7.1.5.1 Positive Impacts

Nuclear power plants have many positive environmental and economical effects. The specific energy production (energy per mass of fuel) is much higher than that of other types of power plants. For example, the electric energy produced from 10 g uranium dioxide is equivalent to the energy obtained from about 1100 m3 gas,
900 dm3 oil, or 5 tons of coal. Conventional thermal power plants emit many pollu­tants such as sulfur dioxide, nitrogen oxide, and carbon dioxide, increasing the greenhouse effect and damaging the ozone layer of the Earth’s atmosphere. Moreover, the ash produced by coal-based thermal power plants can contain radio­active isotopes; in addition, they emit radon, which is also radioactive, along with its daughter elements. Nuclear power plants, however, do not increase the green­house effect and do not damage the ozone layer. In addition, the emission of the radioactive isotopes can be much lower in nuclear power plants than in thermal power plants. For example, a PWR nuclear reactor can emit about 0.01 Bq/s radio­activity, while a coal-based thermal power plant can emit 2700 Bq/s, supposing that coal is rather radioactive.

This is true even if we compare nuclear energy to renewable energy. For exam­ple, the production of 2000 MW of electricity by solar energy requires solar ele­ments with 600 km2 surface area, supposing sunshine over 24 h/day (a condition not existing in real life). In addition, the production of the solar elements and the accumulators use gallium, arsenic, selenium, and other elements which become heavy pollutants when disposed.

7.1.5.2 Negative Impacts

Under normal operation, nuclear power plants emit few radioactive gases. This emission is regularly checked. In accidents, however, the emission of the radioac­tive isotopes can increase. Because of the very strict safety regulation, accidents are very rare, and they are always caused by human faults or natural catastrophes, such as earthquakes (see Section 7.2 for more). A very important problem of nuclear energy production is the safety treatment and storage of the nuclear wastes (discussed in Section 7.3).

There are three natural decay series that include the heavy elements, from thallium to uranium; their initial nuclides are 238U, 235U, and 232Th isotopes, and via alpha and beta decays, they end up as lead isotopes (206Pb, 207Pb, and 208Pb, respectively) (see Figures 4.4—4.6). The half-lives of the initial nuclides are about billion years, which is similar to the age of the Earth (as discussed in Section 6.2.5). The mass number of the members can be given as 4n in the thorium series, 4n 1 2 in the 238U series, and 4n 1 3 in the 235U series. The decay series characterized by 4n 1 1 starts with the 237Np isotope, it can be produced artificially (see Section 6.2.6). Since the half-life of 237Np is 2.2 million years, even if it was present at the time of the formation of the Earth, it has since decomposed.

4.2 Radioactive Dating

As discussed in Section 3.4, stable isotope ratios can be applied for different geological, ecological, and environmental studies, including the dating of different geological formations and groundwater. Besides stable isotopes, radioactive

U-238 Figure 4.4 The main isotopes of the 238U radioactive

a 9 decay series.

a 4.5×109 years

Th-234

Ф a 24.1 days Pa-234

ФР 1.2 min

U-234

Ф a 2.5×105 years

Th-230

фа 8×104 years

Ra-226

ф а 1620 years

Rn-222

фа 3.825 days Po-218

ф а 3.05 min Pb-214

Ф P 26.8 min

Bi-214

a* il P 19.8min Tl-210 Po-214

isotopes or the stable products of different radioactive decays can be used for dating. In these studies, the half-lives of the radioactive isotopes play an important role; the interval, which can be determined by any radioactive dating method, depends on the half-life of the applied radioactive decay. For example, the age of the Earth and the Earth’s crust can be estimated by radioactive dating to be about 5 and 3.6 billion years, respectively. These determinations are based on the fact that the half-lives of different radioactive isotopes are in the range of the age of the Earth and the Earth’s crust. In this chapter, the main methods of radioactive dating will be discussed.

U-235 Figure 4.5 The main isotopes of the 235U

8 radioactive decay series.

a 7.1×10° years

Th-231

Фр 25.6 hours

Pa-231

фа 3.3×104 years

Ac-227

P 14 il а 22 years Th-227 Fr-223

18.2 days a^ 14 p a^ 22 min

Ra-223 At-219

11.7 days a^ 14 P a^ 0.9 min

Rn-219 Bi-215

3.9 s a^ 14 P 7.4 min

Po-215

afc iip 1.8×10-3s Pb-211 At-215

36 min P^ 14a 1.8×10-3s

Bi-211

a*. iip 2.16min

Tl-207 Po-211

4.8min P^ 14a 0.52 s

Pb-207

In nuclear reactions, a target nucleus (A) is irradiated with a particle (x) and a com­pound nucleus (A*) is formed. A compound nucleus can be formed from different target nuclei and irradiating particles (Figure 6.3). Since the nuclear reactions occur

,27Mg+p ■ 24Na+a *’28Al+y 1 26Al+2n

through strong interactions, the lifetime of the compound nucleus is short, and it can emit another particle (y), producing a new nucleus (B) (Eq. (6.6)). The compound nucleus can decompose in different ways.

