Measurement based on the principle of radiation absorption is mainly used to deter­mine the thickness of rolled metals. The intensity of the radiation penetrating a

Counter Recorder

Figure 11.21 Continuous level indication with point radiation sources and a linear detector.

material depends on the elemental composition, thickness, and density of the mate­rial placed in the path of the radiation. For thickness measurements, the material must have a permanent chemical composition and density.

The intensity change caused by a material layer is described by the following formula (see also Eq. (5.46)):

f = exp(-ME)0 (П.27)

10

where I0 is the intensity of radiation entering, I is the intensity of radiation leaving the material, p,(E) is the linear absorption coefficient, and I is the thickness of the material layer.

The upper part of Figure 11.22 shows a measuring arrangement based on radia­tion absorption, while the bottom part shows a measuring arrangement based on radiation reflection. The measuring technique based on reflection is mostly applied when access to both sides of the equipment is impossible due to the mechanical arrangement of the equipment.

Radiation sources used for thickness measurements include gamma — and beta — emitter nuclides. The radiation energy is selected to match the material density. As detectors, ionization chambers and proportional counters are applied.

To obtain the best sensitivity for the thickness measurements by absorption, optimal measuring conditions are applied by selecting the best radiation source (with given p(E)). The conditions of the optimazation measurements can be deduced from Eq. (5.48) as follows: where I0 is the intensity of radiation entering, I is the intensity of radiation leaving the substance, p, is the mass-absorption coefficient, p is the density, I is the thick­ness, and d is the surface density of the paper layer.

The optimal value of д is obtained using the relative measuring sensitivity (Q), which is defined as:

AI

(11.29)

d

For simplification, AI is expressed by the difference quotient of Eq. (11.28):

AI = I1 Ad (11.30)

and

AI = — fiI0 exp(——ud)Ad (11.31)

By substituting Eq. (11.31) into Eq. (11.29), we obtain:

^I0 exp(—ud)Ad

Q = A0 = ^d exp(—^d) (11:32)

d

The differential quotient of the relative measuring sensitivity Q, according to the mass-absorption coefficient p, is:

Q1 = = d exp(-ppl) — pd2 exp(-pd) (11.33)

dp

The function (Eq. (11.33)) has a maximum if Q = 0. From here, d exp(—ppl) = pd2 exp(-pd) (11.34)

From Eq. (11.34), we obtain:

p=d (11.35)

In conclusion, the sensitivity of the thickness measurement is the best if Eq. (11.35) is fulfilled.

For the thickness measurement by backscattering, a similar equation to Eq. (11.35) can be derived from Eq. (5.69).

Examples of applying industrial thickness-measuring systems include the following:

• Continuous thickness measurement on paper-manufacturing machines with beta-emitting radionuclides.

• Continuous thickness measurement of metal sheets on cold rolling machines.

• Thickness measurement of hot rolled steel sheets.

• Thickness measurement of surface layers deposited on thick basic sheets (coatings).

• Thickness measurement gauges on plate glass production lines.

• Thickness measurement of concrete at construction of containers.

The most common method of emission tomography in the world is myocardial perfusion SPECT. The reason is that cardiovascular diseases are the leading causes of death (ahead of tumors), especially in highly industrialized countries (WHO, 2004). It is crucial to identify the cases in which the plaques in coronary arteries cause such a serious stenosis that revascularization (either by angioplasty or by bypass graft surgery) is necessary.

The principle is that stenosed coronary arteries cannot dilate on demand (e. g., when the patient performs physical exercise or takes vasodilator medicine) as healthy coronary arteries do. During exercise, the relative perfusion of myo­cardial regions may be different from when at rest. If the myocardium evenly perfused in a resting state shows a relatively hypoperfused area after stress, it indicates ischemia (a relative shortage of the blood supply) (see Figure 12.11). The patients suffering from active ischemia are those whose cardiac pumping function will probably improve after revascularization, and who are at risk of further cardiac events if left untreated. For myocardial perfusion SPECT, we most frequently use Tc-99m-labeled methoxy isobutyl isonitrile (MIBI) or tetro — fosmin, and less frequently thallium chloride labeled by the potassium-analog Tl-201.

