As discussed in Chapter 2, the nucleons can be in different energy states in the nucleus (see the explanation of the shell model in Section 2.2.2). Therefore, the nucleons may be in excited states as a result of different nuclear processes. The excitation energy can produce the emission of a nucleon or radiation. The emission of a nucleon takes place in the nuclear reactions (Chapter 6), for example, in the (Y, n) nuclear reactions. As discussed in Section 4.4.6, the nuclei of the daughter nuclides can be in an excited state due to a radioactive decay. The excited nucleus may return to a lower excited state or ground state, emitting gamma photons with a characteristic energy.

The gamma photons can excite another nucleus. The cross section of this excita­tion process may be high when the energy of the gamma photon and the excitation energy of the nucleons are very close, for example, when the structure of the emit­ting and absorbing nuclei is similar, such as in the case of isobars, isotopes, or iso­ton nuclei. This process is called “nuclear resonance absorption.”

At first sight, the resonance absorption seems to be simple. However, the recoil of the nuclei during the emission and absorption reduces the energy of the gamma

photons (E0). The loss of energy (Er) can be calculated using the principle of the conservation of momentum:

—Mv = 0 (5.94)

c

where M is the mass of the nucleus, v is the velocity of the nucleus after the emis­sion of the gamma photon, and c is the velocity of light in a vacuum. By expressing the velocity of the nucleus after the emission of gamma photon, we obtain:

Eq_

Mc

The kinetic energy of the recoiled nucleus can be given as:

1 2 e2

Er = Mv2 = 0

Figure 5.27 Overlapping of absorption (left) and emission (right) photons. (A) In the case of gamma radiations, the lines do not overlap because of the high energy of recoil. (B) The overlapping of the emission and absorption lines at electron transmissions (optical spectra).

Besides the energy loss at the emission, gamma photons lose energy again when absorbed in the nucleus of the absorber. Therefore, the energy of the gamma pho­ton (E) after the absorption is:

E = E0 — 2Er (5.97)

As a result of the recoils of the two nuclei, the gamma photon does not have enough energy to excite the nucleus of the absorber. However, resonance absorp­tion can take place even if the gamma lines are so broad that the emission and absorption lines overlap (Figure 5.27).

The natural width of the lines (Г) can be calculated by the Heisenberg uncer­tainty principle:

h

Г t = — (5.98)

where t is the lifetime of the excited state. In nuclear processes, this lifetime is about 10-9—10-7 s, so the natural line width is very small (Figure 5.27, A). Furthermore, the atoms that are emitting radiation have different velocities because of the thermal movement. So, the frequency of each emitted photons (v) is shifted by the Doppler effect, depending on the velocity (v) of the atom relative to the observer:

where v0 is the frequency of the gamma photons when there is no difference in the velocities. Therefore, there is still some possibility of resonance absorption. As the temperature increases, the line width and the probability of the resonance absorp­tion increase.

191Os isotope (half-life, 15 days) emits beta particles, producing an 191mIr iso­tope. This excited nuclide falls into its ground state (191Ir) in 4.9 s, emitting gamma photons with 129 keV energy. Meantime, the nuclear spin decreases from +5/2 to +3/2. Mossbauer performed absorption experiments with this gamma radiation and iridium foil in 1958 and discovered the recoil-free resonance absorption of nuclei. Because he wanted to avoid resonance absorption, he did the experiments at very low temperatures. The unexpected result was that the resonance absorption increased enormously. At the first approximation, it is interpreted by the increased rigidity of the structure of the crystal lattice at low temperatures; i. e., the whole crystal can be considered to be a “recoiled atom.” So, the mass of the crystal can be substituted as M into Eqs. (5.95) and (5.96). As a result, the velocity and the energy of the recoiled atom will be negligible.