Nuclear reactions are classified on the basis of the irradiating and the emitted particles. The characteristic types are listed in Table 6.1.

The term “isotope” was coined by Soddy in 1910, who postulated that elements consist of atoms with the same number of protons but different numbers of neutrons.

If the ratio of the neutrons and protons is different from the optimal ratio associated with the stable state of an atom, the nucleus decomposes, emitting radiation. This process is known as “radioactive decay.” The rest mass of the initial, parent nucleus is greater than the total rest mass of the produced, daughter nucleus and the emitted particle(s). The difference in the masses can be accounted for as the energy of the emitted radiation or particles. The radioactive decay is always exothermal; the emitted energy, however, is usually not released in the form of thermal energy but rather as the energy of the emitted radiation and high-energy particles.

For understanding the radioactive decay, the isobar nuclei (i. e., nuclei that have the same mass number) is a good starting point. The isobars can have odd and even values. The binding energy per nucleon as a function of the mass number gives one parabola for the odd (Figure 3.1A) and two parabolas for the even isobar nuclei (Figure 3.1B—E). In the case of even isobars, the upper and lower parabolas refer to the binding energy of nuclei containing odd or even numbers of protons and neutrons, respectively (the fifth member in Eq. (2.18) can be positive or negative). Thus, the upper parabola is defined by nuclei with odd numbers of protons and neutrons (odd— odd) and the lower parabola by nuclei with even numbers of protons and neutrons (even—even).

In the case of odd isobars, one stable nucleus is at the minimum of the parabola (Figure 3.1A). For this nucleus, the ratio of protons and neutrons is optimal. On the left side of the parabola, the number of neutrons is too high, initiating a radioactive decay in which the number of the neutrons decreases and the number of the protons increases. This process is negative beta decay. On the right side of the parabola, the number of protons is too high, initiating a radioactive decay in which the num­ber of the protons decreases and the number of the neutrons increases. This process is positive beta decay and/or electron capture.

For even isobars, the odd—odd parabolas contain one stable nucleus (Figure 3.1C), whereas the even—even parabolas have one (Figure 3.1B), two (Figure 3.1D), or three (Figure 3.1E) stable nuclei, depending on the relative posi­tion of the odd—odd and even—even parabolas. Similar to odd isobars, the nuclei

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on the sides of the parabolas decompose by negative and positive beta decays or electron capture; as a result of the decays, however, the nuclei go from the even—even parabola to the odd—odd parabola and vice versa. Since the odd—odd parabola is in the upper position, nuclei with odd numbers of protons and neutrons are stable only when they are located in the minimum of the upper parabola and the energy level of this minimum is below the energy level of adjacent even—even nuclei on the lower parabola (Figure 3.1C). This is only the case for four odd—odd light nuclei (2H, 6Li, 10Li, and 14N).

These occurrences of the stable nuclei are summarized by Mattauch’s rule, which states that odd isobars have one stable nucleus, whereas even isobars have two or more stable nuclei, and the atomic numbers of these latter items differ by two. Consequently, if two adjacent elements have nuclides of the same mass, then at least one of them must be radioactive. This rule provides an explanation, for example, why technetium (atomic number 43) does not have stable isotopes.

The parabolas show, too, that the number of radioactive nuclides is much more than that of stable nuclides. Today, we know of approximately 270 stable nuclides and 2000 radioactive nuclides, but the number of radioactive nuclides may reach about 6000. The stable nuclides are listed and classified in Table 3.1.

It can be stated that even—even nuclei are the most frequently stable. The most abundant nuclei of the Earth’s crust are even—even nuclei (16O, 24Mg, 28Si, 40Ca, 48Ti, 56Fe).

One of the most important heavy-charged particles is the alpha particle. As demon­strated by Rutherford, the alpha particle is the nucleus of a helium atom. The energy of alpha particles formed in alpha decay is in the range of 4— 10 MeV.

Alpha particles can interact with orbital electrons, which leads to ionization or other chemical changes; with the nuclear field, where they can be scattered; or with the nucleus initiating nuclear reactions (Table 5.2).

5.2.1 Energy Loss of Alpha Particles

The alpha particles can transfer some of their energy and momentum to the orbital electrons, and their velocity decreases. The energy and momentum transfer can be understood as follows. Let us define a Cartesian coordinate system (Figure 5.2), the horizontal axis (x) of which coincides with the pathway of the alpha particle. In such a system, the electron that participates in the interaction is on the perpendicu­lar axis (y), and the origin of the two axes is where the observation takes place.

During the journey from —o to +o>, the alpha particle transfers p momentum to the electron at distance b. Momentum is a vector with x — and y-components:

1 00

Figure 5.2 The pathway of the alpha particle next to the electron.

