half of the RF period:

/ _ T

v 2

cT, with c the velocity oflight. With P = v/c one

Suppose that particles with energy E„_1 enter gap n and are accelerated by potential V. Thus,

En = E0 + nV (III.3)

where E0 is the input energy into the accelerator. In the non-relativistic limit the condition for synchronism is

n = /2 ЩЙ-

2c M0

where M0 is the rest mass of the accelerated particles. More generally,

Note that the first condition,

defines the initial energy E0 when l0 is given. Thus, linear accelerators behave as velocity filters.

The Wideroe accelerator suffers from two drawbacks:

1. As can be seen in figure III.1, the individual elements of the drift tube are equipotential, which means that the intergap length l has to be much smaller than the radiofrequency wavelength[66]

l < A (III.7)

and, thus

P < 1 (III.8)

which shows that the Wideroe accelerator is effective only at low velocities.

2. As can also be seen in figure III.1, electrical currents of adjacent cells conspire to give very high currents through the structures which hold the drift tubes inside the tank. This would lead to unacceptable deforma­tions of these structures if high accelerating voltages were used. However, low accelerating voltages lead to very long accelerators.

In practice, Wideroe accelerators are only used for the acceleration of slow heavy particles.

The IIASA scenarios foresee a rather modest contribution of nuclear energy to the global energy mix. It seems that the main reason for this shyness is more related to ‘political correctness’ than to economical or technological constraints. It is interesting to examine how much a deployment of nuclear energy limited only by these constraints might limit global warming and resource exhaustion. Such an approach has been followed in reference [30] where it was found that, with a nuclear production potential of 9000 GWe in 2050, the asymptotic temperature increase could be limited to 2 °C, even in the case of scenario A2. We now give the outline of such a treatment for the three scenarios A2, B and C2. We assume that, by 2030, the use of fossil fuels for electricity production will be drastically reduced. Table 2.12 displays the reduction factors used for several geographic aggregates. The same reduction was applied in the three scenarios. Fossil fuels were assumed to be saved by resorting to nuclear power, although renewable energies could equally well be used provided they become reasonably com­petitive. The choice we have made of nuclear power provides a kind of existence theorem for a solution to curb CO2 emissions, and, at the same time, tests the capabilities of nuclear power in terms of fuel availability, wastes produced and capital needs.

Table 2.12 shows different types of reduction factor used for 2050. These reduction factors are used differently in three different scenarios with increasing use of nuclear power:

1. The ‘electric’ scenario, labelled E, assumes that fossil fuels are no longer used for electricity production.

2. The ‘nuclear’ scenario, labelled N, assumes that coal and gas are no longer used in industry, homes or offices (in particular for heating), except in former USSR and China.

3. The ‘hydrogen’ scenario, labelled H, assumes that hydrogen has largely replaced oil in the transportation sector. The share of hydrogen fuel is displayed in the last column of table 2.12.

Table 2.13 summarizes the basic differences between the reference IIASA scenarios and the various more nuclear intensive scenarios considered here. It gives CO2 emission and nuclear power production for 1990, 2030 and 2050.

It appears from table 2.13 that the substitution of nuclear for fossil fuels in electricity production, although it achieves a significant reduction of CO2 emissions, is not sufficient to reach the 3 Gt carbon emission target. In contrast, when fossil fuels are replaced by nuclear power both for electricity

production and for other uses in industry, offices and homes, the target can be reached in 2050 in all the scenarios. The replacement of fossil fuels by electrolytic hydrogen in transportation systems allows further reduction.

While we have noted that known reserves of natural uranium amount to approximately 125 years of production at the present rate, it is clear that a nuclear intensive scenario will put much more strain on uranium resources. This is apparent in table 2.14, where the cumulated needs of natural uranium for the three ‘hydrogen’ scenarios in 2030 and 2050 are displayed. Presently known reserves would only be sufficient up to 2030, while the higher estimate of 17 million tons given by the OECD might allow reaching 2050. The need for a timely development of breeder reactors is, thus, clearly demonstrated and will be discussed later.

The potential of transmutation in a resonance region [57] comes from two effects:

1. The presence of very strong resonances in the nucleus to be transmuted.

2. The fact that the change of energy in very heavy scatterers like lead is very progressive, allowing several chances for the slowing down neutron to interact with a nucleus to be transmuted.