The recoil-free resonance absorption of nuclei can be used in the study of chem­ical states because the oxidation state and the chemical environment influence the energy state of the nucleus via the electrostatic interactions between the electrons and the nucleus. This change, called an “isomer shift” or a “chemical shift,” is by 7—8 orders of magnitude smaller than the characteristic energies of the nuclear pro­cesses. Therefore, a very small change of the very high energies has to be mea­sured. The isomer shift is measured using the Doppler effect: the resonance absorption is created by the relative movement of the sample (absorber) and the gamma radiation source. The relative velocity of the sample and the radiation source corresponds to the degree of the isomer shift, and its dimension is measured in cm/s or mm/s (for example). This small velocity correlates with the small differ­ences of the gamma energies caused by the different oxidation state or chemical environment of the Mossbauer nuclide in the absorber. The most important Mossbauer nuclides are 57Fe, 119Sn, 121Sb, 151Eu, 191Ir, 195Pt, 197Au, and 237Np.

The practical importance of the Mossbauer effect comes from the fact that one of the natural isotopes of iron, Fe-57 isotope, is a Mossbauer nuclide. The gamma radiation source is Co-57 (with a half-life of 9 months), the gamma radiation of 0.0144 MeV of which can excite the Fe-57 isotope. The decay scheme of Co-57 is shown in Figure 5.28.

The isomer shift of the different oxidation states of iron is illustrated in Figure 5.29 by the example of Fe(III) and Fe(II) fluorides. Figure 5.30 shows the Mossbauer spectrum of clay containing iron species.

As seen in Figure 5.29, the iron(III) in FeF3 is in a symmetrical environment and the electron configuration is 3d5, indicating high spin and the presentation of a singlet. The chemical environment of iron(II) in FeF2 is asymmetrical, with 3d6 configuration. The asymmetrical environment interacts with the electric quadrupole moment of the nucleus, resulting in the presentation of a doublet. At low tempera­tures, an inner magnetic field is formed, causing magnetic splitting (the Zeeman effect) with a sextet in the spectrum.

57 Q0 271 days

-7/2

Figure 5.28 Decay scheme of Co-57.

v (mm s-1)

Figure 5.30 illustrates the Mossbauer spectrum of bentonite clay at 74 K before (A) and after (B) treatment with FeCl3 solved in acetones. Segment (A) shows the 12 and 13 oxidation states of iron in bentonite. The sextet in Figure 5.30B refers to the formation of a magnetic phase.

7.4.1 Improvement of the Fission in Nuclear Power Plants

On the basis of their history and technical condition, nuclear power plants are clas­sified into four groups. The first-generation (Generation I) nuclear power plants were developed in the 1950s—1960s in the Soviet Union, the United States, Great Britain, and France. Most of them have been dismantled now; only a few of these reactors are still operating. They do not fulfill today’s safety, technical, and envi­ronmental requirements.

The second-generation (Generation II) nuclear power plants were the improved version of the first-generation plants; they are more safe, economical, and reliable. Most of the power plants in operation today belong to this type. PWRs are the most widespread; they provide about 65% of the total production of nuclear power plants today. The very important difference between first-generation reactors and the PWRs is that in PWR reactors, the total primary circuit (namely, all the con­taminated parts of the reactor) is placed in containment, which is a high-volume, pressure-proof, hermetically sealed building. This creates a new safety barrier in case of an accident.

The third-generation (Generation III) nuclear power plants are improvements over the second-generation plants. For this reason, they are called “evolution nuclear power plants.” The number of evolution power plants is relatively low, there are some third — generation nuclear power plants, for example, in Japan, at this time, but they are being planned and constructed all over the world. Their typical features are as follows:

• They will be built using standard plans, so they can be up and running in a relatively short time (a few years); the operation time, however, is longer.

• Their structure is simpler and more robust than the previous reactors.

• They are safer because of the application of passive protection techniques.

• Their environmental impact is very low.

• The fuel is burned up better, so the fuel cycle is more economical and produces less waste.

The fourth-generation nuclear power plants are called “innovative plants” because they apply new technical solutions and have new safety requirements. The most important trends are as follows:

• In addition to 235U, 238U, and 232Th will be utilized for energy production.