X

where F is the electrostatic force between the alpha particle and the electron in the directions of the x — and y-axis, respectively. The electrostatic force acting between two charged particles can be described by the Coulomb law:

(5-7) (5.8) (5.8) (5.10)

Z means the charge of the alpha particle, Z = 2. As seen in Figure 5.2: b

By substituting Eq. (5.9) into Eq. (5.8), we obtain:

Ze2 2 Ze2 3 Fx = sin2 0 cos 0 and Fy = sin3 0 b2 b2

The variable t of Eqs. (5.5) and (5.6) can be expressed by using an angle 0: b

b

t = — ctg 0

Va

Va sin2 0

Figure 5.3 A cylinder shell surrounding the pathway of an alpha particle.

By substituting Eqs. (5.10) and (5.13) into Eqs. (5.5) and (5.6), we obtain the following:

This means that the alpha particle transfers momentum to an electron as expressed by Eq. (5.15), and the electron moves in the y direction. The kinetic energy transferred to the electron (Ee) is:

where me is the rest mass of the electron.

Equation (5.16) gives the energy, which the alpha particle transfers to one elec­tron. During its pathway, however, the alpha particle can interact with many elec­trons and can transfer energy to them. Therefore, the energies transferred to each electron have to be summed up. The moving alpha particle is surrounded by a cylindrical shell, the volume of which is 2nb X db X dx (Figure 5.3). If the number of atoms with the Z atomic number in a unit volume is n, the total energy trans­ferred to the electrons is:

In Eq. (5.18), bmin and bmax are the minimal and maximal radius of the cylinder, inside which the alpha particle can interact with the electrons.

dE = 4nZ2e4n Z ln bmax dx mev2a bmin

The value of bmin can be determined from the maximal energy transferred to the electron. This value can be calculated from the conservation of momentum (Eq. (5.20)) and energy (Eq. (5.21)):

The left and right sides of Eqs. (5.20) and (5.21) give the momentum and energy before and after the energy transfer, respectively. The rate of the electron (ve) can be expressed by means of Eqs. (5.20) and (5.21):

Since the mass of the electron is much smaller than the mass of the alpha parti­cle, the denominator of Eq. (5.22) tends toward the value 1. The maximal energy transferred to the electron is:

	(5.23)

By substituting Eq. (5.23) into Eq. (5.16), we obtain:

Ze2

(5.24)

The value of bmax can be obtained from the distance where the electrostatic potential is a multiplied by the ionization and excitation potential (I):

Ze2 al
	(5.25)

By substituting Eqs. (5.24) and (5.25) into Eq. (5.19), the energy transferred in a unit pathway can be given as follows:

dE 4Z2e4nn. mev2

dx5 — xmvrZ0 lnd/

When the velocity of the electron is very high, the relativistic mass increase has to be taken into account. In this case, the Bethe-Bloch formula is obtained:

As seen in Eqs. (5.26) and (5.27), the energy transferred to the electrons is inversely proportional to the square of the velocity of the alpha particle (v^); in other words, it is inversely proportional to the kinetic energy. Accordingly, we can observe the broadening of the tracks of the alpha particles in cloud chamber photo­graphs due to the higher energy transfer at the later part of the pathway (Figure 5.7). The energy transfer ends when the alpha particle loses all its energy and transforms to a neutral helium atom as in Rutherford’s experiment (discussed in Section 4.4.1). The ionization effect of the alpha particles of various energies is shown in Figure 5.4. At the same time as the ionization, the alpha particles take up electrons and lose their positive charge. The relative charge of the alpha particles as a function of the alpha energy is plotted in Figure 5.5. When the alpha particles lose their total energy, they lose their charge as well and produce neutral helium atoms.

During its passage through a substance, alpha particles lose energy until the energy becomes close to zero. The distance to this point is called the “particle range.” The range (R) of the alpha radiation depends on the energy of the radiation and the composition of the matter. The range is usually expressed compared to the range in dry air (1 bar, 15°C) (R0):

The range of alpha particles in air is several centimeters. In a more condensed medium, however, it is much lower: alpha radiation is absorbed by a sheet of paper or the dead, upper layer of the human skin.

E (MeV)

Figure 5.6 Determination of the range of alpha particles from the intensity—distance curve.

The stopping power of the alpha radiation is the energy loss per unit distance (—dE/dx), whose dimension is MeV/m. Since the stopping power relates to dis­tance, it is called “linear stopping power.” If it is divided by the atomic density, the atomic stopping power is obtained, whose dimension is MeV X m2/atom.

The relative stopping power (S) of the alpha radiation is the ratio of ranges in air and another medium:

S = RR (5.29)

The range of the alpha particles can be determined from the intensity—distance curve (Figure 5.6). The mean range (Rmean) is the point at which the number of alpha particles decreases into the half, i. e., the range at the inflexion point of the intensity—distance curve. The extrapolated range (Re) is determined by the extrapo­lation of the decreasing branch of the intensity—distance curve.