We give a schematic handling of the effect of a very strong absorption resonance characterized by its width Г, its energy ER and a cross-section ^abs. We want to compute the number of captures in the absorbing nucleus per incident neutron, as a function of the concentration of the absorber. The Breit and Wigner formula reads

with x = (E — ER)/2r. A neutron at energy E enters the medium that has na absorbing nuclei and a macroscopic scattering cross-section of —s. We define —a = naa0 and —(E) = naa(E). The probability that the neutron survives an interaction, i. e. that it is not captured, is Psurv = Es(E)/[Es(E) + -(E)]. After n interactions the survival probability is

1

TT—ER—Ei)]

thus

Eln(1+x2 ) — Ysln

the interval between two successive values of Ei is £ER and that between two xi is, accordingly, 2£ER/r. Using the integral approximation of the sum, and integrating from 0 to i, we get

and

і	(3.64)
Psurv = exp

Slowing down spectrometer

When the value of £ is small, the number of collisions needed to slow down a neutron from 2MeV, for example to thermal energies, becomes large. For lead, for example, 760 collisions are needed. Since the mean free path in lead is around 3 cm, the neutron travels as much as 23 m before reaching thermal energies. It is clear that this process takes a long time, hence the possibility to use this slowing down time to characterize the neutron. We give a schematic derivation of the relation between the neutron velocity and energy and the slowing down time [58]. The average velocity loss at each collision is Av/v = 2 £. The velocity after n collisions is simply

vn = v0(1 — 2£)n. (3.65)

Assuming a constant scattering cross-section and mean free path A, the moment at which collision n occurs is therefore, on the average,

-f1 1 1 1 vA + 1 -11 £ + (1 -1 £)2 + "’ + (1 -1 £)n -1 1 -(1 -1 £)n+1

(3.66)

2 £(1

(3.67)

Using equations (3.67) and (3.65) and dropping the index n one gets

and

E0(1 -1£)2

((2 £/A)ffiEffim)t +(1 -1 £))2 which is commonly written as

K

(t +10)2

with

Practical computer simulations of reactors use either deterministic or Monte Carlo codes. The deterministic codes are the most used for critical reactor simulations, while Monte Carlo methods are almost exclusively used for hybrid reactors. Since our emphasis is on the latter, we only give a short reminder of the deterministic methods. Both methods require extensive neutron reaction data which are obtained from various data bases. The quality of the calculations is often limited by that of the data. Prior to discussing calculation methods, it is fitting to discuss data files and their evaluation.

It is possible to obtain quantities such as neutron fluence as a function of distance (radial or axial). For example, to obtain ф{г) in the core, one has to divide the core cylinder into slices (like an onion). We illustrate that point for the ads file. Some new cylinders have to be added:

13 cz 80

14 cz 60

15 cz 40

16 cz 20

Then, cell 4 is modified and new cells are added:

4 0 -10 13 11 -12 40 imp:n=1 fill=2

104 0 -13 14 11 -12 40 imp:n=1 fill=2

105 0 -14 15 11 -12 40 imp:n=1 fill=2

106 0 -15 16 11 -12 40 imp:n=1 fill=2

107 0 -16 11 -12 40 imp:n=1 fill=2

and, of course, a tally card

F4:N 4 104 105 106 107

5.6.2 Precision

The influence of very long multiplication chains on the accuracy of Monte Carlo simulations has been discussed by the CERN group [80]. M = 1/(1 — k) being the total number of neutrons originating from one initial neutron, these authors give the number N of cascades that have to be generated to obtain a relative error є on M:

with v the number of neutrons per fission. Equivalently, the precision for N cascades is

As an example we take the ATW proposal [127]. It is similar to other proposals like the TRASCO one [126]. The proposal includes an ECR source providing up to 100 mA, a 350 Mcs RFQ accelerating protons up to 6.7 MeV, a normal conducting SCDTL accelerator up to 10 MeV working at 350 Mcs, and super-conducting sections each optimized with respect to the average proton velocity, operating at 350 Mcs up to 210 MeV energy and 700 Mcs from 210 to 1000 MeV. The final energy is 1 GeV for a 45 mA intensity. The most reliable version of the proposal assigns one RF power unit for each cavity with an energy gain of 5.5 MeV.