• Besides electric energy production, hydrogen will be produced by the electrolysis of water. This in itself should have a significant positive environmental impact because right now, hydrogen is produced from natural gas, which also produces carbon dioxide, increasing the greenhouse effect.

• To decrease the quantity of high-level nuclear waste produced, the facilities for the trans­mutation of the long-life radioactive isotopes (see Section 7.3.2) will be included in the nuclear reactor itself.

• The possibility of the production of the nuclear weapons from the spent fuel elements will be significantly reduced.

These aims may be achieved by different reactor types, such as thermal reactors, including the very-high-temperature reactor, the supercritical-water-cooled reactor, and the molten salt reactor; fast reactors, including the gas-cooled fast reactor; and molten metal (sodium, lead, lead—bismuth)-cooled reactors.

The hyperfine structure observable in atomic spectra, including the interactions with nuclei, indicates that the nuclei have spin. The nuclear spin is a vector, and its absolute value is л/I (I + 1)—, where I is the quantum number of the nuclear spin,

2n

simply called “nuclear spin.” Nuclei with even mass numbers have I = 0, 1, 2, 3…,

13 5 11

whereas nuclei with odd mass numbers have I = -, -, -… —. The nuclear spin is

2 2 2 2

the sum of the spins of all protons and neutrons. In nuclear reactions, the conservation of spins also must occur.

Parity is related to the symmetry properties of nuclei. It expresses whether the wave function of a particle is even or odd (symmetrical or asymmetrical), depend­ing on whether the wave function for the system changes sign when the spatial coordinates change their signs.

Even parity: Ф(— x, — y, — z) = Ф(х, y, z)

Odd parity: Ф(— x, — y, — z) = — Ф(х, y, z)

The conservation of parity also must occur for nuclear reactions.

The spin and the parity can be signed together: for nuclei with even parity, a + is written after the value of the spin, while for nuclei with odd parity, a — is written (e. g., 0+ or 7/2—).

The particles can be characterized by statistics describing the energies of single particles in a system comprising many identical particles, which has a close con­nection to the spin and parity of the particles. The particles with half-integral nuclear spin can be described using the Fermi—Dirac statistics. These particles obey the Pauli exclusion principle and have odd parity. These particles are called “fermions.” The particles with zero or integral spin and even parity can be described using the Bose—Einstein statistics. These particles are called “bosons.”

The movement of a charged particle causes magnetic momentum. The unit of measure for magnetic momentum is the Bohr magneton, which describes the mag­netic momentum of an electron:

eh ‘id , ,

MB = ——- = 9.274 X 10—24 J/T (2.14)

4nme

For the nucleus, the mass of the proton can be substituted into Eq. (2.14) as follows:

where T is tesla. The quantity pN expresses the unit of nuclear magnetic momen­tum. The magnetic momentum of the different nuclei is in the range of 0—5pN. Surprisingly, the magnetic momentum of the proton is not equal to the value calcu­lated from Eq. (2.15), but it is about 2.7926 times higher than the calculated value. Perhaps more surprising, the neutrons also have magnetic momentum, which is expressed by —1.9135 pN. This implies that the neutral neutron consists of smaller charged particles known as quarks, as discussed in Section 2.4. The negative sign of the magnetic momentum of the neutron indicates that the spin and magnetic momentums are in opposite directions.

Besides magnetic momentum, nuclei can have electric quadruple momentum too. The formation of quadruple momentum can be caused by the deviation of charge distribution from the spherical symmetry. Quadruple momentums have been determined for many nuclei by I > 1/2. Nuclei I = 0 or 1/2 cannot have quadruple momentums.