The range and the linear pathway of the alpha particles can be seen in cloud chamber photographs (Figure 5.7).

Figure 5.7 Cloud chamber photograph of the pathway of alpha particles. (Thanks to Prof. Julius Csikai, Department of Experimental Physics, University of Debrecen, Hungary, for the photograph.)

5.2.2 Backscattering of Alpha Particles

As discussed in Section 2.2.2, Geiger and Mardsen, led by Rutherford, studied the absorption of alpha radiation in thin gold foil (about 5 X 10_7 m thick). They observed that most of the alpha particles passed through the gold foil without being deflected. However, a very small number of alpha particles bounced back; i. e., they were deflected to 180°. Since the mass of the alpha particles is relatively great, the phenomenon was interpreted on the assumption that most of the space surrounding the atoms was empty; most of the alpha particles could pass through here. Only a few alpha particles were deflected at high angles. This is possible only if there is enormous repulsion between the alpha particles and the deflecting part of the atom. Since the alpha particles are positive, the enormous repulsion proves that the posi­tive charge is found in a very small area of the atom. This means that the entire positive charge and mass are concentrated in this small area of the atom, which is called the nucleus.

When approaching a nucleus, the alpha particles follow a hyperbolic pathway with nucleus in one of the focuses of the hyperbola (Figure 5.8). For the alpha par­ticle, the conservation of both the energy and the momentum applies. Assuming that the Coulomb law is applicable at small distances (<10_10 m):

Figure 5.9 The scattering of an alpha beam with N flux on a nucleus with a Ze charge.

where ma is the mass of the alpha particle, v0 and v are the velocity of the alpha particle before and after the scattering, Ze is the charge of the nucleus, 2e is the charge of the alpha particle, and q is the distance of the nucleus from the original pathway of the alpha particle or, in other words, the collision parameter (Figure 5.8).

The angle of the deflection of the alpha particle is f. As seen in Figure 5.8, the momentum of the original alpha particles and the change of momentum as a result of scattering can be expressed as:

Ap F At 2Ze2 2q 1 4Ze2 _

tg f = — A =———- = ——- = ——— (5.31)

p p q2 Vo p qvop

Ap = FAt has been discussed already in Eqs. (5.5) and (5.6) in Section 5.2.1. At small deflections (around 0 and n), tg <p «tg <p/2. From here:

When the flux of the irradiation alpha beam is N, the number of alpha particles deflected by one scattering atom is (Figure 5.9):

dN = 2nq dq N

The ratio of dN/N = 2nq dq is called the “differential scattering cross section.” Assuming that the thickness of the scattering layer d, the number of the atoms in a unit volume n, and each alpha particle are scattered by only one nucleus:

The ratio dNp/N = nd2nq dq N is the macroscopic scattering cross section. By expressing q and dq by the angle p and substituting into Eq. (5.34), after equivalent mathematical transformation, we obtain:

As seen in Eq. (5.35), the ratio of Np to N at a constant angle p depends on the atomic number and the number of particles in a unit volume.

In addition to the number of deflected alpha particles at a given angle, the energy of the deflected alpha particles depends on the quality of the deflecting atoms:

where Ep and Ea are the energy of the deflected and the original alpha particles and A is the mass number.

One of the main results of the alpha backscattering studies was the experimental determination of the charge of the nuclei, which provides a confirmation about the position of the elements in the periodic table; that is, the atomic number is the number of positive charges. First, Rutherford determined the atomic number of gold, and later Chadwick measured the atomic number of copper, silver, and platinum in 1920.

The atomic number of hydrogen is 1, and its nucleus contains one positive charge, meaning that the nucleus of hydrogen is a proton. The determination of the charges in the nucleus of helium (which is 2) was also significant: the nuclei of helium are alpha particles.

The other important observation was that the alpha particles can get as close as 10-14 m to the center of the scattering atom; at this distance, only the Coulomb repul­sion acts between the alpha particles and the center of the atoms. By substituting the

p 4Ze2

momentum of the alpha particle, p = ma X v0, into Eq. (5.32), we obtain tg =

From this expression, the distance of the closest approach (q) for the alpha particles at a given angle for the different elements can be determined. This value indicates the upper limit of the radius of the nuclei. Similar data for other elements are summarized in Table 5.3.

As seen in Table 5.3, the alpha scattering experiments show that the radius of the nuclei can be about 104 times smaller than the radius of the atoms (^10-1°m). The radius of the proton is about 1.3 X 10-15 m. Therefore, the alpha backscattering studies proved Rutherford’s assumption that almost the entire positive charge and mass is concentrated on the small space of the atom, that is in the nucleus. The resid­ual volume of the atoms is filled with electrons. Since electrons were found to be

Table 5.3 Radii of Several Nuclei on the Basis of the Alpha Backscattering Expression

(Eq. 5.32)

Atom 238U 197Au 107Ag 63Cu 195Pt

r X 1014m 4.0 3.1 2.0 1.2 3.0 even smaller than nucleons, this means that the atom consists of mostly empty space. This model of the atoms is the Rutherford model or planetary model, which became as the quantitative starting point for the term “chemical elements” in the twentieth century.