28Mg can be produced in 27Al(a,3p)28Mg or in 26Mg(t, p)28Mg nuclear reactions. Its half-life is 20.9 h, and it emits negative beta particles.

8.6.3 Aluminum-28

For the nuclear reactions and in the neutron-activation analysis (see Section 9.2.2.1) the target or the sample is placed into aluminum holders since irra­diation of aluminum with neutrons (27Al(n, Y)28Al) produces radioisotope with a short half-life (2.8 min). 28Al emits (3_ and gamma radiation. As a by-product, Na-24 is formed in the 27Al(n, a)24Na nuclear reaction. A negative conclusion of this reaction is the long time gamma radiation of the aluminum sample holders after irradiation in nuclear reactors.

Al-28 can be obtained from 28Mg/28Al generators too.

As mentioned previously, radioactive tracers can be used to determine the local, time, and concentration distribution of the substances, so they are appropriate to use for studying transport processes, including migration, diffusion, and self-diffusion. As discussed in Section 3.2, diffusion is a suitable means of separating isotopes, especially for the isotopes of the light elements. Even heavy isotopes, such as 235U and 238U, were successfully separated for military purposes by diffusion in the United States between 1945 and 1948.

In this chapter, the application of radioactive tracers will be discussed in some arbitrarily chosen solid/gas, solid/liquid, and solid/solid systems. In addition, self­diffusion will be investigated. In the diffusion studies, the application of radioactive isotopes is only one of the available methods, though it is one of the cheapest and most elegant. In the self-diffusion studies, however, the radiotracer method is the only simple possibility. In the diffusion studies, the mixing entropy is maximal; the process is directed by the concentration gradient. The specific activity is con­stant, meaning that the activity is proportional to the concentration. In the self­diffusion studies, the specific activity changes, and the process is directed by the increase in the mixing entropy.

During XRF, an electron is ejected from the K or L electron orbital of the elements to be analyzed. The vacancy is filled with an electron from an outer orbital. The energy difference between the two orbitals is emitted as a characteristic X-ray pho­ton. The energy of the X-ray photons relates to the elements, thus providing quali­tative analysis. The intensity of the X-ray photons provides quantitative analytical information.

The X-ray photons can be produced by the excitation with charged particles (electron microprobe, discussed in Section 10.2.4.2, and proton-induced X-ray emission, discussed in Section 10.2.5.1) or by electromagnetic radiation. Electromagnetic radiation is produced in an X-ray tube, or it may be the gamma radiation emitted by a radioactive isotope. In an X-ray tube, the photocathode emits X-ray radiation. During isotopic excitation, 55Fe (5.9 keV), 109Cd (22—25 keV), 125I (27—31 keV), and 241Am (60 keV gamma energy) are used as exciting sources.

The initial process of excitation with electromagnetic radiation is the photoelec­tric effect. The excitation takes place if the energy of the exciting particle exceeds the binding energy of the electron. The exciting photons transfer their energy to the orbital electron. The energy equal to the binding energy ejects the electron, and the residual part of the energy becomes the kinetic energy of the ejected electron:

Ek = hv0 — Eb (10.35)

where Ek is the kinetic energy of the emitted electron, Eb is the binding energy of the electron, and hv0 is the energy of the exciting photon before the photoelectric effect. This process occurs if the energy of the exciting photon is close to the bind­ing energy of the electron. This means that the electrons are ejected from the K and L orbitals.

As discussed previously, the excited state can be relaxed by the emission of characteristic X-ray photons. The wave number of the X-ray photons is expressed by Moseley’s law (Eq. (5.88)). The wave number (i. e., the energy of the character­istic X-ray photons) increases along with the increase of the atomic number, pro­viding information for qualitative analysis. As seen in Figure 4.12, the light elements practically do not emit characteristic X-ray photons, XRF is useful for the elements Z > 20. For the elements 10 < Z < 19, the X-ray photons are absorbed in air and thus can only be measured in a vacuum.

