The structure of nuclei has been described by different models. At the moment, however, none of them alone explains all experimental observations. A useful review of 37 known models of the atomic nucleus is provided by Cook.

Fermions: Spin 1/2

Name Sign Rest Mass Charge Name Sign Rest Mass Charge Name Sign Rest Mass Charge Interaction

(MeV) (MeV) (MeV)

The alpha model proposes the presence of alpha particles of great stability within the nuclei. This model has been suitable only for the interpretation of alpha decay.

In this process, the nucleus captures an electron from an inner electron shell (K or L shell) resulting in the following transition:

p++ e2 ! n 1V (4.111)

The process is characterized as electron capture, EC decay, or EX decay. EC decay is energetically more desirable than positive beta decay since there is no beta particle emission in EC decay. The neutrinos formed in the electron capture are monoenergetic.

The electron capture is always followed by the emission of electromagnetic radi­ation because the orbital vacancy results in an excited electron state. When the vacancy in the K shell is filled with an electron from an outer, mainly L, shell, the difference between the K and L binding energies is emitted as characteristic X-ray radiation. It is emphasized here that the high-energy electromagnetic radiation is called “gamma radiation” if it is the result of nuclear transition, while if the source of the radiation is the transition of electrons between the extranuclear orbitals, it is called an “X-ray.”

Instead of X-ray radiation, the excitation energy can be transferred to another electron, which is then ejected from the atom. This second ejected electron is called an Auger electron. In this process, the produced nucleus has more than one positive charge, so it can react easily with other substances. The probability of the Auger effect decreases as the atomic number increases. As a result, the ratio of the gamma photons and the Auger electrons depends on the atomic number: for light elements, the Auger electrons are significant, while for heavy elements, the characteristic X-ray is dominant (Figure 4.12).

Furthermore, the electrons captured from the K and L shells, on their pathway toward the nucleus, lose their energy in the nuclear field. This process results in the emission of X-ray radiation called inner Brehmsstrahlung, the spectrum of which is continuous. Thus, as a result of electron capture, both characteristic and continuous X-ray radiations are emitted.

The electron capture results in excited nuclei. This excitation energy may be lost through either the emission of gamma photons or the transition of the excita­tion energy to an electron on the atomic orbital (mainly a K electron) of the same atom, followed by an electron emission. The latter process is called “internal con­version,” and the emitted electrons are conversion electrons. The kinetic energy of the conversion electron is equal to the energy of the gamma quantum reduced by

the binding energy of the electron. This means that the conversion electrons, similar to Auger electrons, have discrete energy.

In some cases, the energy of the electron capture can be measured by using the cyclic process, as shown by the following:

244Am ——! 244Pu (4.112)

a

240Np 244Am

523 MeV

в 0.36 Mevt і AEEC?

a

240U 244Pu

4.65 MeV

4.2.4 Proton and Neutron Decay

Proton decay can take place after positive beta radiation of the light elements, which is followed by proton emission. For example,

Neutron decay, or delayed neutron decay, may occur when a negative beta decay followed by neutron emission takes place. Neutron decay can be observed for the heavier nuclides too. For example,

17N —-— 17O* —-—— 16O (4.114)

Some fission products emit negative beta particles as well as neutrons, for example,

87Br -—— 87Kr* —-— 86Kr (4.115)

127I —-— 137Xe* —^— 136Xe (4.116)

These isotopes are significant in the neutron flux of nuclear reactors, especially when the power is decreasing.

As seen in Figure 2.2, the binding energy in each nucleon has an extremum as a function of the mass number. This means that energy can be obtained by the fission of heavy elements and by the fusion of light elements. (Fission is discussed in Section 6.2.1.)

The most important fusion reactions of the isotopes of hydrogen are exoergic:

The activation energy of the fusion processes, however, is very high. The igni­tion temperature is the lowest for the 2H—3H reaction (see Eq. (6.50) and Figure 6.7), it is about 107 K; the ignition temperature of the 2H—2H reaction (see Eqs. (6.47)—(6.49)) is in the range of 108 K. The H—H reaction requires an even higher temperature, about 1010 K. These reactions take place in stars. Natural fusion reactions will be discussed in Section 6.2.5.

Figure 6.7 The cross section of the D—T reaction as a function of temperature.

All products of the reactions ((6.44)—(6.51)) are inactive, so fusion energy pro­duction should be more desirable than fission energy production. However, there are many technical problems that have not been solved yet, as will be outlined in Section 7.4.

The thermonuclear reactions take place in the hydrogen bomb. The nuclear bomb will be discussed in Section 7.5.