The alpha backscattering studies have analytical importance too. Equations (5.35) and (5.36) show that the number and energy of the scattered alpha particles at a given angle depend on the atomic (Z) number, mass number (A), and quantity of elements (n). This means that the deflection of the alpha particles can be applied to qualitative and quantitative analysis of surface layers. The thickness of the layer depends on the range of the alpha particles. The alpha backscattering spectra of an oxide layer produced on SiC are shown in Figure 5.10.

Radioactive decay is always exoergic since the mass of the parent nuclide is greater than the total mass of the daughter nuclide(s) and the emitted particle(s). The equa­tion expressing the release of energy contains an expression describing the kinetic energy of the particles. When interacting with the environment, the particles slow down and become thermalized, and most of the kinetic energy transforms into ther­mal energy. The decay of the natural radioactive isotopes plays an important role in the heat balance of the Earth. The decay of 1 mol U to Pb releases about

3.3 X 1012 J of energy. The half-life of 238U is 4.5 X 109 years, meaning that the release of this energy is a very long process.

Energy can be produced by nuclear reactions as well. This procedure has a very important practical role since energy can be released in a fairly short time in this way. The binding energy per nucleon can be calculated by means of the liquid-drop model of nuclei by Weizsacker formula (see Eq. (2.17)). As seen in Figure 2.4, energy can be produced by two ways: by fusion of light nuclei (as discussed in Section 6.2.4) or by fission of heavy nuclei (see Eq. (6.21)).

In the fission reaction, two lighter nuclei and some neutrons are formed. The neutrons can initiate additional fission reactions if their energy is relatively low (thermal or slow neutrons). This process can be repeated, producing more and more neutrons. If the quantity of the fissile material reaches critical mass, a contin­uous fission, chain reaction takes place. The principle of the nuclear chain reaction was formulated and patented by Leo Szilard in 1934, and it was experimentally proved by Otto Hahn in 1938. For this discovery, Hahn received the Nobel Prize in Chemistry in 1944.

When the number of fissions increases very rapidly and there are enormously high energy releases, the chain reaction becomes unregulated, as in the nuclear bombs (such as the ones used on Hiroshima on August 6, 1945 (U-235), and Nagasaki on August 9, 1945 (Pu-239), discussed in more detail in Section 7.5). The number of fissions, however, can be controlled. Controlled, sustained chain reactions occur in nuclear power plants.

The first nuclear reactor began to operate in the University of Chicago at 3:45 on December 2, 1942. This was the first artificial nuclear chain reaction. The first nuclear reactor constructed directly for energy production was opened in Obninsk, in the Soviet Union, in 1954. According to the International Atomic Energy Agency (IAEA), more than 440 nuclear reactors were operational as of August 2011, and their number increases continuously; more than 60 nuclear reactors are

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under construction (Figure 7.1). The net electric power of the operating nuclear reactors is 365,837 MW and that of the reactors under construction will be 62,832 MW.

In hydrogeology, the ratios of hydrogen and oxygen isotopes are frequently used. Hydrogen and oxygen are connected by covalent bonds; therefore, the ratios of 18O/16O and D/H are evaluated together. Since a portion of the subsurface water is originated from rainwater, the rate and degree of the accumulation of subsurface water can be determined using the isotope ratios. The IAEA measures the isotope ratios of rainwater monthly for some ten years. These measurements show the geo­graphical distribution of the D/H and 18O/16O and the factors affecting the isotope ratios in water. In the following section, these factors and the related isotope effects are summarized. It is important to note that more than one isotope effect can influ­ence the isotope ratios. The different partial pressure of the isotope molecules, for example, plays a role in all cases when evaporation takes place.

• The effect of height: the ratio of 18O/16O and D/H in the water vapor decreases with the height as postulated by the barometric formula (Eqs. (3.12) and (3.13)). The decrease of 618O and 5D is about —0.12 to —0.5m/100 m and —1.5 to —4m/100 m, respectively.

• The effect of meridians is related to the centrifugation of the isotopes: the 618O and 5D of the water vapor and rain decreases into the direction of the poles because of the decrease in the radius of the meridian (r in Eq. (3.14)).

• The continental effect: the diffusion rate of the lighter isotope molecules of the water vapor evaporated from the oceans is higher, so the ratios of 18O/16O and D/H decreases from the sea side to the inland area of the continents.