In conclusion, the characteristic properties of nuclei are listed as follows:

1. Rest mass

2. Electric charge

3. Spin

4. Parity

5. Statistics

6. Magnetic momentum

7. Electric quadruple momentum (not all nuclei).

4.2.3 Alpha Decay

Alpha decay was discovered by Rutherford, who placed an isotope-emitting alpha radiation into a thin glass foil and put the foil into a glass vessel closed at the bot­tom with mercury. The alpha particles have great energy, so they can penetrate the foil into the glass vessel and transform to helium by reacting with two electrons. The gas, of course, shows helium spectrum under excitation. Helium gas, however, cannot penetrate the foil because of the low energy of the atoms, so helium cannot be detected outside the foil.

Apha decay is characteristic for nuclei with great atomic and mass numbers. Thermodynamically, alpha decay can take place at A > 150, but it is only common at A > 210, except for samarium and neodymium, which have isotopes emitting alpha radiation. Alpha decay is possible when the mass decreases in Eq. (4.75).

Alpha particles consist of two protons and two neutrons. By emitting an alpha particle, the ratio of protons to neutrons changes and the atomic and mass number decreases by 2 or 4, respectively.

AM! A 2 4m + a (4.74)

The alpha particle, consisting of two protons and two neutrons, is very stable because of the filled energy levels for protons and neutrons.

The energy of the alpha radiation is in the range of 4—9 MeV. The energy can be calculated from the difference of the rest masses between the parent nuclide and the daughter nuclide, the alpha particle, and the emitting electrons:

Дт = MA — MA-4 2 ma 2 2mt

where MA, MA-4, ma, and me are the rest masses of the parent nuclide, daughter nuclide, alpha particle, and electron, respectively. Since 1 a. m.u. is equivalent to 931 MeV energy, the energy of the alpha particle can be expressed as:

AE = 931 MeV X Am (4.76)

The energy of the alpha particle, however, is smaller than the value calculated by Eq. (4.76) because a portion of the energy recoil the daughter nuclide. The energy of the recoiling can be calculated on the basis of the law of conservation of linear momentum:

mava + Mv = 0

where ma and M are the masses of the alpha particle and the daughter nuclide, and va and v are the rates of the alpha particle and the daughter nuclide, respectively. The rate of the daughter nuclide is expressed from Eq. (4.77):

The total energy emitted in alpha decay is the sum of the energies of the daugh­ter nuclide and the alpha particle:

E = — Mv2 + — mav2n

By substituting Eq. (4.79) into Eq. (4.78), you get the following:

The energy of alpha radiation can be measured in a calorimeter, so the kinetics of the alpha decay can be studied by calorimetry.

In the case of alpha decay, the decay constants of alpha emitters (A) in a decay series correlate to the radiation energy (E) and the range R of the alpha particles in air. This relation can be expressed by the Geiger—Nuttall rule:

log A = a + b X log R (4.81)

log A = a! + b X log E (4.82)

where a, b, a0, and b are constants in a decay series.

The log A—log E function for the alpha-emitting members of the 238U decay series is shown in Figure 4.7.

The alpha particles have well-determined, discrete energies. An alpha emitter, however, can produce alpha particles with different energies. This phenomenon can be interpreted by the shell model of nuclei: after the decay, the nucleus is in an excited energy state. The energy of the alpha particles is lower than the value cal­culated from the differences of the rest masses (Eq. (4.75)), and the difference cor­responds to the excitation energy of the nucleus. The excited nucleus may return to a lower excited state or ground state, emitting photons with a characteristic energy. These photons are called gamma photons (described in Section 4.4.6). Since the nucleus may return to the ground state via excited states, the emission of an alpha particle can be followed by more than one gamma photon. In the case of intermedi­ate members of decay series, the energy of a small number of alpha particles may be greater than the value calculated from the differences of the rest masses if the parent nuclide has been in an excited state at the moment of the alpha emission.

In Figure 4.8, decay schemes of two alpha emitter nuclides (230Th and 241Am) are shown. Similar schemes are constructed for all radioactive nuclides. All impor­tant information on the nuclides (parent and daughter nuclides), the mechanism of the decay, and the half-life of the parent nuclide can be found. In addition, the ratio of the lines with different energy is given, and the spin and parity (+ or —) are also included.