In Figure 10.16, the energy of characteristic X-ray photons emitted by the differ­ent elements is shown as a function of the atomic number. K and L mean the orbi­tals where the vacancies are formed under the excitation, and а, в, and y mean the outer orbitals from which the vacancy is filled. For instance, Ka means that a vacancy on the K orbital is filled with an electron from the next orbital, L.

The sensitivity of X-ray fluorescence spectrometry is influenced by two factors, both of which relate to excitation: the energy of the exciting photons should be higher than the binding energy but not by too much. The excitation is optimal if the energies exciting and the emitted photons are close, but not identical, so they can be separated spectroscopically in the given measuring system. For this reason, the excitation source and the element to be analyzed should be correlated. Practically, the “light” (Z > 20) elements are excited by low energy, and the K lines are measured. The heavy elements are analyzed using the L lines.

The X-ray fluorescence spectrum of a mixture of Fe2O3, ZnO, KBr, Sr(NO3)2, MoO3, AgNO3, CsNO3, and Nd2O3 is shown in Figure 10.17. The exciting source was the gamma radiation of the 241Am isotope, and the detector is SiLi semiconduc­tor detector (as discussed in Section 14.3). The concentrations of the elements in the mixture are Fe: 0.012523 mol/g; Zn: 0.01229 mol/g; K and Br: 0.008403 mol/g; Sr: 0.004725 mol/g; Mo: 0.006947 mol/g; Ag: 0.005887 mol/g; Cs: 0.005131 mol/g; and Nd: 0.005944 mol/g. Since the exciting energy is about 60 keV, the K lines of all elements are detected. As seen, the light elements (nitrogen and oxygen) do not have lines in the spectrum.

Energy (keV)

As in most spectroscopic methods, the intensity of the emitted X-ray photons provides quantitative analytical information. This intensity is expressed by the fol­lowing formula:

_d + *Si

1 _ e srn Фієд sin Ф2єй

MS,£0 1 Ms, i

sin Ф^ sin Ф2єГГ where Si is the sensitivity for the ith element expressed in the mass unit of the pure element; C, is the concentration of the ith element; d is the surface density of the sam­ple (g/cm2); ^S£o and ^s,; are the mass absorption coefficient of the sample for the exciting radiation and the characteristic X-ray photons of the ith element (cm2/g), respectively; and Ф1б(ї and Ф2б(ї are the angles of irradiation and detection related to the surface of the sample.

Equation (10.36) shows that the intensity versus concentration function should be linear. In most cases, the intensity of the characteristic X-ray and the concentra­tion are not in linear relation because the sample contains other elements (such as a
matrix) which are also excited. This effect is called the “matrix effect.” The matrix effect can influence the intensity—concentration relation in two ways:

1. The intensity is smaller than expected from linear intensity—concentration plot. This is the case if the mass absorption coefficient of the matrix is greater than the mass absorp­tion coefficient of the element to be analyzed. The mean atomic number (Eq. (5.74)) of the matrix exceeds the atomic number of the element.

2. The intensity is higher than expected from the linear intensity—concentration plot. This is the case if the mass absorption coefficient of the matrix is smaller than the mass absorp­tion coefficient of the element to be analyzed. The mean atomic number (Eq. (5.74)) of the matrix is less than the atomic number of the element.

Besides the matrix effect, the intensity—concentration plot is influenced by the so-called internal excitation effect. This means that the studied element is excited not only by the exciting radiation but also by the characteristic X-ray photons of elements with higher atomic numbers. As a result, the intensity increases.

In conclusion, we can say that the matrix effect is significant in XRF. This effect is corrected in different experimental and theoretical ways, as summarized in Table 10.6.

In Figure 10.18, the calibration curves for the X-ray fluorescence spectrum of iron is shown when the mean atomic number of the matrix is smaller (boric acid matrix) or larger (barium nitrate matrix), respectively, than the atomic number of the element in question (iron).

The X-ray fluorescence method is used for the direct analysis of samples without any chemical pretreatment, or after chemical preparations (e. g., separation and enrichment). As mentioned previously in this chapter, all elements from calcium to uranium can be analyzed, or, using a vacuum, even the elements that are heavier than sodium can be measured. The concentration range is from about 1 ppm to 100%.

