Theory of the implementation of the fundamental parameter method for

ISSN 1061-9348, Journal of Analytical Chemistry, 2006, Vol. 61, No. 7, pp. 654–661. © Pleiades Publishing, Inc., 2006. Original Russian Text © G.V. Pavlinsky, A.Yu. Dukhanin, E.O. Baranov, A.Yu. Portnoy, 2006, published in Zhurnal Analiticheskoi Khimii, 2006, Vol. 61, No. 7, pp. 710–717.

ARTICLES

Theory of the Implementation of the Fundamental Parameter Method for the X-ray Fluorescence Determination of Low-Atomic-Number Elements G. V. Pavlinsky, A. Yu. Dukhanin, E. O. Baranov, and A. Yu. Portnoy Research Institute of Applied Physics, Irkutsk State University, bul’v. Gagarina 20, Irkutsk, 664003 Russia Received January 28, 2005; in final form, July 8, 2005

Abstract—The theory of X-ray fluorescence generation in elements with a low atomic number Z is extended to the case of host matrices with high Z. The total contribution of the selective excitation effects is estimated with regard to the cascade K–L transitions. The influence of the elemental composition of the matrix and the excitation conditions on the X-ray fluorescence intensity of the elements in study is assessed. The accuracy of the model of X-ray fluorescence generation in the low-atomic-number elements is confirmed by the agreement of the experimental and calculated intensities of carbon in various carbon-containing compounds. DOI: 10.1134/S1061934806070094

The fundamental parameter method is the most versatile among the methods for the multicomponent X-ray fluorescence analysis of homogeneous substances and materials. It is based on the state-of-the-art theory of X-ray fluorescence generation. This theory assumes only the photoelectric ionization of atoms, which is true for most situations that take place in analysis. However, in the generation of the X-ray fluorescence signal of elements with a low atomic number (boron, carbon, nitrogen, oxygen, or fluorine), the photoelectric ionization of atoms is not the only process. Indeed, the long-wavelength primary radiation of X-ray tubes, which is most efficient for the direct ionization of atoms of these elements, is almost completely absorbed by the beryllium window of the tube. In these conditions, other processes that are negligible in the shortwavelength region of the X-ray spectrum become important. First of all, this is the ionizing effect of photo- and Auger electrons [1, 2] occurring in the irradiated sample. The excitation of atoms by a flux of photo- and Auger electrons is ignored by the current versions of the fundamental parameter method. At the same time, it is known [3] that the contribution of photoelectrons to the generation of the X-ray fluorescence of sodium in rocks can reach 10% with a 0.3-mm-thick window. The present-day theories take into account the ionizing effect of only K and L photoelectrons and K Auger electrons. The ionization of atoms with low Z by photo- and Auger electrons from shells more distant from the nucleus is not taken into account. Another unexplored process in the formation of the X-ray fluorescence intensity of low-atomic-number elements is the selective excitation of their atoms by the radiation of other elements present in the irradiated material. This effect can manifest itself in three ways. These are the ionization of atoms by the K or L radia-

tion of the exciting elements or by the L radiation of these elements arising from intraatomic cascade transitions. The possible effect of cascade transitions on the fluorescence intensity of elements with low Z is mentioned in [4, 5]; however, these papers give no estimates, not even qualitative, of the contribution of the latter process. Cascade transitions transform the primary photons into low-energy photons, which excite the X-ray fluorescence of elements with low Z much more efficiently. Thus, the energy of the L radiation of iron (~0.7 keV) is much closer to the binding energy of K electrons in carbon, nitrogen, oxygen, and fluorine atoms than the energy of the K radiation of iron (~6.4 keV). However, the coefficient of K–L transformation in the cascade transitions governed by the ωL value of the X-ray L fluorescence yield is relatively small (ωL ≈ 0.6% for iron). Therefore, it is hard to assess the importance of this method of selective excitation without calculations. At the same time, it is quite clear that, as the atomic number of the element to be excited increases, the energy of its K absorption edge becomes closer to the energy of iron L radiation and the role of L-selective excitation in the generation of the K fluorescence of elements under study increases. If the energy of the primary photons exceeds the binding energy of electrons in the K shell, then, according to [6], cascade transitions make the major contribution to the generation of L radiation. This contribution drastically decreases when part of the inhomogeneous primary radiation falls between the absorption edges of K and L shells. The relationship between the considered parts of the primary radiation strongly depends on the anode material and the window thickness of the X-ray tube.