Models describing the alpha decay postulate two stages of alpha emission: (1) the separation of the parent nuclide into the alpha particle and the daughter nuclide; (2) the penetration of the alpha particle through a potential barrier that is formed by the joint action of nuclear forces and a Coulomb (electrostatic) interaction of the alpha particle with the remaining portion of the nucleus (daughter nucleus; see Figure 4.9). The range of nuclear forces is very short (see Section 2.2), and at greater distances, the Coulomb interaction is determining. As seen previously, alpha particles have two positive charges. Since the daughter nucleus is also positive, the alpha particle and the daughter nucleus repulse each other. The energy

Figure 4.8 Decay scheme й * 10 1 years of alpha-emitting nuclides (230Th and 241Am).

Figure 4.9 Potential barrier against alpha emission. The center of the nucleus is in the origin, Fs are the wave function at the different spaces, ECb is the Coulomb repulsion energy, r0 is the radius of the nucleus, rx is the outer wall of the Coulomb barrier, and the emitted alpha particle is outside this barrier.

of the repulsion between the alpha particle and the daughter nuclide (the height of the potential barrier) is:

2Ze2

ECb =—— (4.83)

Г0

where ECb is the height of the potential barrier, Z is the atomic number, and r0 is the Coulomb radius. ECb can be determined experimentally in nuclear reactions with charged particles. It is about 20—25 MeV. The energy of the alpha radiation, however, is about 4—9 MeV. According to classical physics, the kinetic energy of the alpha particles is too low to penetrate the potential barrier. Therefore, alpha decay cannot be interpreted by classical physics. The problem of alpha decay can be solved by quantum physics, assuming the wave- particle dual nature. This means that each particle can be described by a wave function, an energy, a linear moment, and a direction of which is the same as those of the particle.

The total energy of the wave or the particle is:

(4.84)

where v is the frequency.

The moment of the wave function (g) is:

g = hk = h = — (4.85)

A c

where A is the wavelength of the alpha particle, its reciprocal (k) is the wave num­ber, and c is the velocity of light in a vacuum.

The intensity of the wave is:

/ = (Ф)2 (4.86)

where Ф is the probability amplitude of the wave function.

The kinetic energy of the particle at a place with U potential can be expressed by the difference between the total energy and the potential energy:

1 2 g2

Ekin = E — U = — mv2 = ^ (4.87)

2 2m

From here,

g = J 2m(E — U) = hk (4.88)

The probability amplitude of the alpha particle is found as follows:

In the nucleus:

2ni(k-r—v-1)

At the place with U potential of the barrier:

ф2 = B2 e2ni(k2r v21)

and over the barrier (outside the nucleus):

Ф3 = B3 e2ni(k3r-v3t)

As a consequence of Eq. (4.88), k is an imaginary number if the potential U is greater than the total energy of the particle (E). When the imaginary k is multiplied by 2ni as in the power, the power of Eq. (4.91) will be a real number. Therefore, the alpha particles can be present outside the nucleus.

The wave functions defined in Eqs. (4.89)—(4.91) can be summarized in the Schrodinger equation. An approximate solution of the Schrhdinger equation for the alpha radiation is discussed here.

When the rate of the alpha particle in the nucleus is v, the number of collisions on the potential barrier in 1 s is nc:

where r0 is the radius inside the nucleus where the nuclear field is homogeneous. The number of hits can be given by means of the de Broglie wavelength (A) of the particle:

When substituting the rate (v) into Eq. (4.92), we obtain:

The ratio of the probability amplitude of the alpha particle existing outside and inside the nucleus is:

ІФ3І2

ІФ1І2

The decay constant is given as the product of the number of collisions and the ratio of the probability amplitude as follows:

A =

4mrg |Ф! І2

When substituting Eqs. (4.89) and (4.91) into Eq. (4.96) and integrating the SchrOdinger equation from r0 to rx, we obtain:

The approximate solution of Eq. (4.97) for heavy nuclei is:

log A = 20.47 — 1.191 X 109 —-—= 14.084 X 106-Z — 2 X —r0 (4.98)

v E

In this way, the decay constant (A) is obtained in seconds. Equation (4.98) is formally similar to the Geiger—Nuttall rule.