The arrangement of an XRF is shown in Figure 10.19. The excitation source is a gamma emitter radioactive isotope. The characteristic X-ray photons induced in the sample are detected by a SiLi semiconductor detector (as described in Section 14.3).

The range (path length) of various types of radiation in body tissues primarily determines the areas of possible application. Body tissues are practically water equivalent at the gamma energies used for imaging (see Tables 12.1 and 12.2).

12.2.4.1 Selection of Radionuclides for Imaging

• Only electromagnetic radiation (gamma — or X-rays) can be detected from outside the patient’s body, as beta radiation (and alpha even more) is adsorbed in a few millimeters of body tissue at most (Table 12.1).

• Besides, gamma energy should be in the range of 60—500 keV. The majority of lower — energy photons will be attenuated inside the patient’s body, while higher-energy photons most likely fly through the detector without any interaction; the counting efficiency is low in both cases.

• An important aspect is the half-life of the radionuclide: several hours (or a few days in some cases) are preferred, so that radioactive material will disappear from the patient’s body shortly after the imaging is completed, thus limiting the radiation dose. If a shorter — lived radionuclide was used, a large proportion of the radioisotope would decay during the procedure of labeling the selected molecule, thus increasing the cost of production.

• If the radionuclide also emits alpha or beta radiation, they unnecessarily increase the patient’s radiation dose while not contributing to image formation.

Similar to that of the atmosphere and hydrosphere, the radioactivity of the litho­sphere originates from both natural and artificial sources. The main sources of natu­ral radioactivity are rocks; their radioactivity determines the radioactivity of the soils formed on the rocks. The radioactivity of rocks depends on their mineral and chemical composition and can be quite different. As a result, the radioactivity also depends on the geographic position. The mean radioactivity is higher in the Northern Hemisphere than the Southern one, and it is also higher in American continent than in Europe.

The most important natural radionuclides in rocks and soils are 40K and the members of the radioactive decay series. Thorium is accumulated in monazite because it has similar chemical properties as the lantanoid elements, which are present in significant quantities in monazite. The mean radioactivities of several isotopes present in rocks are listed in Table 13.2. The standard deviations are rather high due to the wide variety of rocks, which were used to measure the activities, and the varying activities of which lead to high uncertainty in the mean values.

As seen in Table 13.2, the radioactivities of 214Bi and 214Po, as well as 212Bi,

Pb, and Tl, are approximately the same, showing that they are in radioactive

214 214 222

equilibrium. Bi and Po are the daughter nuclides of Rn, which is the daughter nuclide of 226Ra (see Figure 4.4). However, the radioactivity of 226Ra is much higher, proving the emission of the intermediate member, 222Rn, into the atmosphere. The activities of the daughter nuclides of 232Th are approximately the same. In this case, a radioactive equilibrium exists because the half-life of 220Rn (55 s) is too short to escape from the soil.

It is important to note that besides the radioactive isotopes listed in Table 13.2, Pb and Po, the members of the U series, are also important because of the long half-life of 210Pb (21.6 years).

As seen in Section 9.3.2.2, the migration of the radioactive isotopes in the geo­logical formations (as porous solids), including the isotopes present in nuclear waste, is determined by hydrological processes. The migration rate of water pro­vides the upper limit for the migration rate of the water-soluble radioactive nuclide. This actual rate may be significantly lower when the radioactive isotopes can be sorbed on the surfaces of rocks and soils. The sorption is mostly influenced by the chemical species (mainly the charge) of the radioactive isotopes. On the basis of the chemical forms characteristic in geological systems, the radioactive isotopes can be classified as follows:

1. Cations (e. g., 134’137Cs+, 41Ca21, 90Sr21, 54Mn21, 55Fe31, 58’60Co2+, and 59’63Ni2+).

2. Uranium and transuranium elements (U, Np, Pu, and Am isotopes), basically (complex) cations or anions (e. g., 99mTc isotopes as pertechnetate TcO^, 14C isotope as carbonate CO|2, 36Cl", and 129I").

