The alpha particle emitted in this type of radioactivity is a doubly charged ion of helium, 4He2+. All alpha particles emitted by a given nuclide either have the same energy or have at most a few different energy values. Energies are in the range of 2 to 8 million electron volts (MeV), with higher energies associated with nuclides of shorter half-life.

In passing through matter, alpha particles give up their energy and become neutral helium atoms. Their range in sohds and liquids is very short; an ordinary sheet of paper will stop alpha particles; the range-energy curve for air at standard conditions is shown in Fig. 2.1. Because of their short range, alpha particles do not constitute an external hazard to human beings. They are absorbed in the outer layers of the skin before they cause injury. On the other hand, if alpha-emitting elements are taken internally, they are very toxic, because of the large amount of energy released in a short distance within living tissue. For example, 1 X 10"7 g of radium is the maximum amount that may safely be allowed to accumulate in the human body.

Alpha radioactivity is found principally among elements beyond bismuth in the periodic table. All the nuclides important as fissionable or fertile material are alpha emitters, with half-lives and decay energies given in Table 2.1. These half-lives are so long that depletion of these fuel species by radioactive decay is not important, but all these nuclides are toxic, especially plutonium, which is even more toxic than radium.

tGram-atom is that quantity of material whose mass in grams is equal to its atomic mass. Similarly, the mass in grams of 1 gram-mole of material is numerically equal to the molecular weight, and Avogadro’s constant is also the number of molecules per gram-mole.

Figure 2.1 Range of alpha particles in air at 0 C, 760 Toit.

At the end of cycle 1, 64 of the 65 assemblies of lot 1 (called lot 1A) are removed. One of the 1GC assemblies (called lot IB) that had the lowest bumup of the lot 1 group is moved to the central AA position. Residual burnable poison is removed from the remaining lot 2 and lot 3 assemblies, which are shifted to the new positions shown in Fig. 3.21. Sixty-four new assemblies (called lot 4), enriched to 3.2 w/o 235 U and containing no burnable poison, are placed in the positions with heavy borders in this figure. This placement of assemblies was

Center*

line

Assembly Number BOC Burnup, MWd/MT EOC Burnup, MWd/MT BOC Relative Power (Assembly/Average)

Figure 3.20 PWR, assembly power and bumup distribution, cycle 1.

found by Rieck [Rl] to lead to an acceptably low maximum peak-to-average assembly power ratio of 1.34 in assemblies 4DG and 4GD. The burnup at the beginning of cycle 2 is shown in the second row of each square, and the bumup at the end in the third row. The total thermal energy produced in the second cycle is 835.2 GWd, or 20,044 X 106 kWh. At the end of cycle 2, assembly 1GC (called sublot IB) and all assemblies from lot 2 except 2FE (called sublot 2A) are removed.

The neutron flux is the product of the number of neutrons per unit volume and the neutron speed. It has the physical significance of being the total distance traveled in unit time by all the neutrons present in unit volume. It seems reasonable that the rate of reaction of neutrons should be proportional to the distance they travel in unit time. The flux has the dimensions of neutrons per square centimeter per second. Typical values of the flux in nuclear reactors range from around 10u to 1014 л/(cm2 — s).

To specify completely the neutron activity and to choose the proper cross sections for calculating the reaction rate constant, it is necessary to know the distribution of neutron concentration, or neutron flux, with respect to energy. In a thermal reactor the distribution of neutrons in thermal equilibrium with nuclei at an absolute temperature T is similar to the distribution of gas molecules in thermal equilibrium and can be approximated by the Maxwell- Boltzmann distribution

(2.44)

where nM(v) dv = number of thermal neutrons per unit volume with speeds between v and v + dv

nM = total number of thermal neutrons per unit volume m = mass of neutron

к = Boltzmann’s constant, 1.38054 X 10-23 J/K
The most probable speed v0 is that for which nM(v) is a maximum, or

(2.45)

For neutrons in thermal equilibrium at 20°C, the most probable speed from Eq. (2.45) is 2200 m/s.