The radius of the nucleus (r0 in Eq. (4.98)) calculated from the decay constant is always smaller than the radius calculated from the alpha backscattering. For exam­ple, the radius of 238U is 9.5 X 10-15m calculated from the decay constant and 4 X 10-14m from alpha backscattering. The differences originate in a different place than the location of the collision of the alpha particles: in the case of alpha decay, the alpha particles collide at the inner side of the potential barrier, while in the case of backscattering, alpha particles collide at the outer side of the potential barrier.

The nuclear reactions of deuteron are important in the production of isotopes in cyclotrons. They have the advantage that deuteron can easily be accelerated, and it can enter the target nucleus from the direction of the neutron, decreasing the Coulomb repulsion. When, in addition, the emitted particle is a proton,

An (d, p)A+)n (6.31)

the Coulomb barrier decreases to almost zero, so the cross section of the (d, p), or Philips—Oppenheimer reaction, is high. The (d, p) reaction takes place with all ele­ments. For example:

23Na(d, p)24Na (6.32)

The (d, p) reaction is analogous to the (n, Y) nuclear reaction, and the target and product nuclei are the same. Carrier-free isotopes cannot be produced directly. The product nuclide is rich in neutrons, emitting negative beta particles.

The (d, n) reactions are analogous to (p, Y) reactions: the atomic number of the product nucleus increases by 1, so the product is carrier-free and decomposes with positive beta decays or electron captures. For example:

9Be(d, n)10B

12C(d, n)13N (6.34)

56Fe(d, n)57Co (6.35)

Some of the (d, n) reactions, such as

2H(d, n)3He (6.36)

3H(d, n)4He (6.37)

7Li(d; n)24He (6.38)

9Be(d, n)10B (6.39)

are used in neutron sources.

The (d,2n) reactions are strongly endoergic, and they are analogous to (p, n) reactions. This means that there are relatively many protons in the product nucleus and the positive beta decay and electron capture are characteristic. They are used for isotope production as follows:

197Au(d; 2n)197Hg (6.40)

In the (d, a) nuclear reaction, carrier-free product nuclides with positive beta decays or electron captures can be produced mostly in exoergic reactions. For example:

24Mg(d, a)22Na (6.41)

56Fe(d, a)54Mn (6.42)

A special example of the (d, a) nuclear reaction is the 88Sr(d, a)86Rb (6.43)

reaction, where the product (86Rb) emits negative beta particles.

The equilibrium constants of the reactions involving isotope molecules may also be different:

AX 1 BYoAY 1 BX (3.34)

AX01 BYoAY 1 BX0 (3.35)

Both X and X0 mean the isotopes of the same element. The equilibrium con­stants are as follows:

The ratio of the two equilibrium constants is:

K = K = = [BX][AX0]

K0 [BX0][AX]

This ratio gives the equilibrium constant of the isotope exchange reaction:

AX 1 BX0 з AX01 BX

When the equilibrium constant of the isotope exchange is equal to 1, there is no isotope effect—the distribution of the isotopes is the same in both compounds.

The equilibrium constants of some isotope exchange reactions are listed in Table 3.4. The calculated values were obtained from the thermodynamic properties.

As seen in Table 3.4, the equilibrium constants of isotope exchange reactions are close to 1 but frequently not equal to 1. These small differences from 1 have, however, a great theoretical and practical importance because they provide a way to separate isotopes and give important geological information (discussed further in Section 3.4).

As a result of the interactions of matter, beta particles can totally lose their energy and absorb into matter. This process is called “real absorption of the beta radia­tion.” However, during the transmission of beta particles through any substance, the intensity (I) of the beta radiation can also decrease as a result of other processes (e. g., by scattering). These processes have been discussed previously and summa­rized in Figure 5.13. Usually, all decreases in intensity are treated as absorption, regardless of the underlying cause of the decreases.