3. Neutral species (e. g., 3H isotope as water H2O, metallic 110mAg).

Migration takes place in the following geological formations: clay rocks (espe­cially bentonite), granitic rocks, soils, oxides, and other minerals (carbonates, sul­fates, etc.). Since the surface charge of the rocks and soil is usually negative under usual geological conditions (pH, redox conditions), cations usually adsorb on the geological formations, while anions do not. Cesium, then, can occupy a space in the crystal lattices; thus, the sorption becomes irreversible. Other cations adsorb reversibly. In the case of cations of transition metals and transuranium elements, the adsorption is affected by their hydrolytic products. Transuranium elements can form colloids. Cations, except for cesium, readily form stable complexes that increase migration rate. In addition, precipitation, redox processes, and microbial activity can also influence the sorption and, as a result, the migration rate.

Of course, the different migration rate of cations and anions cannot result in the unbalancing of the electric charges. The faster migration of anion is followed by the migration of inactive cations dissolved from the geological formations.

The neutral species are very different behavior. Two extreme cases are tritiated water migrating with natural water (the isotope effect, described in Chapter 3, can be ignored) and Ag-110m reduced to metallic silver, the migration rate of which is practically zero.

Under equilibrium conditions, the sorption of the radioactive isotopes can be characterized by the distribution coefficient. This is the ratio of the sorbed quantity (mol/g) and the equilibrium concentration of the solution (mol/dm3). In Table 13.3,

Table 13.3 Distribution Coefficients of Radioactive Ions on Rocks or Minerals (g/dm3) Rock/mineral 137Cs 45Ca 85Sr 226Ra 60Co 14C 99mTc 131I

the distribution coeffecients of radioactive ions on different rocks are listed. The higher values of the distribution coefficients mean the stronger sorption of the iso­tope on the given rock. The data in Table 13.3 illustrate well the differences in the sorption of cationic and anionic radioactive isotopes.

These isotopes are produced in (n, Y) nuclear reactions. They are of little importance.

8.6.15 Silver Isotopes

Ag-110 is produced from silver by the 109Ag(n, Y)110mAg!110Ag nuclear reaction and isomeric transition. The half-life of Ag-110 is 250 days, and it emits (3_ and gamma radiation. 110Ag is an important polluting isotope of nuclear reactors since the silver in solders is activated.

Ag-111: a carrier-free 111Ag isotope is obtained by the 110Pd(n, Y)111Pd nuclear reaction and the subsequent в -decay. Pd and Ag are separated by the electrolysis of the amine complexes. The half-life of 111Ag is 7.45 days, and it has в_ and gamma radiation.

The rate-determining step of the heterogenous isotope exchange is very frequently the transport of the substances from the bulk to the interface. The transport means the convection, the mixing, and the diffusion of the dissolved substance from the solution phase to the interface, i. e., the reaction zone. The transport in the solution phase has two steps: the movement of the dissolved substances in the bulk solution and through a so-called adhesion layer. The convection and mixing influences only the transport in the bulk phases; in the adhesion layer, only diffusion (called “film diffusion” in this case) is possible, governed by the concentration gradient through the adhesion layer. Mixing decreases the thickness of the adhesion layer.

Since the concentration of the radioactive nuclide in radiotracer experiments is frequently very low, the diffusion plays an important role. In the adhesion layer,
the diffusion can be described by Fick’s first law, assuming that the concentration gradient is constant through the adhesion layer:

dc dc

= D

dt dx

In this case, the net kinetics of the isotope exchange is of the first order.

9.3.3.1 The Empirical Equation of the Heterogeneous Isotope Exchange

The heterogeneous isotope exchange in solid/liquid systems cannot be described by the usual, theoretically correct kinetic equation. For the interfacial reaction of very insoluble salts, an empirical equation was derived by Imre in 1933:

where xt and xN are the relative amount of the radioactive isotope on the solid sur­face at time t and in equilibrium, respectively; kj, k2, k3 are rate constants; and Ab A2, and A3 are empirical coefficients. The members in Eq. (9.112) refer to simulta­neous first-order reactions. However, the equation is formal; thus, it gives no infor­mation on the mechanism of the net reaction. In the literature, there are many frequently speculative interpretations for the rate constants and the empirical coef­ficients. These interpretations include speculation about the rate-determining step. Similar equations are used for the kinetic description of the interfacial processes of the crystalline powder/solution or the metal/solution. The experimental data are interpreted by including additional members to the kinetic equation.