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The goal of this paper is to develop the theory of the fundamental parameter method taking into account the specific features of the analytical signal generation of the low atomic-number elements and to verify experimentally the proposed algorithm for calculating their X-ray fluorescence intensities. THEORY The fundamental parameter method is based on attaining the equality of the experimental and calculated relative intensities of X-ray fluorescence through an iterative process. The inaccuracy in the calculation of the analytical signal intensity is the main source of errors in this method. The X-ray fluorescence intensity NΑ of element A for any process causing the ionization of the inner atomic shells in an infinitely extended sample under polychromatic primary radiation can be calculated from the equation [7] λq

N ( λ )θ A ( λ )dλ -, N A = K A --------------------------------µ' ( λ )

∫

(1)

2. The probability θ A (λ) in the ionization of atoms of element A by photoelectrons of the jth element is Ph

θ A ( λ ) = ω A p Aα Ph

1. The probability θ A (λ) in the direct (photoelectric) ionization of the K shell is dir

θ A ( λ ) = τ A ( λ )ω A p Aα c A , dir

(2)

where τA(λ) is the partial coefficient of the photoelectric absorption of the primary radiation by the K shell of element A; ωA is the X-ray fluorescence yield for the K level of this element; pAα is the emission probability for the α line of the K series of this element; and cA is its content in the sample. JOURNAL OF ANALYTICAL CHEMISTRY

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∑c τ

j qj ( λ )n qj ( c A,

E ),

(3)

j, q

where nq j (cA, E) is the probability that atoms of element A will be ionized by the photoelectron from the q shell (subshell) of the element j with the energy E. Summation in (3) is over all q shells of all j elements whose photoelectrons can ionize the atomic q shell of element A. 3. The probability θ A in the ionization of atoms of element A by Auger electrons. Here, the following should be taken into account: (1) the number of Auger electrons is (1 – ωq) times less than the number of photoelectrons; (2) their energy does not depend on the energy of the primary radiation photons and is governed by the energy difference between the Auger transition levels. Hence, this probability is found from the expression AuL

θA (λ) AuL

λ0

where KA is a factor depending on the irradiated area of the fluorescent sample, on the spectrometer geometry, and on the detection conditions for the radiation of element A; N(λ) is the spectral intensity of the primary X-ray radiation; θA(λ) is the probability of transformation of the primary photon into the fluorescent photon µ(λA) µ(λ) - ; µ(λ) and µ(λA) of element A; µ'(λ) = ----------- + ------------sin ϕ sin ψ are the mass attenuation coefficients of the primary radiation with the wavelength λ and the fluorescent radiation with the wavelength λA, respectively; and ϕ and ψ are the angle of incidence of the primary radiation and the take-off angle of the fluorescent radiation, respectively. The total intensity of the X-ray fluorescence of element A is governed by the sum of expressions (1) over all the possible ionization processes for the atoms of this element. The probability θA(λ) of different transformation processes of the primary photon into the fluorescent one is found as follows.

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= ω A p Aα

∑c τ

j qj ( λ ) ( 1

– ω qj )

j, q

∑r

qjm n ( c A,

E qjm ),

(4)

m

where rqjm is the probability of the Auger transition and Eqjm is the energy of the electron emerging in this Auger transition. Summation in (4) is over the elements j and their shells q whose Auger electron energy is sufficient to ionize the K shells of element A. 4. The probability θ A (λ) in the selective excitation of atoms of element A is found [8] from the expression Enh

θ A ( λ ) = P 1 LP 2 . Enh

(5)

