At the time of the Chernobyl accident, on 26 April 1986, the Soviet nuclear power programme was based mainly upon two types of reactor, the WWER, a pressurized light-water reactor, and the RBMK, a graphite moderated light-water reactor. The RBMK reactors were developed in order to produce both electricity and plutonium for military purposes. For that reason they were built only within the Union.

The Chernobyl power complex, lying about 130 km north of Kiev, Ukraine, consisted of four nuclear reactors of the RBMK-1000 design, Units 1 and 2 being constructed between 1970 and 1977, while Units 3 and 4 of the same design were completed in 1983 (IA86). At 01:23 hr on Saturday, 26 April 1986, reactor 4 exploded. We want to give here a short description of the reactor, especially of the features which explain why the accident was not only possible, but probable, as well as the sequence of events that led to the explosion. This appendix is based on the reports made public on the World Wide Web by the IAEA, the NEA and the IRSN. A thorough description of the RBMK reactors and the circumstances of the accident can be found in Libmann [60].

Proven fossil and nuclear reserves in 1997 were as shown in table 2.3 [21].

Note that if nuclear energy were to rise to 40% of the total energy pro­duction, the reserves, which are estimated presently at 5 million tons of natural uranium at market rates,[2] would decrease to 20 years in the PWR case and to 1000 years in the breeding case. In this last case it would, however, become cost effective to exploit very low grade ores, such as oceanic uranium, so that reserves would be much larger. It would also be

possible to use oceanic uranium, which amounts to nearly 3 billion tons, in non-breeding reactors provided a 50% cost increase of the electricity produced were acceptable [22].

One should take the above reserve estimates with some caution since exploration efforts seem to level off when the estimated reserves amount to about 40 to 50 years. For example, in the case of oil, current reserves have constantly increased since, at least, 1940. However, a recent careful study [23] shows evidence for a decrease of the real estimated reserves starting around 1980. This study predicts a decrease in oil production by 2010.

Taking into account the large reserves of coal, unconventional gas and oil, it seems that fossil fuels could be available at an adequate level during the 21st century. The main limitation to their use will rather, probably, be related to the greenhouse effect.

The first term of the right-hand side of equation (3.18) reads: —div(J(r, v, t)) = Dr2′(r, v, t). The diffusion equation is obtained from the Boltzmann equation when neutrons are assumed to be monoenergetic, or, in other words, to belong to a single group. The integration^ over velocities in equation (3.18) can be dropped, giving

Vi^(r) — ^ ^(Mn S(r, t).

i j ‘

(3.24)

Note that we have replaced ST in equation (3.18) by Xa, since diffusion has no effect on the flux in one-group formalism. Using relation (3.75), equation (3.24) can be written as

= Dr2′(r, t) + ‘(r, t) X Z[j)(r)(ki — 1) + S(r, t). (3.25)

j

Equation (3.25) leads to a few interesting remarks. In an infinite and homogeneous medium, with an evenly distributed neutron source, the

equation should not include derivatives of ‘(r, t), since ‘(r, t) should be independent of r. Thus equation (3.25) simplifies to

‘(t)J2 Zij)(ki — 1) + S(t). (3.26)

v @ t —

Consider first the case that, for t > 0, S(t) = 0 and ‘(0) is finite. Then equation (3.26) has the solution

	(3.27)

which shows that if кж > 1 the flux diverges, while it decreases to 0 for кж < 1. It is time independent only if кж = 1. This condition can never be met in reality. Rather, in critical reactors, a time-dependence of the absorption cross-sections is used, so that кж fluctuates about 1.

Equations (3.26) and (3.27) have a simple interpretation if one defines the neutron lifetime as the mean time separating creation and absorption of the neutron,

then, equation (3.26) reads

@|^= ‘(t) (kl~ 1} + S(t) (3.29)

ot T

omitting the source term. This equation simply expresses that every time a neutron disappears, кж neutrons are re-emitted, the average time between two neutron absorption events being the time T. With the same notations equation (3.27) reads

‘(t)='(0) e^”-1)t/T (3.30)

which shows that the characteristic evolution time of the neutron flux is т/кж — 1|. Other quantities like neutron density n(t), fission rates, specific power W(t), obey the same type of evolution.

Consider, now the case where кж < 1, and S(t) = S0 is time indepen­dent, but positive. The solution of equation (3.26) for stationary states reads

S0

(1—к^)^ .

The number of absorption reactions per second is then

(3.32)

which agrees with equation (3.71).