Fissile material breeding needs neutrons to regenerate the fissile nuclei by neutron captures on fertile nuclei. The neutron balance has to be evaluated precisely to determine if fissile regeneration is possible or not.

For one fission of the main fissile element (233U in the case of the Th/U cycle), 1 neutron is consumed to produce the fission and a are captured on the fissile element (with a = uS^/g™^. Thus, 1 + a nuclei of 233U are lost, and have to be regenerated by 1 + a captures on 232Th. Finally, 2(1 + a) neutrons are consumed to produce one fission and to regenerate the fissile element. The multiplication factor can thus be written as

k = ————————— (11.1)

2(1 + a)T L + Na

where L represents the leakage and the parasite captures on structural materials (normalized per fission) and Na the number of available neutrons per fission, which governs the breeding rates and the capacity for deployment of the cycle. Na can be expressed[53] as

Na = k — 2(1 + a)-L. (11.2)

Breeding is possible only if Na is positive. In this simplified approach, the number of available neutrons per fission Na depends only on the ratio between the probability of capture and fission of the fissile element, and on the probability of escaping the system. If we consider the most optimistic case where L = 0, we can represent Na as a function of the neutron energy (see figure 11.2), for Th/U and U/Pu cycles. The well-known result that breeding cannot be achieved in thermal U/Pu reactor can be seen in this figure. The thorium cycle appears to have fewer available neutrons than the uranium cycle in the fast spectrum, but the figure shows that breeding can be achieved in a thermal or epithermal spectrum. Tables 11.1 and 11.2 show some values of a and Na for a fast and a thermal spectrum, and for different values of the multiplication factor (for L = 0).

In this simplified calculation, we considered neither the other nuclei of the uranium or thorium cycles, nor the fission products, which are respon­sible for additional neutron captures. Fission product poisoning leads to an increasing value of L with time.

The thermal spectrum leads to larger fission product poisoning and requires molten salt reactors. These reactors give the possibility of a [54]

quasi-online treatment and purification of the core; they allow precise reac­tivity control, as well as neutron economy optimization by preventing neutron losses due to captures on fission products. Some of the earliest proposals of hybrid reactors were based on molten salt fuels [1]. Bowman [2] proposed a system operating with a thorium-based molten salt fuel. In order to avoid the fission product poisoning effect, 233 U is obtained via neutron irradiation of a 232Th blanket, followed by online extraction of Pa which is allowed to decay into U. This is made possible by the use of a molten salt (a mix of fluorides) fuel, similar to the fuel used in the Oak Ridge pilot reactor. The fission products are extracted from the fuel and the reactivity can be kept constant over a few years. The main advantage

of the thermal spectrum is the small fissile inventory needed to operate a 1 GWe reactor. In a fast Th/U reactor, the uranium inventory is about 7 tons and the neutron flux about 2 x 1015 n/cm2. In an epithermal molten salt reactor, the uranium inventory falls to 1.2 tons for a neutron flux around 3 x 1014 n/cm2, averaged over all the molten salt. It is clear that the fissile inventory depends strongly on the maximal power density that can be extracted from the core. The values we give seem reasonable, but can be improved in the future by the development of technology and materials. Moreover, in some cases, the volumic proportion of the heavy nuclei can be reduced in order to reach a higher flux, and thus minimized inventories.

We will describe the main characteristics of the thorium cycle with a solid fuel, which requires a fast spectrum. The fuel cycle and reprocessing are simplified, insofar as the reactivity can be kept constant through an adequate 233U regeneration. Some aspects of the thorium cycle in a moder­ated neutron flux and with a liquid fuel are presented later.

The initial 233U concentration can be chosen in order to keep ks constant over 5 years. Figure 11.3 shows the evolution of the source multiplication factor ks, over 100 years of energy generation.

The protactinium effect is specific to thorium based fuels. The decay of 233Pa is relatively slow (27 day half-life), therefore 233U regeneration by neutron capture on Th is delayed by a few weeks. Since Pa is a non­fissile nucleus (of = 0.06 barns in a fast neutron spectrum), this delay induces a reactivity decrease (around 8000 ppm) during a transition phase as shown in figure 11.4. Again, this effect depends on the flux intensity. A lower flux decreases the protactinium effect, but requires a larger 233U inventory.