The value L in (5) is defined by the expression 1 sin ϕ µ(λ) L = --- ----------- ln ⎛ 1 + -------------------------⎞ 2 µ(λ) ⎝ µ ( λ j ) sin ϕ⎠ µ ( λi ) ⎞ sin ψ + ------------- ln ⎛ 1 + ------------------------- , µ ( λi ) ⎝ µ ( λ j ) sin ψ⎠ where µ(λj ) is the mass attenuation coefficient for radiation with the wavelength λj for the K or L fluorescence of the element j; P1 and P2 are the probabilities of the transformation of the primary photon into the photon of the element j and the transformation of the photon of the element j into the photon of element A, respectively. The following holds for the ionization of atoms of element A by K radiation of the element j: P 1 = τ Kj ( λ )c j ω Kj p Kjm ;

P 2 = τ KA ( λ Kj )c A ω KA p Kin ;

for the ionization by L radiation,

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P 1 = τ Lj ( λ )c j ω Lj p Ljm ; 2006

P 2 = τ KA ( λ Lj )c A ω KA p Kin .

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Relative intensity 6 5

L Auger M photo

4

L photo

3 2 1 0

K Auger K photo

N Auger M Auger

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Atomic number Z

Fig. 1. Relative intensity of carbon excited by all types of photoelectrons and Auger electrons as a function of the host atomic number ZH. The primary radiation is generated by the rhodium tube with a 75-µm-thick beryllium window operated at 40 kV.

The following holds for the ionization by L radiation resulting from the cascade K–L transition: P 1 = τ Kj ( λ )c j ( 0.9 – 0.1ω Kj )ω Lj p Ljm ; P 2 = τ KA ( λ Lj )c A ω KA p Kin . We used the above equations to calculate the fluorescence intensity and the contributions of the individual processes to the analytical signal generation of the low-atomic-number elements. RESULTS AND DISCUSSION The analytical signal intensities are calculated for the primary radiation of X-ray tubes with grounded cathodes [9, 10] used in present-day X-ray spectrometers. It is convenient to assess the importance of the individual processes considered above as a function of the atomic number ZH of the host matrix at a low concentration of any of the elements of interest in it. The absolute intensities of the components related to the intensity of one of them are taken as these functions. The components of the X-ray fluorescence intensity of carbon caused by the excitation of its atoms by photoelectrons and Auger electrons from various shells of the elements of the host matrix are shown in Fig. 1. The samples were 0.1% C + 99.9% of the element with the atomic number ZH. All the intensities are given relative to the intensity of carbon directly excited by the primary radiation. It follows from Fig. 1 that the excitation of carbon atoms by photo- and Auger electrons substantially exceeds their direct excitation by the primary radiation photons. Therefore, the use of the existing version of the fundamental parameter method without changing the calculation algorithm for the X-ray fluorescence

intensities of the low atomic-number elements is unacceptable. It also follows from Fig. 1 that a change in the atomic number of the host matrix causes substantial redistribution of the roles of different processes in the fluorescence excitation of carbon. The rather sharp observed maxima of the influence of photoelectrons are attributed to the presence of the characteristic Rh L radiation in the inhomogeneous primary spectrum. As the atomic number ZH of the host matrix element increases, the probability of ionization of its atoms by this characteristic radiation increases proportionally to the photoelectric absorption coefficient, and the number of photoelectrons increases accordingly. In this case, the energy of these photoelectrons and, hence, the count of ionizations caused by one electron decreases. The former factor prevails at low ZH, whereas the latter factor prevails at high ZH. The role of photoelectrons approaches zero when their energy is close to the binding energy of the electrons on a certain q shell. (Broad peaks for the photoelectrons can be explained in the same way as the deceleration component of the primary radiation.) The sharp maxima in Fig. 1 for the influence of Auger electrons are also caused by the characteristic Rh L primary radiation. As ZH increases, both the number and energy of these electrons increase while the generating level can be ionized by this characteristic radiation. As ZH increases further, the Auger electrons of this series do not appear, and their role drops to zero immediately. When the primary radiation is of a deceleration nature, this jump in the contribution of the Auger electrons is naturally absent. The contributions of different types of photo- and Auger electrons to the total X-ray fluorescence intensity of carbon under the primary X-ray radiation of a rhodium tube with a 75-µm-thick window are given in Table 1. One can see from Table 1 that, for carbon, the total contribution of photo- and Auger electrons approaches 90% for most host matrix elements. It also follows from this table that, if only elements with the atomic number ZH < 25 are bound to carbon, only K and L photo- and K Auger electrons can be taken into account in calculating their X-ray fluorescence intensity. The contribution ratio between different types of photo- and Auger electrons substantially depends on the spectral distribution of the primary radiation (which is governed by the anode material, the voltage across the X-ray tube, and the thickness of its beryllium window). The question arises as to whether it is possible to find conditions in which the role of the photo- and Auger electrons in the fluorescence excitation of elements with low Z is small enough to allow the use of the existing versions of the fundamental parameter method when elements with low Z are present in the sample.

