Because 7 ray absorption is by the discrete events described, rather than a slowing down process as with charged particles, 7 rays do not possess a definite range in matter. In all of the events, however, the probability of reaction between the radiation and mat­ter is constant. The constant of proportionality is called the linear absorption coefficient fi. Thus the decrease in intensity of radiation in a beam of intensity I in passing through a thickness of material 6x is given by:

51 = — pi 5x

or 1 = I0 exp(-fix) where I0 is the intensity of the radiation at x = o, generally the material surface. The form of the above equation applies to /3 radiation also because the j3 particles are emitted with a range of energy up to a maximum value for the radioactive source — unlike a particles which are mono-energetic.

In a similar analysis to the expression for radio­active decay in Section 1.4.4 of this chapter, t± = 0.693/(U, where u is the thickness of the material required to halve the intensity of the 0/y radiation. For comparison, 40 mm of aluminium would be required to halve the intensity of 1 MeV 7 ray and 0,2 mm for 1 MeV /3 radiation. An aluminium thickness of 0.002 mm would be more than sufficient to stop 1 MeV « radiation completely. The inverse of д is sometimes referred to as the mean distance travelled by the radiation,

Knowing the axial and radial flux shapes, Section

7.4.1 of this chapter, it may be readily shown using the one group theory that:

Pnl * 1/(1 + (x/L’)2 + (2.405/R’)2M2]

where M is the ‘migration length’ of the neutrons in the reactor material (M2 is sometimes called the ‘migration, ^rea’); it is a measure of the crow-flight range neutrons travel between ‘birth’ by fission and final absorption.

4.6.5 The critical reactor

As keff = k*, Pnl» and using from the expression for Pnl in Section 7.4.3, it follows that for a critical reactor where keff = 1 that:

(k. — 1)/M2 = (x/L’)2 + (2.405/R’)2

The quantity on the left of the equation (к» — 1)/ M2, is determined solely by the core materials and their arrangement within the core. It is called the material buckling Bm2.

Similarly the quantity on the right (x/L’)2 + (2.405/R’)2, is determined solely by the overall geo­metry of the core and is called the geometric buck­ling Bg2. (The term buckling comes from an analogy drawn between the degree of curvature of the neutron flux shape and equations describing a strut buckling

under longitudinal compression.) Hence, the condition for a critical reactor is that:

Bg2 = Bm2

In each fission event the number of neutrons emitted must be an integer. For a very large number of fis­sions of U-235 by thermal neutrons it is known that 2.7% of the fissions give no neutrons, 15.8% give one neutron, 33.9% give two, and so on. The average number of neutrons released per fission is denoted by v and has a value of 2.43 for thermal fission in U-235.

The average number of neutrons released per fis­sion is different for different fissile isotopes. Also, v increases more or less linearly with increasing neutron energy giving an extra neutron for each 7 MeV of neutron kinetic energy.

Table 1.3 gives the values of v for the fission of uranium and plutonium by thermal and 1 MeV neu­trons (or the threshold 1.1 MeV for U-238).

3.3.1 Energy of the neutrons

The neutrons that are emitted at the instant of fission — the prompt neutrons — are found experimentally to have a range of energies. The relative numbers of neutrons of each energy is shown in Fig 1,9.

Fig. 1.9 Energy of the neutrons emitted in fission

This shows a most probable energy of just less than 1 MeV and a long tail on the high energy side, resulting in an average value for the neutron energy of about 2 . MeV. Therefore, although not strictly true for all, the neutrons released at fission may be regarded as being fast neutrons with average energy of 2 MeV.

1.1 Structure of the atom

For our purpose of understanding a nuclear reactor it is sufficient to regard the atom as consisting of a very dense small nucleus, made up of proton and neutron particles, around which move particulate electrons in well defined orbits. This model of the atom has been likened to a planetary system with the sun as the central nucleus and the planets as the orbiting electrons — see Fig 1.1 illustrating the atom of lithium. The structure of the atom is now known to be much more complex than this but the model, despite its simplicity, has enabled the successful development of nuclear power generation.

The masses of a proton and a neutron are similar and much heavier than an electron, by a factor of 1840. As the nucleus contains ail the protons and neutrons, known collectively as nucleons, it follows that the mass of the atom is predominantly concen­trated in the nucleus.