Many heterogeneous isotope exchange processes between metal and the ions of the same element were studied to determine the exchange rate in equilibrium. The Ag/Ag1 system was intensively studied. The kinetics of the exchange process was interpreted as follows:

and

___ VoVj 1 VdVj 1 VoVd

T xN

VoVjVd

where v0 is the rate of surface diffusion, Vj is the rate of the electron exchange, and vd means the rate of diffusion in the solution. Equations (9.113) and (9.114) take into consideration the steps and mechanism of the heterogeneous isotope exchange on the interface of the metal/electrolyte solution. The t versus concentration func­tion gives information on the rate-determining step and the mechanism of the isotope exchange.

The kinetics of the heterogeneous isotope exchange was described by empirical equations that take into account the heterogeneity of the surface. For example:

xt = btn (9.115a)

where b and n are empirical values. These values are interpreted by the energy distribution of the exchange sites of the surface.

The main conclusions of the heterogeneous isotope exchange studies on metal surfaces are as follows:

• In equilibrium between the metal and the solution phase, continuous isotope exchange takes place.

• The rate of the electron exchange (exchange current) can be measured only in special cases; namely, when the rate-determining step of the surface reaction is the electron exchange, such as in the Fe/Fe21 system.

Radioisotopes used for radiotracer labeling are present in micromasses (as low as 1 X 1016g), but they have strong radiation that can be sensitively and rapidly detected. They behave identically with the labeled material during investigation without modifying its characteristics. Radiation is detected outside the equipment or pipes, so sampling or installing instruments into the material flow can be avoided.

The basic requirement for labeling with a radiotracer is that the radiotracer always has to follow the tested material proportionally during its flow; in other words, the mass rate of the tracer has to be identical to the mass rate of the tested material in each phase of the flow.

where x, X, x" are the number of atoms of the tested material in an S, Z, Z" system, y, y, y" are the number of radioactive tracer atoms in an S, Z, Z" system, A is the type of atoms of the tested material (nonradioactive), and A* is the type of atoms of the tracer (radioactive).

While an ideal tracer for every material would be its radioactive isotope, not every element has radioactive isotopes with favorable measuring characteristics. For this reason, a generally accepted and applied method is the incorporation of the radioactive atom into the tracer molecule. However, such chemical labeling is nec­essary only if tested material goes through chemical reaction or phase modification. When studying physical behavior (e. g., flow investigations), the so-called physical labeling is acceptable where the behavior of the material is affected not by chemi­cal but by physical characteristics.

However, a tracer must follow the tested material even in the case of physical labeling (e. g., a salt as tracer dissolved in water to be followed). An extreme case for physical labeling is when a radioactive colloid is bound on the surface of grained materials. In such cases, attention must be paid to the fact that radioactive concentration of fractions will be dependent on the grain size (this is called “sur­face labeling”). Only fractions with the same grain size can be labeled homogeneously.

The relationship between activity of the labeling isotope and the measurable count rate is expressed as follows:

Zrei = 2.22 X 106 Aan (count/min) (11.4)

where /rel is the measured count rate, A is the activity of the labeling radioisotope (цО), a is the number of gamma counts per decay, and n is the total measuring efficiency (n = fєG), in which f is the self-absorption efficiency, є is the counting efficiency, and G is the space angle factor.

For tracer studies, when calculating the minimal required activity of the labeling radioisotope, dilution rate in the industrial equipment and measuring accuracy also must be taken into consideration. While the former is expressed by a simple multi­plication factor, the latter can be deducted from the expected value and standard deviation of the Poisson distribution, which is applied to radioactive decay. Based on these, 1% measuring accuracy needs 10,000 counts, 0.3% accuracy needs 100,000 counts, and 0.1% accuracy needs 1,000,000 counts to be detected (see Section 14.7.1).