Making the assumption that radioelements are conveyed by water, in the presence of sorption and desorption phenomena, elementary transport can be represented as occurring along capillaries. The relevant transport coordinate is the direction perpendicular to the clay layer. At the scale of an elementary length unit, the radioelement losses and gains can be defined as follows:

• Losses are due to:

о Radioactive decay: —Apd(x, t) where A is the decay rate and pd(x, t) is the linear density of dissolved radioelements. о Sorption at the boundaries: —wspd(x, t) where ws is the sorption rate. о Radioelements migrating outside the volume, with the water: —vpd(x, t) where v is the transport speed.[63]

• The gains are due to:

о Desorption at the boundaries: wdpS(x, t) with wd the desorption rate, pS(x, t) the linear density of radioelements absorbed at the boundaries. о The radioelements conveyed by water inside the volume:

v

2 (pd(x — A; t) + p&(x + A; t)) (I.1)

with A an effective transport distance. Note that, in the case where the radioelements have limited solubility, ws and wd depend on the volume density of the radioelements considered.

The time evolution of the density of radioelements dissolved or absorbed can be written as

Let us define D0 = vA, the diffusion constant that characterizes the transport of water in the capillaries. The characteristic radioactive decay times A-1 are larger than 1000 years and the diffusion characteristic times x2/D0 are of the order of 1 year for a 1 cm distance; they are much larger than the sorption and desorption characteristic times that are of the order of 1 s. It follows that the sorption equilibrium can be considered to be achieved and, therefore,

pd(x, t)= d p(x, t). (I.3)

wd + ws

Because of the diffusion and radioactive decay, the sorption equilibrium is not strict. Thus, equation (I.3) cannot be directly applied to equation (I.2) without being careful since the sorption-desorption term can be of the same order as the other terms which gives, for p(x, t),

and identical equations for pd (x, t) and pS(x, t). The diffusion constant D = D0[wd/(ws + wd)] = D0/R, where R is a delay factor that can be large, characterizes the diffusion of the radioelement.

The density profile of the radioelement in the layer being known, it is obviously important to deduce from it the output current. This current is proportional to the water flow in the capillary with a proportionality con­stant that is none other than the concentration of the radioelement consid­ered in the aqueous phase. It follows that, applying Fick’s law, the radioelement current can be written as

Here, the current is expressed as a function of the dissolved radioelement density. This expression is particularly convenient when the density is limited by the solubility of the element considered. In contrast, when the element is soluble, it is more convenient to express the current as a function of the total radioelement density.

Intuitively, one expects that for very high intensities, repulsive forces between the beam particles will lead to defocusing. We give an order of magnitude treatment of this space charge effect. We note that, because of the influence of nearby particles, the assimilation of the mechanical momenta P to the conjugate momenta p is no longer justified, with the consequence that emittance growth is possible.

Let us consider a cylindrical beam of radius a with charge density p. An electric field is created by the charged particles. At a distance r < a this field is

Similarly a magnetic field is created, due to the particle current and per­pendicular to the electric field:

The motion of particles initially at the beam surface obeys mo d2a ql

— 32 dt2 2neaftc

One gets an upper value of the defocusing effect by setting a to its minimum value a0. The variation of a in a drift section reads

where v? is the transverse particle velocity. The second term of the right-hand side of equation (III.80) corresponds to the ‘normal’ beam radius increase along drift sections. One might expect that important beam losses would occur if the space charge induced term is of the order of the ‘normal’ term at the end of a drift section of length L which leads to the relation between l and L:

qlmax(1 — 32)3/2 L_ 4rn’ea0lm0lc2 32

which yields

As an example we take 3? — 10 3, a0/L — 10 3, A — Z — 1, and

32/у/1 — 32 — 0.5 (3 — 0.4) Then

lmax — 26 A.

