As discussed in Section 5.5.3, neutrons are classified according to their kinetic energy as cold, thermal, slow, epithermal, and fast neutrons. Since neutrons are neutral, there is no Coulomb repulsion between them and nuclei. As a result, even thermal neutrons can initiate nuclear reactions. When the neutron collides with the nucleus, an excited nucleus forms, which can emit a neutron whose energy is dif­ferent from the neutron that initiated the process.

The cross section of nuclear reactions with neutrons versus neutron energy is shown in Figure 6.4. The plot has two general features. First, the cross section decreases when the energy and velocity of the neutron increase. This means that at

lower velocities, the neutron spends more time near the nucleus, so the probability of the nuclear reaction increases.

Second, the cross section is enormously high at certain energy values, so-called resonances can be observed. This is explained by the discrete energy state of the compound nucleus. The resonances are observed at the energies equal to any excitation energy of the compound nucleus. The two effects are observed simultaneously.

The most frequent nuclear reactions with neutrons are the (n, Y) reactions:

An (n, y) a +An (6.12)

In this process, the emitted particle, a gamma photon, is also neutral, so there is no Coulomb barrier for either neutrons or gamma photons. Therefore, the (n, Y) reactions are simple, and they take place for each element except helium. They are exoergic, releasing about 8 MeV of energy. The disadvantage of the (n, Y) reactions is that the target and the product nuclei have the same atomic number—only the mass number increases by 1. This means that carrier-free radioactive isotopes
cannot be produced directly by this nuclear reaction; the product radioactive nuclide is diluted with the stable nuclide of the same element. Since the product number is rich in neutrons, it usually emits negative beta particles. An example of (n, Y) reactions, the production of 24Na isotope is shown:

23Na(n, Y)24Na (6.13)

The (n, Y) reactions are applied in the neutron activation analysis and prompt gamma activation analysis (PGAA discussed in Sections 10.2.2.1 and 10.2.2.2).

The competitive reaction of the (n, Y) reactions is the (n, p) reactions:

An (n, p) z _An (6.14)

Since the emitted particle (proton) is heavier than a gamma photon, the (n, p) reactions should have a greater cross section. The proton, however, is positively charged, so its emission is inhibited by the Coulomb barrier of the product nucleus (similar to the emission of the alpha particles, as discussed in Section 4.4.1). As a result, light elements react in the (n, p) reactions, while heavier nuclides prefer the (n, Y) reactions. Similar to (n, Y) reactions, the (n, p) nuclear reactions are also exoer — gic. The atomic number of the product nucleus is reduced by 1, and both the target and the product nuclei have the same mass number. The product nuclear is rich in neutrons, so it is a negative beta emitter. Since the target and the product nuclei have different atomic numbers, they are chemically different, so they can be sepa­rated by chemical procedures. In this way, carrier-free radioactive isotopes can be prepared. For example:

64Zn(n, p)64Cu (6.15)

After irradiation with neutrons, the compound nuclide can emit alpha particles too:

AN (n, a)A _ 2n (6.16)

The (n, a) nuclear reactions are endoergic. Only light elements can react in this way because of the high Coulomb barrier between the alpha particle and the prod­uct nucleus. For example:

6Li(n, a)3H (6.17)

If the reaction (6.17) takes place in heavy water (D2O), the product nucleus, tritium, can react with the nucleus of deuterium as follows:

2H(t, n)4He = a or 3H(d, n)4He = a (6.18)

As a result, the neutron is recovered. The irradiating neutrons are thermal neu­trons with different energies; the produced neutrons, however, are fast and have a well-determined energy, в 14 MeV. This reaction takes place in the hydrogen bomb too.

The (n,2n) reactions such as

A N (n, 2n)A "An (6.19)

are also endoergic, since the mass of the neutron increases when emitted from the nucleus (as discussed in Section 2.2). Since the number of the neutrons in the pro­duced nucleus decreases, the product nucleus decays with positive beta decay or electron capture. Since the atomic number remains the same, carrier-free radioac­tive isotopes cannot be obtained directly. Some examples of (n,2n) reactions are:

63Cu(n, 2n)62Cu, 115In(n, 2n)114In, and 23Na(n, 2n)22Na (6.20)

A very important type of nuclear reaction with neutrons is the fission of heavy nuclei under the effect of thermal neutrons. This is called the “(n, f) reaction.” From the natural nuclides, only the fission of 235U has a high cross section. As a result of this fission, two nuclei with intermediate mass, called “fission products,” and more than one neutron are produced:

235U 1 n! A1n 1 A2n 1(2.4 — 2.8)n (6.21)

The binding energy of the two fission products is less than the binding energy of the target nucleus, meaning that the fission reaction is exoergic, releasing 200 MeV of energy. This energy can be used for energy production in nuclear power plants and has been used in the atomic bombs (see Chapter 7). The fission is usually asymmetric; the ratio of the masses of the fission product is about 2:3. In Figure 6.5, the ratio of the fission products of 235U by thermal neutrons and the

main ranges of elements are illustrated, including the small — and high-fission yields. Strontium and cesium, the most important fission products of the low- and interme­diate-level nuclear wastes, are labeled with bold letters.

A significant number of the fission products are radioactive, and some of them have long half-lives. Therefore, the treatment of the radioactive fission products is a very important environmental and safety problem with the production of nuclear energy.

The fission products can be the parent nuclides of decay series. For example, the simplified scheme of the formation of strontium isotopes is shown in Figure 6.6.

In addition to 235U, three artificially produced isotopes, namely, 239Pu, 241Pu, and 233U, also have high fission cross sections. They can be produced from iso­topes, which are more abundant naturally than 235U: the plutonium isotopes can be produced from the 235U isotope (the ratio of 235U to 238U is 1:139, as detailed in Section 4.3.1), U can be obtained from Th. The nuclear reactions of the pro-

239 241 233

duction of Pu, Pu, and U isotopes are as follows:

238U(n, y)239U -—-! 239Np -—-! 239Pu(n, Y)240Pu(n, y)241 Pu (6.22)

232Th(n, Y)233Th ——! 233Pa ——! 233U (6.23)

The irradiating neutrons are obtained from neutron sources, neutron generators, or nuclear reactors (see Section 5.5.2).

0Br-

*91Rb ——- 91Sr——- 91Y—

1s 9s 58s 10 h 59 days

41.6 4 5.2 4 5.9 4 6.0

P" ,Y no P" ,Y no P" ,Y no P" no

92Br -—— 92Kr —— 92Rb —— 92Sr —-— 92Y —— 92Zr

0.3h 2s 5s 3h 4h

4 0.6 4 4.1 4 6.3 4 6.4

93 в" ,Y 93 в, Y 93 в" ,Y 93 в 93

93Kr—— > 93Rb—— > Sr———- > 93Y—— > 93Zr

1s 6s 7min 10h

41.8 4 4.8 4 6.3

94 P" ,Y 94 P" ,Y 94 P" ,Y 94

94Rb —94Sr ———> 94Y ——-—> Zr

Some isotopes of the transuranium elements have great cross sections for neu­trons, but their produced quantity is too low to be used as fuel in nuclear reactors. Their only application is neutron bombs, which contain 252Cf (see Section 7.5).

Isotope atoms may have some different physical, chemical, geological, and biological properties. In addition, the isotopes are usually present not as free atoms, but in com­pounds, participating in chemical bonds. This means that there are isotope compounds or isotope molecules in which one atom (or perhaps more atoms) is substituted by another isotope. For example, the very simple hydrogen molecule represents six different isotope molecules, which can be written using two different symbolisms:

1H2; lH2H; lH3H
2H2; 2H3H
3H2

and

H2; HD; HT

D2; DT

T2

where D and T mean the isotope of hydrogen with mass number 2 and 3, namely, deuterium and tritium, respectively.

A similar situation exists for oxygen molecules, as follows:

16O2,16O170,16O18O
17O2,17O18O
18O2

The compound of these elements, water, may have 18 different isotope mole­cules. Of course, the relative amount of the isotope molecules is very different, determined by the natural abundance of the isotopes.

The thermodynamic properties of the substances can be characterized by the partition function, combining translation, rotation, vibration, and electron excita­tion. At a constant temperature, the translation energy of the isotope atoms or mole­cules is the same.