Quantitatively, the absorption of beta radiation can only be described with diffi­culty due to the continuous nature of the beta spectra. This means that the energy of beta particles when entering matter can range from zero to the maximum energy of the beta spectrum. Therefore, the expressions describing the beta absorption are usually empirical. It is interesting, however, that the empirical equation of beta absorption (Eq. (5.46)) is similar to the general equation of the absorption of radia­tion (Eq. (5.3)):

-r(E)l

where I0 and I are the intensities of the beta radiation before and after the transmis­sion through the matter, l is the thickness of the absorber, and p(E) is the linear absorption coefficient; its dimension is reciprocal length (e. g., mm"1, cm-1, m-1). The value of the linear absorption coefficient depends on both the energy of the radiation and the atomic number and density of the absorber. By introducing the mass absorption coefficient, it can be avoided to determine the linear absorption coefficient for all maximal beta energies and for all substances. For this purpose, the linear absorption coefficient in the exponent of Eq. (5.46) is divided and multi­plied by the density of the absorber (p). Note that the density is the ratio of the mass and volume (p = m/V = m/(l X S)):

p(E) m

I = I0e p l X S (5.47)

where m and S are the mass and the surface area of the absorber, respectively.

(E)

= p is the mass absorption coefficient, and its dimension is surface area/mass.

p

Since l/l = 1, the mass/surface area (m/S) remains in the exponent of Eq. (5.47). This quantity describes the mass of the absorber on a unit surface area; it is called “surface density” (d); its dimension is mass/surface area (e. g., mg/cm2). This leads to:

I = I0 e"^ (5.48)

The relation of the mass absorption coefficient and the maximum beta energy (Epmax) and the atomic number of the absorber (Z) can be approximated by empiri­cal equations. When Z < 13:

When Z> 13:

= 7.7Z031

p e1.14

E(3max

In Eq. (5.47), Ma is the relative atomic mass of the absorber. For compounds and mixtures, the mass absorption coefficient can be calculated by the mass absorp­tion coefficients of the components, taking into consideration their mass ratio (w):

As seen in Eq. (5.48), the absorption of continuous beta radiation can be described by an exponential equation. However, the monoenergetic (> 0.2 MeV)

electron radiations (e. g., conversion electrons) show the linear absorption curve as a function of surface density. Below 0.2 MeV, the absorption curve of the monoe­nergetic electron deviates more or less from linearity.

To characterize the absorption of the beta radiation (exponential law, Eq. (5.48)), the half-thickness of the absorber (d1/2) is defined. This is the thickness where the intensity of the beta radiation decreases by half:

ln 2

d1=2 =——- (5.52)

As seen in Section 5.2.1, alpha radiation has a well-defined range (R). However, the range of beta radiation can be described only by empirical formulas at different maximal beta energies, such as:

In Eqs. (5.53)—(5.57), the dimensions of range and energy are g/cm2 and MeV, respectively.

Cloud chamber photographs show the differences in the pathways of the beta and alpha particles (Figure 5.7 shows the alpha track, and Figure 5.14 shows the alpha and beta tracks). Since alpha particles are much heavier than beta particles/ electrons, the pathway of alpha particles is linear. Beta particles, however, tend to deviate more or less, depending on their energy. The interaction of gamma radia­tion with matter will be discussed later (in Section 5.4); for now, just note that gamma radiation produces secondary electrons, the tracks of which will be shown in the later discussion.

In the presence or two or more beta emitters, Eq. (5.48) consists of several members:

/ = /10 e-ftd 1 /20 e 1 … 1 In0 e2^nd (5.58)

This means that the absorption of each beta radiation has to be taken into account separately. The mass absorption coefficients and the range of some beta emitters as a function of maximal beta energy are plotted in Figure 5.15. These data are widely applied in the characterization of beta absorption and the planning of shielding against radiation. However, Eqs. (5.49) and (5.50) show that the mass

absorption coefficients depend on the atomic number. Therefore, at Z > 13, the mass absorption coefficients calculated on the basis of Eq. (5.50) are about twice as high as the data in Figure 5.15. It should be noted, however, that the plan of shielding uses the principle of the so-called conservative estimation, which means that the plans consider the worst scenario. Therefore, the application of the lower mass absorption coefficient is permitted or even can be desirable.