The corresponding average current is obtained by taking into account the macro and microstructure of the beam. For continuous beam accelerators the microstructure alone is relevant. Typical duty factors range between 0.01 and 0.1, and we expect space charge limitation to occur for relativistic protons with average currents around a fraction of an ampere. Equation (III.82a) shows that space charge effects increase at low beam energy. For example, with 1 MeV protons one obtains in the same conditions as above

lmax — 0.066 A (III. 84)

with average currents limited to a few mA. Decreasing the drift length L allows us to increase this limit. This is one of the interests of the RFQ low — energy accelerators. For Linacs, L is almost independent of the particle energy, while rather large variable beam diameters are practical. The circular nature of cyclotrons leads to a linear dependence of L on R or, equivalently, on ft. In this case equation (III.82a) can be replaced by

(III.85)

For ft — 0.05 (E — 1 MeV), ft? — 10~3, a0/L — 10~2: (III

[1] Using any kind of energy source, accelerators allow ample production of neutrons which might be used for synthesis of fissile nuclei starting from the fertiles 238U and 232Th.

[2] The market rate is assumed to be less than $150/kg. The OECD organization estimates that the total of assured and plausible uranium underground exploitable reserves amounts to 17 million metric tons.

[3] CO2 emissions are given either in CO2 weight or in weight of the carbon included with the correspondence: weight(C) = weight(CO2) x 42

[4] In an evaluation of the contribution of natural gas to the greenhouse effect, any losses to the atmosphere at each stage, from production to final use, should be taken into account since methane is at least 20 times more efficient than CO2 for inducing a greenhouse effect.

[5] fl radioactivity produces energy in the form of fast electrons (betas) and gamma rays.

[6] Breeder reactors produce more fissile material than they consume, while converter reactors produce as much fissile material as they consume.

^ Np, Am and Cm are produced in relatively small quantities in standard reactors and are thus called minor actinides.

*This cooling time is necessary to allow for the decay of 137Cs and 90 Sr, whose transmutation would be very difficult and costly. Due to their short half-life these isotopes dominate the short-term waste radiotoxicity.

Even in the former USSR the reactors similar to the PWR, the VVER, are considered to be safe by international experts, while the RBMK reactors, such as those of Chernobyl, are unanimously considered unsafe.

[7] Sodium has high heat conduction, low viscosity, similar to water, low melting point and low neutron absorption cross-section. The first sodium cooled reactor was designed to be airborne, and thus the low density of sodium was an important asset.

[9] Minor actinides are weakly fissile by thermal neutrons, but easily fissile by fast neutrons.

[10] However, efficient enough radiation cooling limits the size and power of the reactor to less than about 1 GWth.

[11] The equality between this number and the inventory of the U-Pu breeders is fortuitous.

[12] We have given this derivation of the neutron flux properties because the confusion is often made that it is the number of neutrons crossing the unit surface plane per time unit. In fact it is the number of neutrons crossing a sphere of unit surface cross-section. This is simply obtained if one remembers that each neutron crosses the sphere twice. Equivalently it is the number of neutrons crossing a unit surface circle kept perpendicular to the neutron direction.

[13] This is not strictly necessary for Fick’s law to be valid. It can be shown that the law is valid up to second order.

[14] Lethargy is the logarithm of the neutron energy. Low lethargy media are such that the energy (lethargy) change after a collision is small. This is true for high mass nuclei.

^ We suppose that the slowing down distance for the neutrons is negligible. A better, but still simple, handling can be obtained using the Fermi age theory as described in standard books like that of Lamarsh [55] and Bussac and Reuss [48].

[15] Isotropic elastic scattering

• No inelastic scattering

• Monatomic medium

We consider an infinite medium with a source of N0 neutrons/s. The number of neutrons which are scattered each second from an energy larger than E to an energy smaller than E is evidently equal to N0. A neutron of energy E can be scattered equiprobably to reach energies between E and $E, $ being defined in equation (3.2). It follows that the number of collisions falling below energy E is

[16] For ko < 1.

^ See section 4.3.3 for a more systematic discussion of the different multiplication factors.

[17] A fraction of less than 1% of neutrons is emitted by fission fragments with a delay of up to a few seconds after fission.

[18] We have given a plausibility argument on the sign of the Doppler effect. More elaborate and quantitative treatment can be found in specialized books [55, 48].

[19] In small reactors the decrease of the coolant density leads to an increase of neutron escapes, which may decrease the reactivity.

[20] Some finite value of the power, or neutron flux, is necessary to ensure divergence.

^ This reactivity exceeds 1$, and in principle the slow power rise approximation is not justified. The treatment given here is therefore only indicative. Inserting the additional reactivity by smaller steps would be more justified but would give very similar results.