We have seen that, if the neutron multiplication factor of a reactor assembly keff < 1, the chain reaction cannot be sustained. However, if a source of neutrons is introduced inside the multiplying medium, the initial neutron number is multiplied by a factor which can be very large. Since neutrons are associated with an equivalent number of fissions, a large energy could be produced with a subcritical system, provided a neutron source is available.

If N0 is the number of primary neutrons following, for example, the interaction of a proton with a target surrounded by a multiplying medium, chacterized by a multiplication factor k[27] the total number of created neutrons, after multiplication, is

N0

(1 — k ■

The number of secondary neutrons (produced after at least one multi­plication) is

kNp

(1 — k) •

5.6.1 Reactivity calculation

As already mentioned, in a Monte Carlo calculation, a particle is followed from its birth to its death. The code ends with the end of the last history. But in an (over-)critical system, neutron chains are infinite and such a calculation will not end. A special technique has been developed to handle this case. It allows us to obtain keff and tallies for any criticality. Starting with Ns source neutrons, the calculation develops over cycles (or neutron generations); within a cycle, a neutron is followed from its birth to its death but, this time, fission is considered as a cycle termination (like a capture). Within a cycle, at each collision point, the number of fission neutrons stored n is randomly sampled in order to have a mean value

-f W

keff

(W is the neutron weight, of-is/t is the microscopic fission/total cross-section and keff is estimated from the previous cycle, or is a user-given value if this is the first cycle). For the next cycle, M =^2 n particles (M ~ Ns) are emitted on the corresponding collision site. The effective multiplication factor can be obtained as

where Nifis and Niabs are respectively the number of fissions and of absorp­tions in generation i. In MCNP, three methods (based on cross-section calculations) are used to obtain the effective multiplication factor keff and the final value is a weighted average of the three factors.

To perform a critical calculation, one has to use the KCODE card (in the Data block card); a ‘starting’ source has to be defined to initiate the first cycle. The syntax is

KCODE Ns kexpt N NT

where Ns is the number of ‘source’ neutrons per cycle, kexpt is the expected value of the effective multiplication factor, NT is the total number of cycles (~100 is usually sufficient) and NI is the number of initial cycles that are
excluded from the keff (and tallies) calculation, in order to give enough time for the fission source to be established.

Going back to the example of the uranium sphere in a light water cylinder, the input file could be

First simple geometry c

c Cell cards c

1 1 -18.75 -1 $ the inner sphere

2 2 -1.0 -2 3 -4 1 $ the cylinder without the sphere

3 0 #2 #1 $ exterior

c

c Surface cards c

1 SO 5 $ centred sphere with R=5 cm

2 CZ 20 $ infinite cylinder with R=20cm

3 PZ -20 $ bottom plane intersecting the cylinder.

4 PZ 20 $ top plane intersecting the cylinder.

c Materials

M1 92235.60c 1 $ 235U

M2 1001.60c 2 8016.60c 1 $ H2O

c Source : kcode

sdef pos 0 0 0 erg 2.5

kcode 1000 1 10 80

totnu

Suppose that this input file is named lstgeo. MCNP can be run by enter­ing the command

mcnp n=1stgeo

At the end of the run, the keff value is displayed on the screen and is stored in the lstgeo file (after the short table summarizing the number of source neutrons, captures, escapes,…). The result is keff = 0.840 ± 0.003 (with more cycles, the precision is improved). If the sphere radius is changed from 5 cm to 6.312 cm, keff = 0.999 ± 0.003 which corresponds to the critical case.

Then, using the S(a,@) treatment (by inserting an MT2 lwtr.07 card after material M2), we see that keff is reduced to keff = 0.974 ± 0.003; this demonstrates the importance of this treatment for thermal neutrons.

Linear accelerators

The highest-power operational proton linear accelerator is that of Los Alamos Meson Factory (LAMPF).

The LAMPF accelerator. The LAMPF accelerator accelerates protons up to 800 MeV. It is 800 m long. It provides 1ms bunches of protons with an intensity of 15 mA and with a repetition rate of 120 Hz. Correspondingly its duty cycle is 12%, and its average beam power 1.4 MW.

The accelerator comprises three parts:

1. The source and injector. The source can provide either H+ or H~ ions. The largest intensity of 30 mA is obtained with the H+ ions. The ions are accelerated at 750 kV by a Cockroft-Walton electrostatic accelerator. The continuous beam is then bunched at 201.25 MHz.