Any of the above-mentioned isotope effects can be used to separate the isotopes. Distillation, gas diffusion, centrifugation, electromagnetic separation, electrolysis, and chemical isotope exchange are widely used methods for isotope separation. A newer, novel method of doing this is laser isotope separation (LIS).

The LIS technique was originally developed in the 1970s as a cost-effective, environmentally friendly way of supplying enriched uranium. The method is based on the fact that different isotopes of the same element absorb different wavelengths of laser light. Therefore, a laser can be precisely tuned to ionize only atoms of the desired isotope, which are then drawn to electrically charged collector plates.

The isotope separation is characterized by the separation factor. In a two- component system, the separation factor (a) is defined as:

Xf(1 ~ X0) = R, (1 — Xi)Xo Ro

where X0 and X, are the molar fraction of one of the isotopes before and after sepa­ration, respectively.

In addition, 1 — a is called the enrichment factor.

Since the degree of the isotope effects is usually small, one separation step is frequently not enough to reach a high enough enrichment. In this case, a multistage process in cascade can be applied. The enrichment factor of a separation cascade (A) is proportional to the number of stages (n):

A = an = —1 (3.42)

R0

By increasing n, the enrichment increases proportionally.

The enriched isotopes are used for the production of fuels and moderators of nuclear reactors and nuclear weapons, for analytical purposes (e. g., NMR, Mossbauer spectroscopy), and for the preparation of targets in the production of radioactive isotopes. In Table 3.5, the most important enriched isotopes are listed. Beside enrichment, the depletion of the isotopes can be important for special applications. Depleted 64Zn is used in nuclear industry. The addition of zinc to the cooling water inhibits the corrosion and the formation of 60Co (discussed in Section 7.3) from the steel of the reactor, decreasing the workers’ radiation expo­sure. Natural zinc contains 48% 64Zn; however, the gamma emitter 65Zn isotope is produced by (n, y) nuclear reaction of 64Zn (discussed in Section 6.3). To avoid the production of 65Zn, depleted MZn (<1%) is produced by centrifugation and applied in nuclear reactors.

The beta particles may scatter both on the orbital electrons and in the nuclear field. Since the beta particles are much lighter than the alpha particles, the degree of the scattering of the beta particles is much higher than that of the alpha particles, resulting in very important measuring and analytical consequences.

Layer of the sample (g/cm2)

Figure 5.18 Intensity as a function of thickness in the method of constant specific activities.

The backscattering of the beta radiation as a function of the thickness of the scattering medium can be described as follows. Let us take a beta emitter on the bottom of a ring of lead-shielding, and a scattering medium with d thickness and arrange them as illustrated in Figure 5.19.

Then let us irradiate the surface area (F) of the medium with a beta radiation with I0 intensity. Because of the absorption of the beta radiation, the intensity decreases when passed through a distance x, and the intensity reaching the dx unit thickness is:

dIx = I0e2^ (5.66)

Let v be the ratio of beta particles that are backscattered from the dx thickness:

v dIx dx = v I0 e~Fx dx (5.67)

The backscattered beta particles are absorbed again when returned through the x thickness. Therefore, the intensity of the backscattered beta particles reaching the surface (F) can be expressed again by the absorption law. The energy of the back — scattered beta particles may be lower than the energy of the original beta particles, so the values of the mass absorption coefficients may be different when the beta particles pass in (Fin) or out (^out). In backscattering studies, the resultant effect of the two mass absorption coefficients is observed, so we can assume that

Fin 1 Fout Fb:

dI = vI0 e2(Fin+F“t)x dx = vI0 e2Fbx dx (5.68)

The total backscattered intensity can be obtained by the integration of Eq. (5.68) for the total thickness (d):

■d V

dI = —10[1 — e-M>d] 0 Mb

As seen from Eq. (5.69), the backscattered intensity tends to a limit as a function of the thickness. This limit for the infinite thickness of the sample is:

The backscattering of beta radiation can be characterized by the backscattering coefficient (Rf):

Rf can also be expressed in percent.

The energy of the backscattered beta particles is less than the energy of the orig­inal particles (Figure 5.20).

Similar to Eq. (5.65), the half-thickness of the backscattered beta radiation can be defined. The backscattered intensity of beta radiation of aluminum, zinc, and lead is shown in Figure 5.21 as a function of the half-thickness. As seen, the back — scattered intensity depends on the atomic number of the scattering media.

