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

pr r 2e	(III:73)
with a maximum value
E — pa a 2є.	(III.74)
p is related to the beam intensity by
I = ‘wpa2 ^c	(III:75)
and
E 1 a 2^eaPc	(III:76)

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.