The neutron kinetic energy E is related to the neutron speed by

From Eqs. (2.45) and (2.46) the energy E0 at the most probable speed is

E0 = kT (2.47)

and for thermal neutrons at 20°C, E0 has the value of 0.0253 eV.

By means of Eq. (2.46), the speed distribution, Eq. (2.44), can be transformed into an energy distribution,

nM(E) dE = nM Ш3/2 ЕУ2є-е^т dE (2.48)

where nM(E)dE is the number of neutrons per unit volume with energies between E and E + dE.

The distributions presented in Eqs. (2.44) and (2.48) can be written in dimensionless form in terms of the most probable speed u0 and the energy E0 at the most probable speed as follows:

пм(Фо) = J_ M2e-W

nM n/tt Vo) K ’

The left side of Eq. (2.48) is the fraction of the total thermal neutrons that have a speed ratio v/v0, per unit increment in speed ratio v/v0. Similarly,

nM(E/E0) _ _2_ пм ^/n

Dimensionless flux distributions may be obtained by multiplying the neutron density distribu­tions by the neutron speed ratio vjv0:

The dimensionless neutron density and flux distributions, Eqs. (2.49) to (2.52), are plotted in Figs. 2.10 and 2.11.

Despite the inability of these equations to represent accurately the concentration of higher plutonium isotopes, the reactivity-limited bumup attainable from fuel initially containing 3.2
w/o 235 U can be predicted quite satisfactorily from the results of this table. Reactivity p, calculated from these concentrations by Eq. (3.35), is listed in the last row of Table 3.15 and is plotted against burnup В at the circled points in Fig. 3.30. The points are represented quite accurately by the straight line

p = 0.17 — 8.16 X 10‘6Д (3.82)

The reactivity equals zero at a burnup of 20,833 MWd/MT. This is in excellent agreement with the reactivity-limited burnup of 21,085 MWd/MT for batch irradiation of 3.2 w/o fuel in this reactor obtained by Watt [W2] using the computer codes CELL [B2] and CORE [К1].

In addition to classifying nuclear reactors as thermal or fast, they may be characterized by their purpose, by the type of moderator used to slow down neutrons, by the type of coolant, or by the type of fuel. The principal purposes for which reactors may be used are for research, testing, production of materials such as radioisotopes or plutonium, or power generation. This text is concerned mainly with power reactors.

The most effective substances for slowing down neutrons are those elements of low molecular weight that have low probability of capturing neutrons, namely, hydrogen, deuterium (the hydrogen isotope of atomic mass 2, chemical symbol D), beryllium, or carbon. Examples of moderators containing these elements are light water (H20), heavy water (D20), beryllium oxide, and graphite.

In many types of thermal power reactors, moderator, fuel, and coolant are kept separate in the reactor. Figure 1.7 is a schematic diagram of a nuclear power plant utilizing such a reactor. Table 1.2 lists five examples of reactors with separate moderator, fuel, and coolant and gives references where more detailed information about these reactors may be obtained. In this type of reactor, fuel and moderator ordinarily remain in place in the reactor and only coolant flows through the reactor to remove the heat of fission. Hot coolant flows from the reactor to a steam generator, where it is cooled by heat exchange with feedwater. The feedwater is converted to steam, which drives a steam turbine. The steam then is condensed, preheated, and recirculated as feedwater to the steam generator. Coolant, after being cooled in the steam generator, is returned to the reactor by the coolant circulator. The steam turbine drives an electric generator.

When H20 is used as coolant, the same material serves also as moderator, so ihat the reactor structure can be simplified. Figure 1.8 is a schematic diagram of a pressurized-water reactor, in which the coolant and moderator consist of liquid water whose pressure of 150 bar (2200 lb/in2) is so high that it remains liquid at the highest temperature, around 300°C (572°F), to which it is heated in the reactor. The main difference in principle from Fig. 1.7 is

that there is no separation of coolant from moderator in the reactor. The pressurized-water reactor is one of the two types of power reactor in most common use in the United States. More information about it is given in Chap. 3.