In the fission process, two or three neutrons are formed (Figure 7.4), 99% of which are emitted within a very short time. These are called “prompt neutrons,” and their mean half-life is about 10 4 s. Some fission products also emit neutrons (via neu­tron decay, as discussed in Section 4.4.4), and these are delayed neutrons. The amount of the delayed neutrons is less than 1% of the total numbers of the neu­trons. Delayed neutrons play a role in the neutron balance of the reactor. If all the neutrons were produced as prompt neutrons, the whole fission products would be complete within a very short time and could not be controlled. Therefore, the nuclear reactors are planned in such a way the controlled chain reaction can be ini­tiated in the same time as the delayed neutrons.

The reactors are controlled by control rods. They are fabricated from very good neutron absorbers, such as boron (as boron carbide) or cadmium. The control rods are inserted among the fuel rods (Figure 7.3). The reactivity is controlled by the movement of the control rods. When the reactor starts, the control rods are raised. The power is measured by neutron detectors. When the power reaches the desired value, the control rods are stopped. So, the power does not continue to increase, the reactor becomes critical.

There are three different types of control rods:

1. Safety rods, whose function is the fast stop of the reactor in an emergency. In normal operation, they are totally raised.

2. Shim rods, which are used for coarse control and/or to change reactivity in relatively large amounts. They equalize the changes of the reactivity resulting in burn-up, poison­ing, or breeding. Another tool for the equalization of reactivity is that neutron absorbers (usually boric acid) are dissolved in the coolant, whose concentration can be varied as required. Of course, this method can be applied only in the water reactors.

3. Regulating rods, which are used for fine adjustments and to maintain the desired power or temperature.

Radioactivity (A, also known “absolute activity”) is defined as the number of decompositions in a unit time. Radioactivity is in proportion to the initial quantity of the radioactive nuclei:

A = — dN = AV = AN0 e-At = A0 e-At (4.12)

It is important to note that the activity—time function (Figure 4.12) is analogous to the number of the radioactive nuclei—time function (Eq. (4.8)).

The unit of radioactivity is the becquerel (Bq), which describes the number of decomposition/disintegrations that take place in 1 s (1 Bq = 1 dps = 1 disintegrations per second). An earlier unit of radioactivity was the curie (Ci), which is the number of decompositions in 1 g of radium in 1 s. The relation between the two activity units is 1 Ci = 3.7 X 1010 Bq. Besides these two, dpm (which means “disintegrations per minute”) is frequently used for practical purposes.

Radioactivity is usually measured not by identifying the radioactive nuclei but by counting the emitted particles. In theory, to achieve accurate activity measure­ment, all particles emitted in 4n spatial angles should be taken into consideration. In practical applications, however, it is more common that the radioactive intensity (I), a quantity proportional to the radioactivity, is measured. The proportionality factor is the measuring efficiency (k):

I = kA = kAN (4.13)

The intensity—time function, of course, is similar to the activity—time function (shown in Eq. (4.12)):

I = I0e-At (4.14)

Obviously, this relation is valid so long as the measuring efficiency (k) stays constant for all measurements.

The units of intensity are as follows:

• cpm means counted particles per minute,

• cps is counted particles per second.

As mentioned several times previously (Chapter 2), the neutrons are the basic parti­cles of nuclei. They can be present as free neutrons as a result of neutron decay (see Section 4.4), but it is a very rare phenomenon. Neutrons, however, can be pro­duced by nuclear reactions (Chapter 6) and can be widely used in different scien­tific and practical applications. Thus, in this chapter, the basic concepts of the interactions of neutrons with matter will be discussed.