2. The 0.75 MeV protons are injected into a drift tube Linac (DTL or Alvarez). The DTL is 16.7m long. It accelerates protons up to 100 MeV. It works at a 201.25 MHz frequency. The Q value of the accelerating structures reaches 5 x 104. The drift tubes are enclosed in three tanks. Each tank shelters 165 drift tubes. Each tank is fed with an RF unit with 2.7 MW peak power. Beam focalization is achieved with 135 quadrupoles arranged in the focusing-defocusing-focusing-defocus — ing (FDFD) mode.

3. The 100 MeV protons are further accelerated up to 800 MeV in a side — coupled Linac (SCDTL). This is 726.9 m long. It works at a 805 MHz frequency. Only one out of four micro bunches is filled with particles. The Q value of the accelerating structures reaches 2.4 x 104. The 5000 cavities are enclosed in 104 tanks. The RF power is provided by 44 units with 1.25 MW peak power each. Beam focalization is achieved with 204 quadrupoles arranged in doublets.

The main characteristics of LAMPF are given in table 6.2. From the value of the shunt resistance and the phase angle of 64°, using equation (16.22) of Appendix III,

V2

P = ^ (6.21)

2Psh

one obtains a thermal RF power of approximately 10 MW, to be compared with a beam power of 12 MW,[44] and thus an RF-to-beam efficiency of 55%. The total power of the RF power units amounts to 65 MW, which gives a line-to-RF efficiency of 34% and a line-to-beam efficiency of 19%.

The diffusion equation in a multiplying medium reads

where

The solution of the diffusion equation is

, , u(r) (A e—7r + B e7r) ,

‘(r) =——— =———————— where 7

rr

We now apply the method to media 2 and 3.

SCKCEN, the Belgian Nuclear Research Centre, and IBA s. a., Ion Beam Applications, are developing jointly the MYRRHA project, a multipurpose neutron source for R&D applications related to the ADSR concept.

The design of MYRRHA needs to satisfy a number of specifications such as the following. [61]

‘{E > 0.75 MeV) = 1015 n/cm2/s at the locations for MA transmutation; ‘{E > 1MeV) = 1013 to 1014n/cm2/s at the locations for structural material and fuel irradiation;

‘{Eth) =2 x 1015 to 3 x 1015 n/cm2/s at locations for long-lived fission product (LLFP) transmutation or radioisotope production.

• Subcritical core total power, ranging between 20 and 30 MW.

• keff < 0.95 in all conditions, as in a fuel storage, to guarantee inherent safety.

• Operation of the fuel under safe conditions: average fuel pin linear power <500 W/cm.

In its present state of development, the MYRRHA project is based on the coupling of an upgraded commercial proton cyclotron with a liquid Pb-Bi windowless spallation target, surrounded by a subcritical neutron multiplying medium in a pool type configuration. The spallation target circuit is fully separated from the core coolant as a result of the windowless design presently favoured in order to utilize low-energy protons without reducing drastically the core performances. The core pool contains a fast — spectrum core, cooled with liquid Pb-Bi, and several islands housing thermal spectrum regions at the periphery of the fast core. The fast core is fuelled with typical fast-reactor fuel pins. The central position is left free to house the spallation module. The proton beam will be impinging on a molten lead or tungsten target.

The accelerator parameters presently considered are 5 mA current at 350 MeV proton energy. The positive ion acceleration technology is envisaged, resting on a two-stage accelerator, with a first cyclotron as injector accelerating protons up to 40 to 70 MeV and a booster further accelerating them up to 350 MeV.

Following the Lagrangian formalism, the conjugate momenta of the spatial coordinates x, are defined as

with у, = dx,/dt. The electromagnetism Lagrangian reads [187]

L = m0)c2(1 — (1 — ф2)1/2) — дф + qv ■ A

where ф is the electric scalar potential and A the magnetic vector potential. From equations (III.43) and (III.44) the conjugate momentum vector is obtained:

(III.45)

The six-dimensional space (x, p) is the phase space. The Liouville theorem states that, in an energy conservative system, the density in the phase space is invariant. In other words, particles initially enclosed in a small phase volume dV = dx • dp remain in an equal volume throughout their motion. If A can be considered to be constant over the phase volume, then dx • dP is also conserved. It follows that the normalized emittances defined in equations (III.42) are such that

ex. ey. ez = constant. (III.46)

If the coupling between the transverse and longitudinal motions can be neglected, then

ex. ey = constant. (III.47)

In many instances,[70] in particular when fields are locally linear, coupling between the two transverse coordinates can also be neglected, and thus both transverse emittances are constant:

axagx = constant; (III.48)

emittances are usually expressed in mm-mrad.