The values of Rf and the atomic number (Z) are in strict correlation. The back — scattered intensity, or Rf versus Z function, cannot be calculated exactly; empirical correlations are usually applied. One of them is as follows:

In = ki Zk2 (5.72)

where Inj is the scattered intensity at an angle, k1 = 0.0415 INJ2n and k2 = 2/3. Another Rf versus Z function is the so-called Muller formula:

R = aZ 1 b (5.73)

where a and b are constants for the elements in a given period of the periodic table (Table 5.5). Hydrogen is a special element; it can be fitted into the system by a hypothetical atomic number, which is —7.434. This can be explained by the fact that the ratio of nucleons to electrons is usually 2, while in the case of hydrogen, this ratio is only 1.

Equations (5.73) and (5.74) are also valid for compounds and mixtures if the mean atomic number is applied. The mean atomic number can be defined as:

n

niAiZi n

Z = i51 = XiZi (5.74)

niAi i51

i=1

where Z, is the atomic number of the constituents, n, is the number of the atoms, A, is the mass of the atoms, and x, is the mass ratio of the ith atom in the compound or mixture.

Equations (5.73) and (5.74) can also be applied for solutions; the Rf versus x, function is linear. Therefore, the Rf versus x, function is suitable for the concentra­tion measurement of solutions. In addition, by extrapolating the Rf versus x, func­tion to x, = 1, the backscattering coefficient (Rf) of the pure solid substance is obtained.

In conclusion, the measurements of the backscattered intensities of beta radia­tion give information on:

1. The thickness of the scattering matter or the thickness of thin layers on a thick plate (Section 11.3.4).

2. The mean atomic number.

3. The concentration of solutions.

The analysis of uranium ores shows that the ratio of 238U:235U isotopes in the natural uranium is constant (139:1), and the concentration of 235U is about 0.7%. There is just one uranium pitchblende, in Oklo (Gabon), in which the ratio of 238U:235U is higher than the usual value, the concentration of 235U is below 0.5% ( U: U > 200:1). Studies of this uranium mine have shown that the concentra­

tion of the rare earth elements is also higher, and they show similar ratios to the fission products of 235U. For example, natural neodymium contains 27% 142Nd, while the Oklo ores contain less than 5% 142Nd. The 143Nd content, however, typi­cally is 12%, while its concentration in the Oklo samples is 24%. Neodymium formed in the fission of 235U contains 29% 143Nd and no 142Nd isotope.

These values indicate that the fission of 235U could be taken place a very long time ago; that is, a natural nuclear reactor could have been present long ago. Natural water probably acted as the moderator. Based on the composition of the fis­sion products and the uranium content, the properties of the natural reactors are estimated to be as follows: the neutron flux was <109 neutron/cm2 s in the core of the reactor, and its power was less than 10 kW about 2 billion years ago. It con­sumed about 6 tons of 235U, and produced about 1 ton of 239Pu.

A radioactive decay is described as branching when one parent element decom­poses to two daughter nuclides. This type of decay can be characterized by two decay constants and half-lives as follows:

B

A

2 B2

where A is the parent nuclide, and B2 are the daughter nuclides, and Ai and A2 are the decay constants for the production of B1 and B2, respectively. Examples of such decay are the decomposition of the Pb isotope into Po and Tl, the decay of 64Cu isotope to 64Zn and 64Ni, and the disintegration of the 40K isotope into 40Ca and 40Ar isotopes.

Since during branching decay, the quantity of the parent element decreases via two independent processes, the rate of decay of the parent element can be defined by the sum of the two decay constants:

dN

— — = (Ai 1 A2)N dt

From here,

By the integration of Eq. (4.16): ln N = —(Ai 1 A 2)t 1 constant

Assuming that at t = 0, N = N0:

N = N0e2(Al+A2)t (4.18)

Equation (4.18) is similar to the kinetics of the simple radioactive decay (Eq. (4.8)), except that the sum of the individual constants is used as the decay con­stant. In those cases, when the daughter elements formed through different decay mechanisms or the energy of the emitted radiation is sufficiently different, the values of the decay constants can be determined separately. The proportion of the decay constants will determine the relative quantity of the daughter nuclides formed.

In most cases, however, both daughter elements are formed via beta decay, the spectra of which is continuous (see Section 4.4.2), and the decays are very difficult or impossible to separate. In this case, the ratio of the quantity of the daughter ele­ments can be calculated as follows.