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Table 1. Contribution of photo- and Auger electrons to the generation of X-ray fluorescence of carbon under the primary radiation of a rhodium tube with a 75-µm-thick beryllium window Z of the host K photo, % 5 11 17 23 29 35 41 48 53 59 65 71 77 82

9.71 38.59 12.89 14.09 6.95 2.23 0.77 0.19 0.00 0.00 0.00 0.00 0.00 0.00

L photo, % M photo, % N photo, % K Auger, % L Auger, % M Auger, % 0.00 4.61 12.11 39.33 51.63 39.73 15.79 18.49 14.62 12.28 10.32 8.40 6.09 4.41

0.00 0.00 0.00 0.00 0.00 15.70 22.02 44.44 49.38 49.81 41.28 27.49 22.63 20.37

0.00 0.00 0.00 0.00 0.00 0.00 0.00 8.72 11.60 15.99 22.01 27.85 25.92 22.42

First, according to [11], let us examine the possibility of decreasing the thickness of the beryllium window of the X-ray tube to amplify the long-wavelength component of the primary radiation and, thus, to increase the role of direct photoelectric excitation of the lowatomic-number element atoms. Figure 2 shows how the thickness of the beryllium window of the X-ray tube affects the contribution of direct ionization of carbon atoms by the primary radiation of the tube with a rhodium anode. It follows from Fig. 2 that this contribution becomes governing only when the window thickness is no more than 5–7 µm. At a thickness of more than 50 µm, the contribution of direct ionization of carbon atoms becomes about 20–30%. That is, in present-day X-ray tubes with a thin beryllium window, the main contribution to the X-ray fluorescence excitation of carbon is still made by photo- and Auger electrons emerging in the irradiated material.

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0.00 0.00 0.00 0.82 13.09 28.97 49.80 14.49 12.55 10.79 9.39 7.87 5.89 3.77

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5.49 16.27 27.18 37.84

9.71 58.20 75.53 72.80 80.25 88.66 89.48 86.48 88.16 88.87 88.49 87.89 87.71 88.81

to 5 kV, the intensity decreases approximately 12 times). Therefore, the effect of photo- and Auger electrons on the generation of the analytical signal of elements with low Z cannot be eliminated by choosing the excitation conditions. This effect should be taken into account to ensure the validity of the fundamental parameter method. The role of photo- and Auger electrons in generating X-ray fluorescence of carbon, nitrogen, oxygen, and fluorine calculated in [1] is shown in Fig. 3.

The other way is to decrease the energy of photoelectrons and, therefore, their ionizing power by decreasing the high voltage across the X-ray tube. Table 2 shows how the voltage affects the intensity of the small amount of carbon in the iron matrix. It follows from Table 2 that, as the voltage across X-ray tube decreases, the contribution of photo- and Auger electrons to the total fluorescence intensity of carbon slightly decreases. However, their effect cannot be eliminated completely, because the energy of L photoelectrons remains relatively high. At the same time, decreasing the voltage across the tube below 5 kV is hardly advisable, because the analytical signal intensity of carbon noticeably decreases (as the voltage decreases from 40 JOURNAL OF ANALYTICAL CHEMISTRY