The boiling-water reactor is the other type of power reactor in common use in the United States that uses H20 as coolant and moderator. In this type the water in the reactor is at a lower pressure, around 70 bar (1000 lb/in2), so that it boils and is partially converted to steam as it flows through the reactor. Coolant leaving the reactor is separated into water, which is recycled, and steam, which is sent directly to the turbine as illustrated in Fig. 1.9. Comparison with Fig. 1.8 shows that the boiling-water system differs from the pressurized-water system in having no external steam generator, the reactor itself providing this function.

In a fast-breeder reactor it is impractical to use water as coolant because it is too effective a moderator for neutrons. Liquid sodium is the coolant most extensively investigated for fast

reactors; helium gas has also been proposed. Fast reactors need a higher ratio of fissile to fertile isotopes than thermal reactors to support a chain reaction; a mixture of 20 percent plutonium and 80 percent 233 U is typical for a fast-reactor fuel. Mixed dioxides or mixed monocarbides are possible fuel materials. Although natural boron, which contains around 20 percent of the strong neutron-absorbing isotope 10 B, is satisfactory for control material in thermal reactors, concentrated 10 В is preferred for some fast reactors.

The molten-salt reactor differs from all reactors thus far described in that it uses a liquid solution of uranium as fuel and removes heat from the reactor by circulating hot fuel to an external heat exchanger. No reactor coolant is employed other than the fuel itself. The molten-salt breeder reactor (MSBR) uses as fuel a solution of UF4 in a solvent salt consisting of mixture of BeF2, 7LiF, and ThF4. Separated 7Li is required instead of natural lithium because the 7.5 percent of * Li in natural lithium would absorb so many neutrons as to make breeding impossible. The MSBR is a thermal reactor that breeds 233 U from thorium; neutrons are thermalized by means of graphite moderator blocks, fixed in the reactor, containing channels through which the molten salt flows.

Table 1.3 summarizes the materials used for the principal services in pressurized-water and boiling-water reactors, the high-temperature gas-cooled reactor, fast reactors, and the molten-salt reactor, and indicates which materials are fixed in each reactor and which flow through it.

Consider the general radioactive decay chain

ATj —— ►ЛГ,—— >N3——- ►——————— ►———- >N,——- ► • • •

with Ni atoms of the first member at time zero and none of the other members present at that time. The differential equations are

Nt =N? e~Kt (2.144)

To find the inverse transform of Eqs. (2.143b) to (2.143/) it is necessary to express the denominator as a sum of partial fractions. For Eq. (2.143/) this would be

To find a specific coefficient Aj, multiply each side of Eq. (2.145) by (X;- + s):

І

—— ————— — Aj + (X,- + s) £

її (x* + *) кФІ

k*i

and let s approach —X;. When s = — X,-,

(2.147)

(2.148)

Because the inverse transform of 1 /(X/ + s) is e V,

i — X — f

Ni = ZV? X,X2 • • • X,_, 2 —’—————————- O’ > 1) (2.149)

/=i ri(x*-w

k=

k*i

which is the Bateman equation (2.17). The product term ї’кФІ (X* — X,) has no meaning when the / species is the initial member of the chain, so Eq. (2.149)necessarily applies only to the daughter species, i. e., />1.

Ions of different valences of a metal behave like different elements with respect to extract — ability. The difference between Ce3+ and Ce4+ in Table 4.2 is one example. Another is afforded by Pu4+ and Puvi022+, which are readily extracted by TBP in kerosene, whereas Pu3+ has a very low distribution coefficient [G3]. Consequently, by adjusting the oxidation — reduction potential of the aqueous phase to control the proportion of an element in different valence states, it is possible to vary its distribution coefficient between wide limits. This is the means by which plutonium is stripped from aqueous solutions containing plutonium and uranium in sections C and D of Fig. 4.5 illustrating the Purex process. Addition of a reducing

agent such as a ferrous salt to the aqueous stripping solution entering section C reduces plutonium to Pu3+ and renders it readily back extracted into water. Uranium remains as U0)3+ in the organic phase.