The proton carries a unit of positive electrical charge (1,6022 x 10“ 19 coulomb) and the neutron is electrically neutral. Each electron carries a unit of negative charge equal in magnitude to the positive charge on a proton. The number of orbital electrons is equal to the number of protons in the nucleus so that their charges balance and overall the atom is electrically neutral. (Should an atom lose or gain one or more planetary electrons it is then left with a residua] electric charge and the atom is said to be ionised.) Thus the number of protons in the nucleus determines the number of orbiting electrons. It is the

Fig. 1.1 Structure of the atom

electrons, and in particular the outermost orbiting electrons, that give the atom its chemical properties. Chemical reactions are due to electron interactions.

The atom is extremely small — a drop of water contains several thousand million million million atoms. Even so, an atom consists mainly of empty space. Atomic diameters, the diameter of the electron orbits, are of the order of 10~’° m whereas the diameter of the nucleus is of the order of 10"14 m, ten thousand times smaller. If Fig 1.1 was shown to scale, and taking the scaled diameter of the nucleus in the figure to be of the order of 1 cm, the electrons would need to be shown orbiting the nucleus at a distance of 100 m or so.

The moderator must not only have the good ‘nuclear physics’ characteristics of a high moderating ratio but also of course other desirable properties. For example, the moderator material must be:

• Readily available.

• Cheap.

• Machineable if a solid.

• Chemically compatible with reactor materials.

• Unchanged by neutron bombardment.

• Non-flammable.

• Non-corrosive.

• Non-toxic.

5,5 Possible moderator materials

From the previous sections the ideal moderator should have the following features:

• Solid or liquid.

• High scattering cross-section.

• Low absorption cross-section.

• Low mass number.

• Low cost.

• Where appropriate — ‘good’ mechanical, chemical and other physical properties.

In identifying materials that may be suitable for use as a moderator only the lighter elements need be con­sidered — up to oxygen say:

• Hydrogen, helium, nitrogen and oxygen are gases.

• Lithium is chemically very active and in nature contains 7 7 Щ Li-6 for which 6C = 70 b.

• The highly toxic beryllium is difficult to fabricate; BeO has been considered but large scale production is also difficult.

• Boron in nature contains 20% boron 10 for which ffc = 3836 barns.

• Carbon in the form of graphite is cheap, an easily machined solid, is relatively inert chemically and has the ‘nuclear’ properties of a reasonably low mass number (12.1), low ac (0.004 b) and accept­able cts (4.8 b).

Of the elements then, only carbon in the form of graphite is suitable for use as a moderator.

However, the potential of the low mass of the hydrogen nucleus may still be utilised in the form of ‘light’ water, H2O, or ‘heavy’ water, D2O:

• Light water is cheap, a liquid, has acceptable chem­ical behaviour and a high scattering cross-section (- 100 barns). Unfortunately it also has a highish

absorption cross-section (0,66 barns) with the re­sult that reactors using light water as moderator need slightly enriched (~ 3%) fuel to counter the neutron losses.

• Heavy water has nearly identical physical proper­ties to light water but, although the scatter cross — section is reduced to about 13 barns, the absorption cross-section is extremely low (0.001 b) giving heavy water its pre-eminence in terms of moderating ratio, see Table 1.7. But heavy water is expensive. It is obtained by isotopic separation from natural water which contains about 1 part of D2O in 6500 of H2O. Thus a decision whether or not to use heavy water as a moderator is a matter of balance between its excellent moderating property and the very high production costs.

To sum up this section on moderators, only three materials can be regarded as being viable for use as a moderator in commercial nuclear power reactors: light water, heavy water and graphite.

1.2 Interaction of radiation with nuclei

In Section 1.5 of this chapter the interaction of radia­tion with the orbital electrons of atoms was discussed. Radiation may also interact with, and induce changes in, the nuclei of atoms. Such interactions are called nuclear reactions and may be represented by X + x — Y + y. The target nucleus X reacts with the incident particle x to give a product nucleus Y and particle y. The reaction may also be written X (x, y) Y. The particles x and у may include the photon ‘particle’ of 7 rays.

1.3 The compound nucleus

A model to explain what takes place during a nu­clear reaction was put forward by the Danish scientist Niels Bohr in 1936 and is known as the theory of the compound nucleus. Bohr postulated that a nuclear reaction occurs as two distinct events:

• An incident particle is absorbed by the target nu­cleus to form a compound nucleus.

• The compound nucleus disintegrates by ejecting a

particle to leave a final product nucleus.

Bohr’s postulation that the two steps are independ­ent and may be regarded as separate processes is in agreement with the observed facts of nuclear transmutation.