[21] This value was given in the Rasmussen report. Following the Three Mile Island accident improvements were made both on the equipment and on the operational procedures so that a value of 10~5 is claimed to be more representative.

^ One possibility for containment disruption is a secondary, explosive, hydrogen reaction with air. Hydrogen is produced from the thermal decomposition of water in contact with the hot zir — calloy casting of the fuel elements. More efficient hydrogen extraction from within the reactor building is one of the major safety improvements considered in future reactor design.

[22] fission of heavy nuclides

• ft or a decay

[23] This condition is satisfied for both fast and thermal reactors. However if a reactor could be made to operate in the resonance region, it might be incorrect. In this case, the fertile part would, progressively, disappear in favour of the fissile part. To our knowledge, no system based on this property has yet been proposed. It would lead to a very high breeding ratio.

^ Here, we neglect the effect of radiative captures in fissile nuclei except for the difference between V = v4’s) /4(s and v. Thus, in the evolution equation 4(f) = 4(as).

* In the following considerations the value of k^ may be significantly larger than unity. In such cases additional neutron absorbers or leakage will ensure subcriticality.

[24] In the case of the U-Pu cycle effects due to captures in 239Np, analogous to those due to 233 Pa in the Th-U cycle, are ten times smaller, because of the much shorter half-life of 239Np.

[25] D. Heuer, private communication.

[26] We shall see in the next section that such a high thermal flux may be difficult to accept, because of the ‘protactinium effect’.

^ Electro-breeding denotes the process by which the fertile to fissile conversion is achieved by the spallation neutrons produced by the accelerator.

[27] In principle k depends on the properties of the source neutron (energy, spatial distribution) and on the geometry of the reactor and is usually noted ks (‘k source’). For simplicity we keep the notation k, at this stage.

[28] This energy corresponds to the kinetic energy of the fission fragments and their prompt de­excitation (neutrons and photons). In the total energy balance one should also include the beta radioactivity energy which amounts to approximately 20 MeV. However, in the FEAT [122] experiment, which is the only existing direct measurement of k, only the fission kinetic energy was measured.

^ The thermodynamic efficiency of present PWRs is around 0.33, while that of gas combined cycle turbines reaches 0.5. Lead (and sodium) cooled reactors can reach efficiencies of 0.4. High temperature gas reactors can reach even higher efficiencies of 0.5.

[29] This law is also known as a chi-square distribution with n degrees of freedom.

^ Note that widths Г(о) should not be confused with the Г function.

* The Wigner law shows that levels with same spin and parity repulse each other. They do not follow the random Poisson distribution, and reflect quantum chaos.

[30] http://www. nndc. bnl. gov; another important site is that of the Nuclear Energy Agency, http://www. nea. fr.

[31] S is known as the scattering function of a, the reduced momentum transfer, and 0, the reduced energy transfer.

^ In fact, this is an approximation for ^AEn/kT > 2; a more general formula is used when this number is smaller than 2.

[32] Note that the sum made by MCNP takes into account correlations between histories, therefore the arithmetic summing of all bins is not equal to the sum bin given by MCNP.

[33] This is not necessary, but lines must not stop beyond column 80.

[34] One can also use the VOL card where all cell volumes have to be given.

[35] One can find at NEA (http://www. nea. fr) the MCB (Monte Carlo Continuous Energy Burnup) code which is a general-purpose code that can be used for calculation of nuclide density evolution with burn-up or decay, including keff calculations of critical and subcritical systems and neutron transport calculation together with all necessary reaction rates and energy deposition (at six dif­ferent temperature). MCB is compatible with MCNP-4C and complete burn-up calculations can be done in a single run that requires preparation of a single input file with a very few more data lines compared with a regular MCNP input.

[36] For example, in the study of Ref. [77], it was found that the average neutron capture cross­section for the nuclei produced by the fission of 233U was equal to 0.15 barn.

[37] ‘Smaller’ is rather ambiguous. The black nucleus cross-section for n-n interactions is w(2r + (Acm/2w))2 and comes close to the classical limit for, say, Acm/2K = 0.5r or A = кг/2.

[38] As said in the previous subsection these include a high-energy part like the Bertini cascade, an evaporation part like the Dresner EVAP, and, possibly, a pre-equilibrium part.