All the beta-radioactive nuclides important in nuclear reactors decay by emitting negative electrons. The daughter nuclide then has an atomic number one higher than the parent, as in the example of f|Sr given in Sec. 2.1. Beta emission differs from alpha emission in that beta particles from a particular nuclide undergoing decay have all energies between zero and a maximum energy characteristic of that nuclide. Figure 2.2 is an example of how beta-particle energies are distributed. The average energy is usually around one-third the maximum. This distribution of energy is explained by postulating that a second particle, the neutrino, is emitted along with the electron and that the sum of the energies carried by the electron and the neutrino equals the maximum observed beta energy. The average neutrino energy is thus about twice the average electron energy.

Neutrinos carry no charge, have little if any mass, and have practically no observable effects. Their range in matter is so great that their energy cannot be utilized. They have no present practical importance.

Beta-radioactive isotopes are known for every element. The half-lives and maximum energies of a few of the most important are listed in Table 2.2, together with their source.

Maximum energies range from 10,000 eV to about 4 MeV. Half-lives range from microseconds to billions of years, with large half-lives tending to correlate with lower energies.

The dependence of range of beta particles in aluminum on energy is shown in Fig. 2.3. Although beta particles have a range greater than alpha particles, they can be stopped by relatively thin layers of water, glass, or metal. The range of beta particles in tissue is great enough, however, to cause bums when the skin is exposed. Beta-active isotopes that may become fixed in the body are very toxic. 90Sr, which becomes fixed in bone, is an example. Those, like 8SKr or 14 C, that are turned over quickly by the body, are much less toxic.

Figure 3.22 shows the placement of assemblies at the start of cycle 3, with new assemblies containing 3.2 w/o 23SU placed in positions near the edge of the reactor, with heavy borders.

This refueling pattern is somewhat similar to modified scatter refueling. The maximum relative power, at 5DG and 5GD, is now 1.36. The average burnup in cycle 3 is 9894 MWd/MT. The total thermal energy is 866.5 GWd or 20,796 X 106 kWh.

If the energy dependence of a cross section is known, the total rate at which neutrons react with a nuclide is obtained by integrating the flux, cross-section product over all possible energies:

Total reactions with neutrons per unit volume per unit time = f ф{Е)о(Е2) dE

‘o

(2.54)

It is convenient to determine an effective cross section a for the nuclide, so that when a is multiplied by the total thermal flux фм the proper reaction rate is obtained:

If the cross section is one that varies inversely with the neutron speed, as in the case with many of the absorption cross sections, then

o(E) = o(E0)(^^J for l/v absorbers (2.56)

where o(E0) is arbitrarily chosen to be the cross section at the energy E0 = kT corresponding to the most probable neutron speed. We shall first assume that all neutrons are in a Maxwell-Boltzmann thermal equilibrium, so that ф(Е) = Фм(Е). The integral in Eq. (2.55) is then transformed to the variable E/E0, and Eqs. (2.52) and (2.56) are substituted to yield

The integral can be evaluated in terms of the gamma function, which in this case has a value of Vrr/2:

— _ g(£~o)s/ff for yjv abSorbers in a Maxwell-Boltzmann distribution (2.58)

Tables (see App. C) usually list values of the thermal absorption cross sections for mono­energetic neutrons of speed 2200 m/s. Because this happens to be the most probable speed for neutrons in thermal equilibrium at 293.2 K, the effective cross section at temperature T (K) can be obtained from

Figure 2.10 Neutron density and flux distributions with respect to speed ratio.

The effective cross section obtained from Eq. (2.59), when multiplied by the total flux of thermal neutrons, will give the proper value of reaction rate with thermal neutrons for a 1/u absorber. However, many of the most important nuclides entering into reactor calculations (e. g., the fissile nuclides) are not 1/u absorbers, and the integration of Eq. (2.54) must consider dependence of the neutron spectrum and the cross section on neutron energy (or speed). In refined calculations this integration is done stepwise by dividing the energy scale encountered in reactors (0 to ~ 12 MeV) into energy groups. An effective cross section is determined for each group and is multiplied by the flux of neutrons in that group to determine the group reaction rate. Digital computers are normally employed.