The sum of the quantities of the two daughter elements is equal to the quantity of the decomposed parent element at any time:

B1 + B2 = N0 — N = N0(1 — e-(A1+A2)t) (4.19)

The rate of the formation of the daughter elements is:

^ = A1N = A1N0 e-(A1+A2)t dt

^ = A2N = A2N0 e-(A1+A2)t dt

By integrating Eqs. (4.20) and (4.21) from t = 0 to 00:

we obtain:

, = A1

!1 = ‘

A2

B2 — AT+X N0

The ratio of Eqs. (4.24) and (4.25) is:

Therefore, in branching decay, the ratio of the quantities of the daughter ele­ments is equal to the ratio of the decay constants. By determining the quantities of the daughter elements, the ratio of the decay constants can be calculated.

Equations (4.20) and (4.21) have been integrated from t = 0 to oo, but the same results are obtained by the integration over any time interval.

Neutrons can be produced in different ways:

• In neutron sources.

• In neutron generators.

• In nuclear reactors.

• By nuclear spallation.

In neutron sources, neutrons are mostly produced by (a, n) nuclear reactions (as discussed in Section 6.2.3). The alpha particles are obtained from an alpha emitter radioactive isotope such as Ra-226, Pu-239, or Po-210. These isotopes are mixed with a light element (the binding energy of neutron is relatively low), mainly by beryllium. The neutrons are produced in the reaction as follows:

9Be(a, n)12C (5.100)

The neutron yield of these neutron sources is 106—108 neutrons/s.

The radium—beryllium (RaBe) neutron source has undesirably high gamma radi­ation, and therefore it is no longer used.

Neutrons can be produced by the spontaneous fission of 252Cf. The yield of the commercial 252Cf neutron sources is about 107—109 neutrons/s.

Neutrons can be produced by (Y, n) nuclear reactions (see Section 6.2.2). Gamma photons can initiate nuclear reactions if their energy is higher than the binding energy of the target nucleus. For example, the 24Na isotope has high-energy gamma photons. The gamma photons can initiate nuclear reactions with deuterium, lithium, beryllium, and boron. For example:

Be(Y, n)8Be (5.101)

2H 1 y! n 1 p1 (5.102)

Therefore, when a salt containing an 24Na isotope is dissolved in heavy water (D2O), a mobile neutron source can be produced (as described in Section 6.2.2).

In neutron generators, the isotopes of hydrogen are used in nuclear reactions. Mostly deuterium, tritium nuclei, or the mixture of these nuclei are accelerated in linear accelerators, and the metal hydride target containing deuterium, tritium, or both is bombarded by the accelerated nuclei. The nuclear reactions (described fur­ther in Section 6.2.4) are:

2H 13H! 4He 1 n (5.103)

2H 12H! 3He 1 n (5.104)

The energy of the neutrons produced in neutron generators is about 14 MeV. The yield of the neutron is about 108—109 neutrons/s.

Neutrons can be produced in cyclotrons by (p, n) nuclear reactions. For this reac­tion, lithium or beryllium is used as target material.

The neutron production in nuclear reactors will be discussed in detail in Section 6.2.1 and Chapter 7. Thus, it is not detailed here; we will just mention that in the fission reaction, high-energy gamma photons are also produced, which initi­ate the reaction (5.102). This reaction produces extra neutrons, which affects the neutron balance of the nuclear reactors.

The greatest neutron yields can be obtained by nuclear spallation. Spallation is a nuclear reaction in which photons or particles with high energy (e. g., protons with GeV) hit a nucleus, resulting in the emission of many other particles (such as neu­trons or light nuclei) or photons. The target is a heavy element (e. g., mercury, tung­sten, or lead). Recently, there are only a few spallation neutron sources all over the world.

The lifetime of free neutrons is short; they transform into protons, beta particles with 0.728 MeV, and antineutrinos.

0n mo, ! 1p 1 в0.782 MeV 1 0V (5Л°5)

10.25 min

The half-life of the reaction is 10.25 min.

8.1 History of Radioactive Tracer Methods

Radioactive tracing was discovered by George Hevesy in 1911. He was working in Rutherford’s laboratory, where radium was prepared from uranium ore by copreci­pitation with lead chloride. His goal was to separate RaD from lead chloride. At that time, the term “isotope” had not been defined; the decay series of uranium was described as in Figure 8.1. All radioactive isotopes were considered to be “new radioactive elements,” and correspondingly named after the parent element and the place of the given product in the decay series. Thus, RaD indicated the sixth ele­ment of the Ra decay series. As comparison, refer to the “modern” decay series of 238U shown in Figure 4.4.