When a nuclear particle enters a nucleus its energy is quickly shared amongst all the nucleons of the nucleus. The compound nucleus is said to be in an excited state. The magnitude of the excitation is the sum of the kinetic energy of the incident particle and its binding energy within the compound nucleus. The excess energy is shared in a random manner. At a given instant the excitation energy may be shared by several nucleons and at a later time it may be shared by some other nucleons; or again it may be concen­trated in one nucleon or group of nucleons. When this happens the one nucleon, or group of nucleons, may have enough energy to break away from the com­pound nucleus. The compound nucleus disintegrates into the product nucleus and an outgoing particle.

As a result of the random manner in which the excitation energy is distributed, the compound nucleus has a lifetime which is relatively long (about 10-l4s) in comparison with the time it takes a particle to travel across the nucleus (about 10-,7s for ‘slow’ neutrons). Thus during its relatively long lifetime the compound nucleus can ‘forget’ how it was formed and so have a disintegration which is independent of its formation.

It should be noted that in a nuclear reaction the product nucleus may be radioactive and subsequently suffer further disintegration through radioactive decay.

The loss of neutrons through leakage may be reduced by surrounding the core with a suitable material such that some of the neutrons are ‘reflected’ back into the core. The material should have a high scattering cross-section, low absorption cross-section and, for a thermal reactor, contain a low mass number element so that leaking fast neutrons are not only returned into the core but slowed down in the process. A good moderator is therefore a good reflector and in many thermal reactor designs they are the same material.

The effect of the reflector is illustrated in Fig 1.16; an unreflected reactor is referred to as a ‘bare’ reactor.

The advantages of using a reflector are that:

• Because of the better neutron economy, the critical size is reduced and hence also the mass of fuel required.

• The neutron flux, and hence power production, across the core is more uniform leading to a gain in the overall power output.

So far we have been discussing the properties of neu­trons that are released within 10“14 s of fission occur­ring. These are known as the prompt neutrons. There are a few more neutrons, the delayed neutrons (see Fig 1.7), that arise as part of the radioactive decay of some of the fission products; these neutrons do not appear immediately but at a rate characteristic of the decay of the precursor fission product.

For example, bromine 87 is a fission product. This decays by beta emission to krypton 87 with a half life of 55 s. Figure 1.10 shows that there are two modes of decay. About 30% of the Br-87 decays directly to the ground state of Kr-87, emitting a beta particle with maximum energy of 8 MeV. However, about 70% of the Br-87 emits a beta particle with maximum energy of only 2.6 MeV, resulting in an excited Kr-87 nucleus. The excited Kr-87 may then

either drop to the ground state emitting a 5.4 MeV gamma ray or, rarely — 3%, transform from Kr-87 to Kr-86 by the emission of a neutron. This is a delayed neutron and the original bromine 87 is called a delayed neutron precursor.

A number of delayed neutron precursors have been identified. In analytical work the precursors are di­vided into six groups, each group covering a defined range of half life values. For our purposes we can regard the precursors as leading to an overall mean delay time; it will be seen later that the delayed neutrons are absorbed in the reactor relatively quickly and hence the mean delay time is effectively the mean lifetime of the delayed neutron precursors.

The value of the delayed neutron lifetime т depends on the fissile material as does the delayed neutron yield that is, the fraction of the total neutrons released by fission which are delayed. Table 1.4 gives the thermal fission values of (3 and r for the fissile materials U-235 and Pu-239. They are also given for U-238 but of course for fission by fast neutrons, which makes an important contribution to the overall fission rate in fast reactor systems.

It may be seen that although the fraction of de­layed neutrons is small — 99.32<7o prompt, О. бв^о delayed for U-235 — their lifetime of several seconds is very long compared to the 10 “3 s or less that is the typical lifetime of prompt neutrons. Delayed neu­trons have a profound effect on the kinetics of re­actor behaviour and indeed without delayed neutrons the control of nuclear reactors would not be possible.

As has been stated the nucleus consists of protons and neutrons, the nucleons.

The number of protons is called the Atomic Number and is designated by the letter Z. All atoms of a par­ticular element have an unique atomic number — the number of protons defining the element. Elements that occur in nature have atomic numbers ranging from 1 for hydrogen to 92 for uranium, the heaviest element that occurs naturally. Elements with atomic numbers greater than 92 have been produced artifi­cially — the transuranic elements. It will be seen in subsequent chapters that ^the element plutonium, atomic number 94, has a particular significance in nuclear power generation and is inevitably produced in an operating nuclear reactor. (Strictly, plutonium should also be referred to as a naturally occurring element. It has been found at different locations around the world where fortuitous conditions at some time in the history of the earth allowed uranium de­posits at these places to behave as a nuclear reactor and thus produce plutonium which is still present but in very small quantities.)