[39] In principle k depends on the properties of the source neutron (energy, spatial distribution) and on the geometry of the reactor and is usually denoted ks (‘k source’). For simplicity we keep the notation k, at this stage.

[40] From here on we call neutron multiplicity the number of neutrons per incident particle Nn/p rather than the average multiplicity of the neutron distribution {Mn). The difference between the two numbers in the work by Hilscher et al. comes from cases where the incident particle did not suffer any nuclear reaction. In practice, for targets thick enough, the two measurements converge.

[41] Qualitatively, the importance of a neutron, in a multiplying medium, measures the number of its descendents, relative to that of an ‘average’ fission neutron.

^ Private communication.

[42] A recent experimental measurement [121] of neutron multiplicities for deuterons and protons of 200 MeV confirms that, for uranium, the two projectiles are equally neutron prolific.

[43] This is only true on the average for a neutron chosen randomly according to the flux distribu­tion of the adjoint reactor.

[44] Here one should consider the power during the macro-pulse rather than the average power.

[45] Proton energy should be larger than 600 MeV, in order to optimize the number of neutrons produced per MeV of incident energy.

• Beam power should exceed 10 MW, possibly more. This means that for 1 GeV protons currents larger than 10 mA are needed.

• Beam losses should be made very small in order to minimize accelerator structure activation.

• High beam availability is required. This is obvious for long-lasting shut­downs due to equipment breakdown: this requirement is analogous to

[46] Possible designs of accelerators for ADSRs have been described in several workshops of the NEA on ‘Utilization and Reliability of High Power Proton Accelerators’.

^ Maximum intensity has been shown by the SIN physicists to increase as where Eg is the energy gain per turn.

[47]Nd -! 149Pm -! 149Sm (stable).

2h 54 h ( )

[48] Note that our notations are slightly different from those of reference [138]. In particular we count stages starting from stage 0 as the aqueous feed input.

[49] The first molten salt reactor studies were done in the context of the airborne nuclear reactor project. It was based on the fluoride mixture NaF : ZrF4 : UF4 using uranium highly enriched in 235U. The operation temperature was around 500 °C.

[50] Here we follow closely the treatment given in the thesis of Lemort [155].

[51] More precisely this is a Daniel cell.

[52] Here we assume that the concentrations are kept constant by an external input or with a salt bridge.

[53] Burn up of 66 GWd/ton, a conservative figure for fast reactors.

• 5 years cooling before reprocessing and reuse.

• Reprocessing losses of 0.1% for uranium and plutonium and 1% for neptunium, protactinium, americium and curium.

• The composition of the wastes was that obtained after five cycles (irradia­tion + cooling). It is close to the asymptotic composition.

• No special treatment of the curium isotopes, although fuels with a signifi­cant amount of 244Cm might be difficult if not impossible to manufacture, due to the high spontaneous fission rate of this isotope.

The calculations were based on the coupling between the MCNP Monte Carlo code and an evolution code, as described in chapter 5.

The gain for thorium is around 50 as compared with the uranium cycle during the first ten thousand years after discharge. After this period, the radiotoxicity of thorium spent fuel increases because of 233U decay and both cycles are equivalent. The main radiotoxic elements of the thorium

ОТІ

fuel cycle are Pa and U, produced by (n,2n) reactions on Th and 233U. This is a significant difference from the uranium cycle, for which radio­toxic elements are produced by successive neutron captures. When compared with the once-through PWR (BWR) scenario, the radiotoxicity gain is around 2000 for the thorium cycle during the first ten thousand years. Note that, in a breeding scenario (thorium and uranium cycle), most of the

[54] The divergence of Na for vanishing k is due to the fact that, for k = 0, there are no fissions in the subcritical medium.

[55] Shorter cooling times might be obtained with pyrochemical reprocessing, yielding correspond­ingly smaller inventories.

[56] The fact that fission products are not separated is due to the extremely simple uranium extraction method, by sublimation of UF6.

[57] The fact that fission products are not separated is due to the extremely simple way of extracting uranium by sublimation of UF6.

[58] All costs are given in euro (c). The American dollar is around 1 euro.

[59] The costs are based on 1992 values. They may be misleading for a comparison between gas and nuclear energy because of the drop in gas prices and in gas turbines since then.