The rotation energy (Er) of a diatomic molecule can be expressed by the Schrodinger equation for a rigid rotor:

@2Ф @2Ф @2Ф 8п2р т „

—- :т 1 ——— :т 1 ——— т — 1 ——- ;;— £гФ — 0

@x2 @y2 @z2 h2

The solution of Eq. (3.1) is:

where Ф is the wave function, I is the moment of inertia, ш is the angular speed, L is the moment of impulse, and J is the rotation quantum number. The moment of inertia of a diatomic molecule is expressed as follows:

I — теУ 1 m2r| — ^ (3.3)

where T and r2 are the distance of the center of mass from the atoms with mj and m2 mass and p is the reduced mass, i. e.,

m1 m2

P —

m1 1 m2

The reduced mass can be very different for the isotope molecules, and this differ­ence will affect the chemical properties. For example, the masses of the TH and D2 molecules are very similar, but the reduced masses are rather different: 3/4 and 1 for TH and D2, respectively.

The vibration energy of a diatomic molecule (Ev) can be expressed by the SchrOdinger equation of a harmonic oscillator:

where

where v is the vibration quantum number, ш can be defined as:

А, і

2nc і

In this equation, k is a constant (a spring constant in classical physics), x = r — re, and r and re are the mean and the shortest distance between the two atoms, respectively.

When comparing the ratio of the vibration energies for two isotope molecules, look at the following equation:

Ev1 = /і2

Ev2 І1

The electronic excitation can be characterized by the wave number (v*) of spec­trum lines. It can be described by Moseley’s law. For the hydrogen atom, it is:

where Ma and me are the masses of the nucleus and the electron, n1 and n2 are the main quantum numbers of the electron shells involved in the excitation process, and Ry is the Rydberg constant. As seen, the reduced mass of the atom appears in Eq. (3.9), which may be different when the isotope is not the same because of the different masses of the nuclei.

All expressions of the rotation, vibration, and electronic excitation energies con­tain the reduced masses, which are different for isotope atoms and molecules. This difference in the reduced masses is responsible for the isotopic effects, namely, the different physical, chemical, and other properties of the isotopes and isotope molecules.

1.1.1 Physical Isotope Effects

At a given temperature, the thermal (kinetic) energy of ideal gases is the same, independent of the chemical identity of the gas. So, the kinetic energy (Ekin) of the different molecules of hydrogen isotopes (H, D,T) is:

Ekin = 2 RT = 2 mHvH = 2 movD = 2 mTvT

Since the ratio of the masses of the isotopes is mH:mD:mT = 1:2:3,

. . =,. 1 . 1 V”VDVT = ‘.pf. pf

This difference in the velocity of the isotope molecules influences all the proper­ties involving the movement of gases, for example, diffusion and viscosity.

In gas columns, such as the atmosphere, the isotopes separate because of their different masses. This separation can be calculated by the following barometric formula:

Mgh

ph = po e RT

where p0 and ph are the pressure at the level of a reference level (zero level) and at the height h, respectively, M is the molar mass of the gas, g is the gravitational constant, h is the height related to the reference level, R is the gas constant, and T is the temperature (in kelvin).

For two isotopes/isotope molecules with different mass numbers (Mi and M2):

The partial pressures, of course, are proportional to the concentrations of the iso — topes/isotope molecules.

A similar expression can be deduced for the centrifugation of the isotope mole­cules, substituting g X h with (шг)2, where ш is the angular speed and r is the dis­tance from the rotation axis:

p2 p20 («7-«1)(шг)2

p1 p10

As seen in Eqs. (3.13) and (3.14), the degree of the isotope effects is determined by the difference of the masses. It means that these effects are observed for all iso­topes, including heavy elements. Therefore, the centrifugation can be applied to the

235 238

separation of isotopes of heavy elements, for example, U and U.

In electric and magnetic fields, the charged particles move along a curved path. The deviation from the initial direction is proportional to the specific charge of the moving particle.

In electric fields,

where X is the deviation, k is a constant, E is the strength of the electric field, v, e, and m are the speed, the charge, and the mass of the particle, and e/m is the specific charge (mass-to-charge ratio).

In magnetic fields,

He

Km

vm where Y is the deviation, Km is a constant, and H is the strength of the magnetic field.

The specific charge of isotopes with different masses and the same charge is dif­ferent; therefore, they move along differently curved paths in the same electric or magnetic field. The mass spectrometers utilize this process for determining the mass of particles. Isotopes can also be separated in macroscopic quantities using the deviation from the straight line in electric and magnetic fields.