0.00 15.01 50.53 18.56 8.58 2.03 1.10 0.15 0.00 0.00 0.00 0.00 0.00 0.00

Total contribution, %

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Idirect /I 1.0 0.8 0.6 0.4 0.2 0

50

100

150

200

250

300 dbc, µm

Fig. 2. Contribution of the photoelectric excitation of carbon atoms to its total intensity as a function of the thickness of beryllium window of the X-ray tube. Carbon content is 0.1% in silicon matrix; rhodium anode operated at 40 kV. 2006

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Table 2. Contribution of different components of the total X-ray fluorescence intensity of the small amount of carbon in the iron matrix (0.1% of carbon and 99.9% of iron) as a function of the voltage across the X-ray tube (Rh anode, thickness of Be window 75 µm) Their contribution (%) at Excitation processes Directly Selectively Effect of K photoelectrons Effect of L photoelectrons Effect of Auger electrons Photo + Auger electrons Total intensity

5 kV

10 kV

20 kV

30 kV

40 kV

40 9 ~0 51 0 51 100

33 7 2 51 9 60 100

30 5 6 49 10 65 100

27 5 8 49 11 68 100

24 4 12 46 14 72 100

One can see from Fig. 3 that the contribution of photo- and Auger electrons to the total X-ray fluorescence intensity of a small amount of a fluorescent element decreases eight times or more as its atomic number increases from Z = 5 (boron) to Z = 9 (fluorine). One should note that all the above results deal with the contribution of photo- and Auger electrons at a low content of the elements of interest in the host matrix. This contribution decreases as the content of element A increases and becomes minimum for the single-element samples. For these samples, direct ionization of atoms by primary radiation is amplified only by their ionization by their own photoelectrons. In this case, the contribution of photoelectrons (X-ray tube with rhodium anode and 75-µm-thick window, 40 kV) becomes small Rc 100 90 80 70 60 50 40 RN 60 50 40 30 20 10 0

Boron

Nitrogen

5 11 17 23 29 35 41 47 Zr

Rb 80 70 60 50 40 30 20 RO 50 40 30 20 10 0

and is 17.7% for carbon, 9.0% for nitrogen, 4.8% for oxygen, and 2.5% for fluorine. Let us consider selective excitation effects. Table 3 shows the contributions of each of the three modalities of this effect (see introduction to this paper) to the X-ray fluorescence intensity of carbon, nitrogen, oxygen, and fluorine at their low content in the iron matrix (99.9% iron and 0.1% of the element under study). It follows from Table 3 that the selective excitation of atoms by iron L radiation is the most important among the three modalities of the selective effect. In this case, the contribution of K–L cascade transitions to the L radiation is not so important. The total contribution of the selective excitation effect grows rapidly with the atomic number of element A (from several percent for carbon to 22.6% for fluorine). For the host matrices

Carbon

RF 20 Oxygen

Fluorine 10

5 11 17 23 29 35 41 47 Zr

0

5 11 17 23 29 35 41 47 Zr

Fig. 3. Total contribution of photo- and Auger electrons of various single-element matrices with atomic number Zr to the X-ray fluorescence intensity of light elements (mass fraction 0.1%). The excitation conditions were the following: molybdenum anode, 0.3-mm thick beryllium window, 40 kV, ϕ = 50°, and ψ = 30°. JOURNAL OF ANALYTICAL CHEMISTRY

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Table 3. Contribution of the individual modalities of selective excitation to the generation of the analytical signal of 0.1% of low-atomic-number elements in various host matrices. The X-ray tube with a tungsten anode and a 75-µm-thick beryllium window operated at 40 kV Contribution of the individual processes (%) Matrix