For simplified reactor calculations a “one-group” approximation can be employed. Westcott [W4] has developed a convention such that the total reaction rate with Nt atoms of nuclide і is given

Total reactions per unit volume per unit time = Nj$o (2.60)

where $ is defined in terms of some arbitrary reference speed v as

where n is the total density of neutrons in the reactor.

The reference speed v is arbitrarily chosen as 2200 m/s, which is the most probable speed for a Maxwell-Boltzmann distribution at temperature T = 293.2 K. The cross section о is now the specially defined effective cross section that, when multiplied by the “2200 m/s flux” $, gives the proper reaction rate constant.

From Eqs. (2.54), (2.60), and (2.61),

/ 0(£M£) dE

In the Westcott formulation the energy distribution ф(Е) is treated as a Maxwell-Boltzmann energy distribution Фм(Е) of thermalized neutrons on which is superimposed an epithermal distribution Фе(Е) of nonthermalized neutrons, so that

The epithermal flux distribution фЕ{Е) can be approximated by a 1 /Е energy dependence above some lower cutoff energy of fjkT, and it can be normalized to the integrated thermal flux фм by a factor 0. Then

фЕ{Е)ЛЕ = фмЩ-<Ш (2.66)

where Д is the unit step function at цкТ energy. A typical value of ц for a well-moderated reactor is 5.

By substituting Eqs. (2.64) and (2.66) into (2.63),

m dE = фм (ф^г е-Е/*т + dE (2.67)

which will be used in solving the integral of Eq. (2.62).

To solve Eq. (2.62) we also need to formulate the total neutron density n as the sum of the densities of Maxwell-Boltzmann neutrons nM and epithermal neutrons nE

n = nM + nE

To obtain n from a flux distribution,

-/

Jo

where Eq. (2.46) was used to change from v to E.

Using Eq. (2.64) in (2.69) to obtain nM:

(2.70)

and using Eq. (2.66) in (2.69), we obtain nE:

JfikT

From Eqs. (2.70) and (2.71),

»e _ 40 пм ч/щї

We now substitute Eqs. (2.67), (2.68), and (2.72) into (2.62) and perform the integration. The results can be written in the form
and

The fraction nE/n in Eq. (2.75) is a parameter specified by the reactor designer. For a purely thermal spectrum nE — 0, so that r = 0 and a = a^oog — For the pressurized-water reactor consiaered in Sec. 6.4 of Chap. 3, the epithermal ratio r is estimated to equal 0.222. When a varies inversely with v, a(E) = o22ao/kf/E, so that g = 1 and s = 0.

The factor g is called the “non-1 /и correction factor.” It becomes greater than unity for a cross section that decreases with increasing neutron speed less rapidly then 1 jv, and it becomes less than unity when the cross section decreases more rapidly than 1 /v.

Values of g and s for 333 U, 335 U, and 239 Pu, as a function of the thermalization temperature T, are listed in Table 2.7. More detailed compilations are available in published reports [Cl, Wl, W4, W5].

The Westcott g and s factors can also be used to determine the effective thermal cross section a, such that when multiplied by the integrated Maxwell-Boltzmann thermal flux <pM the proper reaction rate with a nuclide is obtained, as already defined by Eq. (2.55). From Eqs. (2.55) and (2.62), a is related to a by

Фм

By substituting Eqs. (2.45), (2.68), and (2.70) into (2.77),

(2.78)

or, using Eq. (2.75) to introduce the spectrum parameter r:

(2.79)

3 is the effective cross section defined by Eq. (2.79), which is used later in this text (cf. Sec.

4 and Chap. 3).

The Westcott formulation for the effective cross sections a and a is useful only for well-moderated thermal reactors, where the approximations of the neutron spectra are more reasonable. Even in such reactors, more detailed calculations of actual neutron spectra and effective cross sections are necessary for precise reactor design. The Westcott cross sections are not applicable to fast-spectrum reactors, where neutron moderation and thermalization are suppressed.

(See footnotes on page 52.)