• The effect of temperature is associated with the change of the partial pressure of the iso­tope molecules, as shown in Eq. (3.18). When the temperature increases by 1°C, the 618O

of water vapor increases by 0.5m. It results in several effects. Seasonal effects are 1816 1816 observed: in winter, the O/ O and D/H ratios decrease. The difference of O/ O

isotope ratios can reach 10m. The effect of temperature is shown in the isotope ratio of the rainwater, i. e., in Central Europe, warm, Mediterranean rainwater originating from the south usually contains more heavier isotopes than rainwater coming from the north.

• There is a linear relationship between the S18O and SD values. The function is called the Global Meteoric Water Line—GMWL:

SD = 8S18O 1 S (3.46)

The slope of the straight line is 8, while the intercept, which is the mean value of S, is 10. S18O = 0 represents the Standard Mean Ocean Water, in which the abundance (Rstandard) of hydrogen isotopes is about 10 times lower than that of oxygen isotopes. It should be noted that the mean value (10) includes fairly high differences: in North America, this value is 16m, while in Mediterranean areas, it is 122m. The difference comes from the partial pressure of the isotope mole­cules. The slope of the GMWL, however, is independent of geographical location, except that when water evaporation is significant, the slope is in the range of 3—6. As before, this fact can be explained by the effect of the temperature: when the temperature increases, the heavier isotope molecules evaporate more quickly.

• At high temperatures, isotope exchanges can take place between water and rocks (Eq. (3.45)). This is a chemical isotope effect, which causes the increase of the 18O/16O ratio in water and simultaneously the decrease of this ratio in the rocks. Since the oxygen content of the rocks is much higher than the hydrogen content, the change of the hydro­gen isotopes can be neglected.

• The isotope ratio allows for the possibility of finding the leakages in the aquifers. The S values are additive, so they can be used to study the communication between the aquifers when the composition of water is very similar, but the isotope ratios are different. Because of the additive character of S18O and SD ratios, the degree of mixing, if any, can be calculated.

The classical theory of the scattering of electromagnetic radiation is valid only when hv«mc2, i. e., at small energies. At higher energies, the wavelength of the scattered radiation changes: the frequency of gamma photons decreases, meaning that gamma energy is lost. This is called “inelastic” or “incoherent” scattering. This process was first studied by Compton.

The process is interpreted as follows. The gamma photons with hv energy encounter an electron. By inelastic collision, part of their energy is transferred to the electron and the direction of the pathway of the gamma photon changes. The process can be described quantitatively by assuming a coordinate system, the x-axis of which is the direction of the pathway of the gamma photon; the y-axis is perpen­dicular to the x-axis. The electron is placed where the axes intersect, in the origin (Figure 5.23). The energy of the gamma photon before and after the collision with the electron is hv or hv0. The energy of the electron before and after the collision is m0c2 and mc2, respectively. m0 is the rest mass of the electron; m is the mass of the moving electron. Before the collision, the momentum of the gamma photon is hv/c in the direction of the x-axis, and zero in the direction of the y-axis. The momen­tum of the electron before the collision is equal to zero in both directions of the coordinate system. After the collision, the electron gains momentum, which is pe cos^ in the direction of the x-axis and pe sin^ in the direction of the y-axis.

By applying the conservation of energies and momentums for the collision of the electron and the gamma photon, we can define the equations as follows:

кщ = hv + Ec

hv0 hv

= cos V + Pe cos p

c c

hv

0 = —- sin V — Pe sin p

c

Ec = mc2 — Ш0С2

The relation between the rest mass of the electron and the mass of the moving electrons (m0 and m) is:

where c is the velocity of light in a vacuum, and v is the velocity of the electron. By substituting Eq. (5.79) into Eq. (5.78), we obtain:

Ec=m, c2(/—J ‘)

The momentum of the electron (pe) can be expressed as:

v

The momentum of the electron (pe) can also be expressed from Eqs. (5.76) and (5.77):

42 2

(5.82)

In addition, pe2 can be obtained by means of Eqs. (5.78), (5.80), and (5.81), using

E = hv. By equivalent mathematical transformation, we obtain the following:

and

By multiplying Eq. (5.84) by the Planck constant (h), we obtain the following:

hv0 _ hv = —0—^ (1 _ cos в) = E0 _ E = —(1 _ cos в) (5.85)

m0c2 0. 51

In this equation, m0c2 means the energy equivalent of the rest mass of the electron;

i. e., 0.51 MeV.

The change of the energy of the primary gamma photon can be obtained using Eq. (5.85):

A— = —2(1 _ cos в)

E0(1 _ cos в) 10.51

As seen from Eq. (5.86), the energy of the photon as a result of Compton scat­tering depends on the energy of the primary photon (hv0) and on the angle. The highest change of the gamma energy can be observed at 180°; the energy does not change at 0° (no scattering). Compton scattering has an important effect on the gamma spectra (see Section 14.2.1 and Figure 14.5).