The number of nucleons in a nucleus is called the Mass Number and is designated by the letter A. Therefore,

A = Z + N

where N is the number of neutrons.

It is usual to denote an element as Ac where:

ZE

• E is the chemical symbol for the element.

• Z is the atomic number = number of protons.

• A is the mass number = number of nucleons

= total number of protons and neutrons.

Atoms of a particular element which, by definition, have the same number of protons in their nucleus may however have different numbers of neutrons. For example, the nucleus of the element uranium has by definition 92 protons but can have 142 or 143 or 146 neutrons, represented by 234 It, 235 , r or 238 rr

92 U 92 u 92 U’ Again, a chlorine nucleus — which must have 17 protons — can have 18 or 20 neutrons: 35 r,, 37 r]

17U 17L

Atoms with the same atomic number Z but different mass number A are called isotopes. 234 n, 235 ,, and

92 U 92 U

chemical symbol gives the same information as the atomic number, may be written in
the form U-234, U-235, U-238 respectively) are there­fore all isotopes of uranium. Similarly Cl-35 and Cl-37 are isotopes of the element chlorine. Appendix A lists the naturally occurring isotopes of the elements, from hydrogen to uranium, and gives other useful infor­mation (technetium and promethium are included but they are not found in nature). It will be noted in the appendix that for stable elements of low mass number the number of neutrons is about equal to the number of protons. For the heavier elements, however, the number of neutrons increases rather faster than the number of protons until there are about one and half times as many neutrons as protons in the uranium nucleus. This is shown graphically in Fig 1.2.

One may qualitatively understand why heavier ele­ments have disproportionately more neutrons than protons by considering what holds the nucleus to­gether. The gravitational forces between the nucleons are very small indeed and negligible compared to the electrostatic repulsion forces between the positively charged protons, tending to disrupt the nucleus. How­ever there is a strong attractive force between the nucleons — between proton and proton, proton and neutron, neutron and neutron — known as the strong nuclear force. The effect of having neutrons mixed w’uh the protons means that the latter are further apart on average and therefore the electrostatic repul­sion force, which is inversely proportional to the square of the distance of separation between the charges, is reduced. Equally importantly the total attractive force is increased by the presence of the neutrons. The heavier the element the more necessary it is to reduce the electrostatic forces and increase the nu­clear forces.

4.3 Reactor design considerations

Section 4.4.2 of this chapter indicated that a value for кgreater than unity may theoretically be pos­sible in a reactor design in which natural uranium fuel is combined with a moderator material where the elastic scattering collisions convert fast neutrons of the fission events into thermal neutrons. However it is not clear at this stage how to proceed further with specifying the reactor design to ensure k« > 1 or indeed if к» > 1 is achievable in any real practical design. The designer will need to know, for example, how much moderator is needed relative to the fuel; how should the fuel be dispersed in the moderator and what effect the option of slight fuel enrichment may have on the design.

In other words, an understanding is required of what factors determine the value of k® in order to optimise these factors and hence maximise k®.

4.4 The neutron life cycle

The infinite multiplication factor k® is a measure of the number of neutrons in one generation of the chain reaction relative to the number in the previous generation. In order to gain an understanding of the factors that make up the overall value of k® it is therefore sensible to ‘observe’ a generation of neu­trons until they have been replaced by the next gen­eration of identical neutrons; that is, to study the neutron life cycle.

It will be seen that the neutron life cycle may be regarded as comprising four distinct stages. A mea­sure of the ‘efficiency’ of each stage is useful and this is done by dividing the number of neutrons at the end of a particular stage by the number at the beginning of that stage. The four stages of the neutron life cycle are illustrated in Fig 1.13.

The mathematical analysis of the neutron life cycle of Fig 1.13 can be referred to in the literature; never­theless it is useful to consider qualitatively each of the four stages. As it is a cycle it is not imperative to ‘start* at aiiy particular stage — let us here start with Пі thermal neutrons in the moderator.

To clarify matters further, numerical values are given in the brackets for the number of neutrons at each stage (n 1 = 1000) and for the four factors. These values do not represent any particular reactor design.