selective excitation direct excitation

K-

Fe Ni Zn

21 20 21

0.3 0.2 0.1

Fe Ni Zn

43 42 47

0.7 0.5 0.2

Fe Ni Zn

58 58 64

1.2 0.9 0.4

Fe Ni Zn

65 64 70

1.8 1.3 0.6

LCarbon 4 3.9 3.3 Nitrogen 8.7 9.5 8.5 Oxygen 14 15 14 Fluorine 19 20 18

considered (iron, nickel, and zinc), the total contribution of the selective excitation process to the X-ray fluorescence intensity of low-atomic-number elements changes only slightly. The dependence of the effects of selective excitation of low-atomic-number elements on the material of the X-ray tube anode is shown in Table 4. It follows from Table 4 that the main contribution to the selective excitation is made by L radiation of iron atoms in all cases. The contribution of the other two components of the selective excitation effect is governed by the location of the characteristic lines of the primary radiation. Strong characteristic Cu K radiation efficiently excites the K shell of iron atoms. Therefore, the contribution of iron K and K–L radiation to the selective excitation of atoms of elements with low Z is relatively high and is approximately 30% of the total selective effect, as one can see from Table 4. Characteristic W L radiation efficiently excites the K shell, but, at the same time, W M radiation excites only the L shell of iron atoms. Therefore, the contribution of iron K and K–L radiation to the selective excitation of atoms of elements with low Z decreases to ~15%. Characteristic L radiation of Rh anode does not excite the iron K shell. Therefore, the contribution of K and K–L radiation in the selective excitation is only 4– 7%. JOURNAL OF ANALYTICAL CHEMISTRY

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K– L-

Σ

excitation by photo- and Auger electrons

0.4 0.4 0.2

4.7 4.5 3.6

74.3 75.5 75.4

0.8 0.9 0.4

10.2 10.9 9.1

46.8 47.1 43.9

1.3 1.5 0.6

16.5 17.4 15

25.5 24.6 21.0

1.8 2.1 0.9

22.6 23.4 19.5

12.4 12.6 10.5

Therefore, the total contribution of the selective excitation effects (see Tables 3 and 4) is rather high, and all three components of this process should be taken into account to ensure the validity of the fundamental parameter method in this case. The validity of the proposed algorithm of forming the X-ray fluorescence intensity of low atomic-number elements is verified by comparing the calculated and measured intensities of carbon in various matrices. The experiment was performed using an ARL-9800 X-ray spectrometer with a 3GM X-ray tube with a grounded cathode. The anode material was rhodium. The thickness of the beryllium window was 75 µm. The crystal analyzer was AX-16. The spectrometer operated at 40 kV and 20 mA. We used a series of carbon-containing compounds whose composition was chosen in such a way as to ensure as wide a variation of the contribution of photoand Auger electrons to the excitation of carbon atoms as possible. Among the materials under study was graphite. In graphite, the above-mentioned electrons make the lowest contribution to the fluorescence excitation of carbon. Mixtures of carbon-containing compounds with other substances were not used in order to minimize the effect of the particle size of the irradiated material on the measured intensity of the analytical line of carbon.

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Table 4. Contribution of the selective excitation of atoms of low-atomic-number elements to the generation of their X-ray fluorescence for different X-ray tube anodes. The content of these elements in iron matrix is 0.1%. Beryllium window thickness is 75 µm; the voltage across the tube is 40 kV Contribution of the individual processes (%) Anode material

selective excitation direct excitation

Fe K-

Cu W Rh

6.2 21 25

0.5 0.3 0.1

Cu W Rh

17 43 51

1.4 0.7 0.3

Cu W Rh

30 58 69

3 1.2 0.4

Cu W Rh

41 65 76

5 1.8 0.6

Fe LCarbon 2.4 4 4 Nitrogen 7 8.7 8 Oxygen 14 14 12 Fluorine 21 19 15

Experimental and calculated relative X-ray fluorescence intensities of carbon in its various compounds are given in Table 5. The intensities are related to the analytical line intensity of carbon in graphite. The samples Table 5. Experimental and calculated intensities of carbon in various compounds relative to its intensity in graphite Calculation Compound