The transformation of the nuclei and the electron orbitals may result in electron emission. As discussed in Section 4.4.2, the negative or positive particles (namely, electrons or positrons following the transformation of the nuclei) are called nega­tive or positive beta radiation, respectively, and they have continuous spectra. The transformation of the atomic orbital can also produce electrons, as discussed in Section 4.4.3. These electrons, such as Auger and conversion electrons, have dis­crete energy. In addition, electromagnetic radiation can produce photo, Compton, and pair electrons, as discussed in Section 5.4.

The rest mass of the beta particle is 0.51 MeV, which is much less than the rest mass of the alpha particle. Therefore, at the same energy of the radiation, the veloc­ity of the beta particle is much higher than that of the alpha particle. Because of the high velocity, the relative increase in the mass often has to be taken into account.

When beta radiation interacts with matter, the electrons in the matter may get excited or ionized, and the direction of the pathway of the beta particle may change as a result of elastic and inelastic collisions. In addition, the kinetic energy is partly or totally transmitted to the matter. When the beta particles interact with the nuclear field, Bremsstrahlung is emitted, which has a continuous spectrum. The inner Bremsstrahlung has been discussed in Section 4.4.3.

The beta particles can be scattered and absorbed, eventually losing all their energy (Table 5.4).

The controlled and uncontrolled chain reaction of the fission of 235U is used in nuclear power plants and weapons, respectively. The neutron balance is quantita­tively determined using the effective neutron multiplication factor (k), which is the average number of neutrons produced from one fission that cause an additional fission:

k = —

Up

where Up and ns are the number of primary and secondary neutrons, respectively.

One of the most important properties of the fission is that two to three neutrons per fission are released (see Eq. (6.21)), which can initiate new fission steps. The

condition of the sustainable chain reaction is that at least one of the released neu­trons should initiate an additional fission. If the average number of the neutrons ini­tiating new fission (k) is 1, the released energy becomes constant. In a stationary state, this is the case in nuclear reactors.

When the number of neutrons initiating additional fission is more than 1, the released energy exponentially increases. This is the case, for example, at the startup of nuclear reactors, or when the power produced by nuclear reactors is to be increased. Nuclear weapons are designed to operate in this way.

In fission reactions, the energy of the released neutrons is usually high (1—2MeV); the additional fission, however, can be initiated only by slow or ther­mal neutrons (<0.1 eV). For this reason, the velocity or energy of the neutrons has to decrease, which significantly influences the neutron multiplication factor. When the fissile material is assumed to be in an infinite quantity, the multiplication factor (kN) is given by the so-called four-factor formula as follows:

k® = єpfП (7-2)

In this equation, є is the fast fission factor, which takes into consideration that the fast neutron can initiate another fission to a small degree (by 1—3%); P is the resonance escape probability, the fraction of neutrons escaping capture while slowing down. The value of p usually ranges from 0.6 to 0.9 and is increased by all factors assisting the slowing down of the neutrons (e. g., by the improvement of the moderators) by decreasing the size of the fuel and by increasing its distance from the fuel rods. The thermal utilization factor, f, is the ratio of the thermal neutrons initiating additional fission to the number of thermal neutrons captured by another reaction (e. g., by nuclides other than fissile ones). And n is the thermal neutron yield, that is, the number released in the fission process.

If the size of the fuel is finite, the effective multiplication factor (keff in Eq. (7.1)) is used; keff<k®. At keff< 1, the chain reaction stops because of the continuous decrease of the neutrons. The reactor is subcritical. When keff = 1, the rate of the chain reaction is constant, and the reactor is critical. When keff > 1, the number of the neutrons, and, as a consequence, the number of fission reactions, increases and the reactor is supercritical.

A characteristic property of the reactor is the reactivity (p):

keff ~ 1
keff

The value of p can be negative, zero, or positive, depending on whether the reactor is subcritical, critical, or supercritical, respectively. Since the fissile material is continuously used up by fission, the fission products can also capture neutrons, and a certain excess of reactivity is required for the critical operation.

7.1.1 The Main Parts of Nuclear Reactors

The very simple scheme of a nuclear reactor and the connecting energetic units are shown in Figure 7.2. The arrangement of the fuel and control rods in the reactor vessel is shown in Figure 7.3.

Figure 7.3 The arrangement of the fuel and control rods in the reactor vessel.

Coolant to heat
exchanger

Fuel rod Control rod

The most important parts of the nuclear reactors are the fuel elements, modera­tor, reflector, cooling system, control rods, and shielding. In the following sections, the material, properties, and operation of these parts will be discussed.