He attempted to separate RaD by many chemical methods but did not succeed. The separation factor was found to be 1 in every method, so Hevesy concluded that RaD could be suitable for labeling lead. Nowadays, we already know that RaD is a radioactive isotope of lead (210Pb); that is, lead and RaD are chemically the same. Using RaD as a radioactive tracer, Hevesy, with F. Paneth, determined the solubility of lead salts (sulfide and chromate) which are very little; the solubility products are about 10-33 mol2 dm-6. In 1943, Hevesy received the Nobel Prize in Chemistry for “his work on the use of isotopes as tracers in the study of chemical processes.”

Similar to radioactive isotopes, stable isotopes can also be used as tracers. The determination of the stable isotopes, however, requires expensive instrumentation (nuclear magnetic resonance and mass spectrometers), while it is much simpler and cheaper to measure radioactive isotopes. In addition, the radioactive isotopes can be measured easily in very small quantities. Depending on the decay constants, as small quantities as 10-16—10-6g of the radioactive isotopes can be detected. The application of the radioactive tracers/indicators is independent of the physical and chemical properties (pressure, temperature, chemical species, etc.) because the energy of the nuclear radiation is 6— 8 orders of magnitude higher than the energy of the aforementioned physical and chemical effects. Since the radioactive isotopes are chemically the same as the studied inactive isotopes, they do not change the studied system. They can be applied in dry analytical methods. If the radioactive indicator is chemically pure, no contaminants are added to the investigated system.

Nuclear and Radiochemistry. DOI: http://dx. doi. org/10.1016/B978-0-12-391430-9.00008-1

© 2012 Elsevier Inc. All rights reserved.

The liquid-drop model is based on the constant density of nuclei, independent of the number and quality of nucleons. The phenomenon is analogous to a liquid drop in which the molecules are subjected to the same van der Waals forces, indepen­dent of the size of the drop.

According to the liquid-drop model, the nucleus can be imagined as a rather compact, spherical structure (similar to a liquid drop), the constituents of which are subjected to strong interactions acting in a very small range (about 10—15m). Really, this is the nuclear force, the energy of which is approximately proportional to the mass number (A).

When the binding energy of a nucleus is calculated using this model, the nuclear energy has to be taken into account first. Let’s suppose that the nuclear energy between two nucleons is U0. In the closest geometric packing of spheres, one nucleon has 12 neighbors (the coordination number is 12). It should mean —12U0, the total energy for one nucleon; however, each nucleon is considered twice (nucleon pairs are investigated), so only —12U0/2 = — 6U0 is the nuclear energy for one nucleon. For a nucleus with a mass number of A, the total nuclear energy is —6U0A. This energy is shown as volume energy in Figure 2.4.

Of course, the peripheral nucleons have only six neighbors, decreasing the nuclear energy. Considering the thickness of peripheral layer a, the volume of this layer is 4R2na (surface X thickness). Since the volume of one nucleus is 4R3n/3 and the number of nucleons is A, the volume of one nucleon is 4R3n/3A. The num­ber of nucleons in the peripheral layer can be obtained by dividing the volume of the layer with the volume of one nucleon: 3aA/R. Therefore, the surface energy (Es) of the nucleus can be expressed as:

The nuclear energy corrected by the surface energy can also be seen in Figure 2.4.

Since the protons repulse each other, the energy of repulsion has to be taken into account. The energy of the electrostatic repulsion can be expressed as in Eq. (2.11). So, the binding energy per nucleon has to be corrected with the electro­static repulsion too (Figure 2.4):

The calculated binding energy per nucleon (Eq. (2.17)) is not accurately equal to the experimental value. The differences can be explained by two things. The first is that each value of A has a value of Z for which there is maximum stability (see Eqs. (2.10) and (2.10a)). When the ratio of the protons and neutrons is different from the maximum stability, a so-called asymmetry energy must also be taken into account because of the slightly different interaction energies of proton—proton, proton—neutron, and neutron—neutron pairs. The second is that the nuclei with even—even proton and neutron numbers are more stable than nuclei with odd—odd, even—odd, or odd—even nuclei. For even—odd and odd—even nuclei, this effect is taken to be zero, and for even—even nuclei, it is a negative number (increasing sta­bility), whereas for odd—odd nuclei, it is a positive number (decreasing stability), and it will be discussed in Chapter 3.

The total semiempirical formula by Weizsacker for the binding energy per nucleon is as follows:

where

and

— = ±a5A3/4 (2.20)

A

and y and a5 are constants. As seen in Figure 2.4, the binding energies calculated by Eq. (2.18) agree well with the experimental values.