Experiment

Li2CO3 NaHCO3 Na2CO3 MgCO3 K2CO3 CaCO3 K3Fe(CN)6 K4Fe(CN)6 Y2(CO3)3 · 3H2O La(CH2COO)3 Tm2(CO3)3 · 3H2O Pb(CH3COO)2

0.088 ± 0.004 0.071 ± 0.004 0.049 ± 0.004 0.050 ± 0.004 0.074 ± 0.006 0.088 ± 0.006 0.140 ± 0.006 0.121 ± 0.006 0.051 ± 0.006 0.274 ± 0.006 0.063 ± 0.006 0.286 ± 0.006

with without photo- and photo- and Auger Auger electrons electrons 0.087 0.069 0.044 0.051 0.077 0.085 0.139 0.120 0.054 0.286 0.072 0.298

0.080 0.041 0.029 0.035 0.044 0.054 0.084 0.073 0.019 0.093 0.015 0.034

Fe K–Fe L

Σ

excitation by photo- and Auger electrons

0.6 0.4 0.1

3.8 4.7 4.2

90 74.3 70.8

1.7 0.8 0.3

10.1 10.2 8.6

72.9 46.8 40.4

3.5 1.3 0.4

20.5 16.5 12.8

49.5 25.5 18.2

5.1 1.8 0.5

31.1 22.6 16.1

27.9 12.4 7.9

were prepared by compressing the material to be analyzed into the boric acid cell. The table also gives the values of the relative intensities calculated without taking into account ionization of carbon atoms by photoand Auger electrons. One can see from Table 5 that the experimental data on the X-ray fluorescence intensity of carbon in its various compounds agree well with the intensities calculated taking into account the effect of photo- and Auger electrons and hardly resemble the intensities calculated without this effect. These discrepancies are extremely large for hosts containing high-atomic-number elements (Y, La, Tm, or Pb), where the ionization of atoms by photo- and Auger electrons from the shells distant from the nucleus is governing. Thus, neglecting the effect of photo- and Auger electrons in calculating the intensity of carbon in Pb(CH3COO)2 gives a result differing by eight times from the experimental value. CONCLUSION Our study demonstrated that generation of longwavelength X-ray fluorescence is a rather complicated process that goes beyond the pure photoelectric effects taken into account by the existing versions of the fundamental parameter method. The achieved agreement of the calculated and experimental X-ray fluorescence intensities of carbon indicates sufficient validity of the

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proposed theory of the method in the case of its implementation for X-ray fluorescence analysis of homogeneous materials containing low-atomic-number elements. REFERENCES 1. Pavlinsky, G.V. and Dukhanin, A.Ju., X-ray Spectrom., 1994, vol. 23, p. 221. 2. Pavlinsky, G.V. and Dukhanin, A.Ju., X-ray Spectrom., 1995, vol. 24, p. 293. 3. Borkhodoev, V.Ya., Rentgenofluorestsentnyi analiz gornykh porod sposobom fundamental’nykh parametrov (X-ray Fluorescence Analysis of Hard Rocks by the Fundamental Parameter Method), Magadan: Ross. Akad. Nauk, 1999.

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4. Mantler, M., Adv. X-ray Anal., 1996, vol. 40, p. 625. 5. Weber, F.A., Da Silva, L.B., Barbee, T.W., Ciarlo, D., and Mantler, M., Adv. X-ray Anal., 1997, vol. 39, p. 821. 6. Pavlinskii, G.V., Kitov, B.I., and Tyumentsev, V.N., Appar. Metody Rentgenovskogo Anal., 1987, vol. 36, p. 49. 7. Pavlinskii, G.V., Appar. Metody Rentgenovskogo Anal., 1992, vol. 41, p. 83. 8. Sherman, J., Spectrochim. Acta, 1955, vol. 7, no. 5, p. 283. 9. Pavlinsky, G.V. and Portonoy, A.Yu., Radiat. Phys. Chem., 2001, vol. 62, nos. 2–3, p. 207. 10. Pavlinsky, G.V. and Portnoy, A.Yu., X-ray Spectrom., 2002, vol. 31, no. 3, p. 247. 11. Dukhanin, A.Yu., Pavlinskii, G.V., Portnoi, A.Yu., and Kyun, A.V., Analitika Kontrol, 2002, vol. 6, no. 4, p. 383.

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