The safe operation of nuclear power plants is very important. People are very sensi­tive to all events, both usual and unusual, relating to the nuclear power plants, including their construction, operation and radioactive wastes. The safe operation is controlled by the IAEA. In 1990, an International Nuclear Event Scale was intro­duced, which evaluates events other than normal operations. The scale is as follows:

Level 1: Anomaly

• Overexposure of a member of the public to radiation in excess of statutory annual limits.

• Minor problems with safety components with significant defense-in-depth remaining.

• Low-activity lost or stolen radioactive source, device, or transport package.

Level 2: Incident

• Exposure of a member of the public to radiation in excess of 10 mSv. (The units of radio­active doses will be discussed in Section 13.4.1.)

• Exposure of a worker to radiation in excess of the statutory annual limits.

• Radiation levels in an operating area of more than 50 mSv/h.

• Significant contamination within the facility into an area not designed for.

• Significant failures in safety provisions but with no actual consequences.

• Found highly radioactive sealed orphan source, device, or transport package with safety provisions intact.

• Inadequate packaging of a highly radioactive sealed source.

Level 3: Serious Incident

• Exposure to radiation in excess of 10 times the statutory annual limit for workers.

• Nonlethal deterministic health effect (e. g., burns) from radiation.

• Exposure rates of more than 1 Sv/h in an operating area.

• Severe contamination in an area not designed to handle it, with a low probability of sig­nificant public exposure.

• Near accident at a nuclear power plant with no safety provisions.

• Lost or stolen highly radioactive sealed source.

• Misdelivered highly radioactive sealed source without adequate procedures in place to handle it.

Level 4: Accident with Local Consequences

• Minor release of radioactive material unlikely to result in implementation of planned countermeasures other than local food controls.

• At least one death from radiation.

• Fuel melt or damage to fuel resulting in more than 0.1% release of core inventory.

• Release of significant quantities of radioactive material within an installation with a high probability of significant public exposure.

Level 5: Accident with Wider Consequences

• Limited release of radioactive material likely to require implementation of some planned countermeasures.

• Several deaths from radiation.

• Severe damage to the reactor core.

• Release of large quantities of radioactive material within an installation, with a high probability of significant public exposure. This could arise from a major accident or fire.

Level 6: Serious Accident

• Significant release of radioactive material likely to require implementation of planned countermeasures.

Level 7: Main Accident

• Major release of radioactive material with widespread health and environmental effects requiring implementation of planned and extended countermeasures.

The most important nuclear accidents and their impacts are briefly presented here.

1957, Windscale (Great Britain): In a plutonium breeding reactor, graphite heated up. As a result, some fuel rods filled with natural uranium were melted and

131 133 137 89 QO

radioactive isotopes ( I, Te, Cs, QSr, 9 Sr, and noble gases) were emitted into the environment. About 700 km2 was contaminated. The effect on human populations could not be detected. The emitted radioactivity was 4 X 1016 Bq. The average effective dose in the area of the power plant was 0.8 Sv. This incident was categorized as a Level 5 accident on the International Nuclear Event Scale.

1Q7Q, Three Mile Island (USA? Pennsylvania): Because of the coincidence of some technical, mechanical problems and human mistakes, the reactor got out of control, and the active zone melted. 131I and radioactive noble gases were emitted into the environment. The polluted coolant was emitted into the Susquehanna River. The effect on human populations could not be detected. The dose evaluation showed that one more instance of cancer was expected for the 2 million inhabitants within 20 years (the usual number of cancer is 350,000). The emitted radioactivity was about 1015 Bq. The average effective dose in the area of the power plant is not known. This event was treated as a Level 5 accident on the International Nuclear Event Scale.

1986, Chernobyl (Soviet Union, today Ukraine): This accident took place during a test to determine how long turbines would spin and supply power to the main cir­culating pumps following a loss of the main electrical power supply. The automatic shutdown mechanisms did not permit some of the operations. For this reason, the operators, who were not well trained in this type of reactor, switched them off, and a sudden power increase boiled up the coolant water. Water vapor is less able to absorb the neutrons than liquid water, so the neutron flux increased. This resulted in an increase of power. By the time the operator attempted to shut down the reac­tor, the control rods were too high to stop the chain reaction. In addition, the con­trol rods were made of boron carbide with graphite tips. The graphite tips initially displaced coolant before neutron-absorbing material (boron) was inserted and the reaction slowed. As a result, the power continued to increase, and the total volume of coolant boiled up. At the same time, the reactor prompt became critical. The interaction of very hot fuel with the cooling water led to fuel fragmentation, along with rapid steam production and an increase in pressure. The overpressure caused a steam explosion that released fission products into the atmosphere. Seconds later, a second explosion occurred, in which the hydrogen produced from the reaction of the graphite moderator and zirconium cladding with water blew up. In this process, the graphite moderator could react with the oxygen of the air, graphite ignited. Since the reactor was over-moderated (Figure 7.5) and the moderator/fuel ratio decreased, the multiplication factor continued to increase.