2.3.1 Standard reactors

Most existing energy producing reactors are of the light-water cooled type, either pressurized water (PWR) or boiling water (BWR). Although other types of commercial reactor like the heavy-water CANDU have interesting characteristics, our discussion focuses on the light-water reactors. The power of commercial reactors ranges between 600 and 1500 electric MWatts (MWe), with thermodynamical efficiencies close to 33%. As an example, we consider a 1000 MWe reactor.

Each fission produces approximately 200 MeV (185 MeV at the moment of fission and 15 MeV produced by subsequent fl radioactivity).[5] Accord­ingly the fission of 1 kg of a fissile isotope typically produces 80 TJ (or

1900 toe). A 3GW (thermal gigawatts) reactor, yielding 1 GWe (electrical gigawatt), produces annually about 7TWhe for an availability of 80%. It burns annually about 1 ton of fissile isotopes which is equivalent to two million tons oil equivalent (toe). More precise numbers are given in table 2.15, where material inventories at loading and discharge are given [36].

In table 2.15 a burn-up of 33GWd/ton (gigawatt-day/metric ton) is assumed. The table shows the following interesting features:

• The amount of 235U which has disappeared equals 674 kg. This accounts not only for the fission of this nuclide, but also for its neutron captures, at least the 111 kg of 236U produced.

• This means that at least 383 kg of the higher isotopes, mostly 239Pu, have contributed to fission. This can also be considered as an indirect fission of the 238U isotope, which lost 673 kg corresponding essentially to the production of plutonium. Of these 673 kg only 286 kg are found in the form of plutonium isotopes and minor actinides.

• The mass balance between the initial and discharge inventories is not exact. This is due to the mass equivalence of the energy produced (about 1 kg) and to the neutrons captured in the structure elements and cooling water (2 kg corresponding to approximately 0.5 neutron per fission).

The nuclear wastes to be considered can be divided into three categories:

1. The plutonium and minor actinides, with very high radiotoxicities due to their dominant alpha-decay. They have long lifetimes, up to 25 000 years for 239Pu and more than two million years for 237Np. They would require either long-term underground disposal or transmutation. In the latter case they can only disappear by fission (this is usually called incineration). The fission of 280 kg of plutonium and minor actinides would produce

about 2 TWh of electrical energy. This means that at least one incinerating reactor for four PWRs would be needed if one wants to completely incin­erate the plutonium and the minor actinides.

2. The long-lived fission products, nuclides with lifetimes longer than 1000 years which decay by emission. The main fission fragments involved are shown in table 2.16, together with the amounts produced yearly by a 1 GWe reactor.

3. The medium-lived fission products, essentially 90Sr and 137Cs, which have very high activities at discharge and small neutron capture cross-sections. It does not seem realistic to transmute them and they would, then, set a minimum duration of around 300 years during which the wastes are radioactive and require supervised storage.

The inefficient use of uranium in current thermal reactors has con­sequences on the amount of mining required, as well as on the level of resources. In the absence of recycling, each 1 GWe reactor requires annually about 100 tons of fresh natural uranium. Typically currently used uranium ores have grades around 0.25% [37]. This means that a 1 GWe reactor requires the extraction of 40 000 tons of ore, to be compared with the two million tons of oil which would be needed to produce the same amount of energy. The rather large amount of mill tailings is associated with radioactivity due to the descendents of uranium, especially to a con­tinuous flow of radon during a long period (75400 years as defined by the half-life of the parent 230Th). This radon gas escapes more readily from the tailings than from the unmined uranium ore.

The uranium reserves are estimated around 5 million tons at costs close to the present. The present world power production is about 350 GWe, requir­ing an annual 40 000 tons of natural uranium. Thus the present known reserves are estimated to last 125 years. Again, it is not a problem as long as the present small contribution of nuclear power to the overall energy pro­duction is maintained. However, as in the wastes case, should the nuclear share increase to a 30% level, the reserves would be reduced to approximately 40 years, no more than the oil reserves. One should note, however, that there is a very large reserve of uranium in sea water, amounting to about three billion tons [37], at a concentration of 3.2 parts per billion. It seems possible to extract this uranium at a cost ten times higher than the current cost, which would increase the cost of the produced electricity by 50%.