[60] A European group chaired by Professor C Rubbia, including state and industry representatives from France, Italy and Spain, is working on the layout of a possible demonstration facility.

[61] Achievement of the neutron flux levels required by the different applica­tions considered in MYRRHA:

[62] This dose is close to a Dari. The Dari is the measuring unit proposed by G Charpak and R L Garwin. It corresponds to the internal radiation received due to the radioactive elements present within the human body. The Dari is the absolute minimum of radiation that a human being is liable to be exposed to.

[63] The surface diffusion of the radionuclides must also be included. The transport speed, v, has to be taken as an effective speed.

[64] Such tests were pursued on many reactors, on the recommendation of the IAEA, following sabotages of the power lines converging to nuclear power plants which had been perpetrated in West Germany by anti-nuclear activists.

[65] Some recent electron accelerators use a recycling scheme where particles are recycled several times through the accelerating structure. This technique is made possible because relativistic electrons have the constant light velocity. However, this technique does not apply easily to protons and other nuclei, because their velocity depends on their energy.

^ The reader can find more detailed developments on accelerator physics in many good books. The following was largely inspired by three books. High-Energy Accelerators by S Livingston, Interscience Publishers (1954) is a very simple and clear introduction to the physics of accelera­tors. Principles of Cyclic Particle Accelerators by J J Livingood, D Van Nostrand Company (1961). Principles of Charged Particle Acceleration by S Humphries, Wiley (1986).

[66] Typical frequencies of Wideroe accelerators are a few Mcs, and RF wavelengths are around 100 m.

[67]

and, with typical values of Q

0.5 for

[70] This appears to be rarely the case for cyclotrons.

Rather than specializing about one-third of the PWRs into MOx-PWRs, it seems that replacing uniformly the traditional 3.5% 235U enriched fuel elements by elements where about two-thirds of the fissile nuclei would be U and the remaining one-third 9Pu and Pu would allow a stabilization

of the plutonium inventory. A practical method to do so might be to mix UOx and MOx needles in a fuel element [40, 41]. Minor actinides should be extracted at each reprocessing, since these cannot be incinerated in thermal reactors.[9] The minor actinides could then be incinerated in fast [42] or hybrid [43-46] reactors.

Recently, it has been proposed [41, 47] to incorporate special annular fuel rods highly enriched in plutonium in standard PWR reactors. The PWR reactor could then consume 160 kg of plutonium per year instead of producing 200 kg as in present standard PWRs. Incinerating minor actinides in dedicated fuel elements also seems possible. Such a solution would be attractive, at least as long as uranium reserves do not command breeding reactors. Indeed, with minor modifications, the existing reactor system could be run with a stable plutonium and maybe minor actinide inventory.

Another solution, proposed by Rubbia [45], among others, is to replace depleted uranium by thorium. Thorium has neutronic properties close to those of 238U. The incineration of plutonium would be associated with the production of 233U. The proposed system could burn, annually, around 1.2 tons of plutonium while producing 0.7 ton of 233U. This nucleus could then either be a substitute of 235U in standard PWR fuel, or be a part of a new fuel based on the mixture 232Th-233U. Such a fuel could be reprocessed as many times as desired in PWR reactors, at variance with the U — 9Pu mixture. The main difficulty of such a scheme would be the fuel element fabrication: irradiation of U produces a significant amount of U by (n, 2n) reactions on 233U, whose decay is accompanied by intense high-energy gamma activity which would require large biological shielding for fuel fabrication. The whole fuel cycle would have to be redesigned.

A renewed interest has been displayed in gas cooled high-temperature reactors (HGTRs or HTRs). In these reactors the fuel is composed of tiny fissile-fertile particles (uranium, thorium, and plutonium carbides) coated with carbon and Si-C layers. Such fuel allows operation at very high temperatures (up to 900 °C). Even after a loss of coolant accident the fuel retains its integrity, its temperature being limited only by radiation cooling to an admissible value (more than 2000 °C).[10] The high stability of the fuel would allow very high burn-ups of more than 200MWd/ton. With such burn-ups a very efficient incineration of plutonium should be obtained, while reducing the amount of spent fuels or reprocessing wastes. However the question of the fate of the extremely active and minor-actinide-rich spent fuel remains open.