In the Chernobyl accident, about 5% of the fuels were emitted into the atmo­sphere. This contained the fission products (e. g., 131I and other iodine isotopes, 134Cs and 137Cs, Sr isotopes, noble gases), uranium, and transuranium elements. A significant number of the radioactive isotopes were bounded to aerosols. The aerosols and the gaseous isotopes were spread over several thousand miles, at first in a northwest direction, then toward the south. The emitted radioactivity was about 2 X 1018 Bq. The average effective dose in the area of the power plant was 6—16 Sv, in Chernobyl, 0.2—1 Sv, in Kyiv. This incident was categorized as a Level 7 accident on the International Nuclear Event Scale.

2011, Fukushima (Japan): The fourth strongest earthquake in history, followed by a tsunami, occurred in Japan. As a result, the power supply of the nuclear reac­tors was destroyed. The emergency instruments stopped the reactors; however, the decay of fission products that had already been produced continued and overheated the reactors. About 75% of the fuel melted. The operators tried to cool the melted fuel by adding sea water, which caused hydrogen explosions in four reactors (three operational and one nonoperational). The activity measurements of the radioactive isotopes (iodine-131 and cesium-137) showed the released radioactivity was about 15% of the released radioactivity in the Chernobyl accident. At first, the accident was assigned to Level 4 of the International Nuclear Event Scale, and then raised to Level 5 and then raised again to Level 7.

The age of rocks can be estimated by means of the radioactive decay series. Let us suppose that at the time of the rock formation, only the initial isotopes of the decay series had been produced, and the lead isotopes at the end of the decay series had been formed only from the initial uranium and thorium isotopes. As a result, the concentration of the lead isotopes (with 206, 207, and 208 mass numbers) in the rock is determined by the age. To derive the relationship between the ratio of the lead isotopes and the age of the rock, the 238U decay series is used. According to the kinetics of the simple radioactive decay:

238N = 238N0 e~A238t (4.63)

where 238N and 238N0 are the quantity of the 238U isotope at the time of the mea­surement and at the time of the rock formation (t = 0), respectively, A238 is the

Th-232 Figure 4.6 The main isotopes of the 232Th radioactive

a 10 decay series.

a 1.41 x 10 years

Ra-228

Ф в 5.7 years

Ac-228

ФР 6.13 hours

Th-228

Ф a 1.91 years

Ra-224

Фа 3.64 days

Rn-220 Фа 55 s Po-216

фа 1.58×10-1s

Pb-212

ФР 10.6 hours

Bi-212

a14 iip 0.6 min

Tl-208 Po-212

3.1 min p^ 14a 3×10-7 s

Pb-208

238 238

decay constant of U, and t is the age of the rock. As the quantity of U isotope decreases, the quantity of the stable nuclide (206N), the 206Pb isotope, increases:

206N = 238N0 — 238N = 238N0(1 — e-A238t) (4.64)

Equation (4.64), however, cannot be used directly because the quantity of the 238U at t = 0 is not known. This unknown quantity can be neglected if the ratio of the quantities of the first (238U) and last (206Pb) members of the decay series is expressed by dividing Eq. (4.64) by Eq. (4.63):

206N = 238n0(1 — e-A238t) = 1 — e-A238г = A2381 _

238N 238^0є-А238 t e-A238t e

Similar equations can be described for the other decay series, namely, for the ratio of the 207Pb/235U and the 208Pb/232Th isotopes:

where the Ns are the quantities of the isotopes (the mass numbers of which are signed in the upper indices), and the As are the decay constants.

By measuring these ratios, the age of the geological formations can be deter­mined. However, the chemical behavior of uranium, thorium, and lead, as well as the intermediate members of the decay series, is different, so they may have been leached from the rock differently. So, in this form, Eq. (4.65) can be applied only when the loss of the members of the uranium or thorium decay series can be neglected. If not, the different leaching of the uranium and lead can be neglected if the ratio of the lead isotopes is taken into consideration. The ratio of the uranium isotopes in nature is U: U = 1:139, determined as follows:

When Eq. (4.67) is divided by Eq. (4.65) and Eq. (4.68) is substituted, we obtain:

207N = ^_ (eA235t — о (469)

206N 139 (eA2388 — 1) 9

As a conclusion, the age of rocks can be estimated by means of Eq. (4.69) from the ratio of lead isotopes determined by mass spectrometry since the decay con­stants are known and thus t can be calculated. The advantage of this method is that it gives the right results even if the lead has been leached from the rock because the leaching does not change the ratio of the lead isotopes.

With these dating methods, the quantity of 206Pb, 207Pb, and 208Pb are measured by mass spectrometry. For more accurate measurements, these quantities are related to the quantity of the 204Pb isotope as a reference nuclide, which is not radiogenic.