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Ab initio calculations of XRF intensities in non-homogeneous matrices
Spwrochimica
Arm,
Vol.
188.
No.
516. pp. 835-842.
0584-8547/83
1983. 0
Prinlcd in Great Britain.
Ab initio calculations
53.00 + .oO
1983. Pergamon Press Ltd.
of XRF intensities in non-homogeneous
matrices
B. VREBOSand J. A. HELSEN Katholieke Univesiteit Leuven (KUL), Departement Metaalkunde. G. de Croylaan (Receiued
2, B-3030Heverlee,
Belgium
30 November 1982)
Abstract-The physical state of a multiphase solid material affects the intensity of spectral lines in quantitative X-ray fluorescence analysis. The current correction algorithms for absorption and secondary fluorescence do not account for this effect. In the present paper, the relative influence ofa second phase in a solid on the XRF intensity of Ka-lines with respect to the homogeneous case is demonstrated by a Monte Carlo simulation of X-ray fluorescence in a (hypothetical) binary, fully segregated alloy of a particular geometry.
1. I Nra0oucr10~ CAREFULANALYSIS of analytical data obtained from XRF-spectra in our laboratory or published in the current literature shows that, in spite of the application of correction algorithms and the use of a sufficient number of standards, the precision is not always as good as counting statistics or instrument quality leads us to expect. When the same sample of a rolled copper-zinc-aluminium alloy is spectrometrically analyzed after different heat treatments, the compositioncalculated by the Lachane-Trail1 algorithm (or by the Philips’ a-program) is apparently changing. Wet chemical analysis, however, showed the composition to be constant; the surface finish (grinding on Sic-paper nr. 800) was the same after each treatment (Table 1). The application of any other correction algorithm is not able to account for the apparent change in composition. The aging of lead-tin alloy standards is another problem that has long been recognized in spectrochemical analysis (Table l), ref. [ 11. By digging into the literature, more examples of this kind have been found, especially in the days of the beginning of the generalized application of XRF-analysis. MICHAELIS and KILDAY[2] found in 1962 that surface finish could minimize but not eliminate apparent differences in concentration of silicon in hypereutectic aluminium alloys, obtained by different casting techniques, but with the same nominal chemical composition. Particle size effects of particulate samples such as soils, cement, dusts and minerals were recognized by different authors. Some examples are described by ANDERMANN and ALLEN [3,4] and GUNN[5] in 1961. CLAISSEand SAMSON [6] and BERNSTEIN [7,8] in 1962 and 1963; Table I. Apparent change in composition by aging of alloys at IOWC Alloy
Change in I!,,,
63 7” tin in soldera 10% tin in soldera (1) 69 ‘;,, copper in Cu-Zn-Al
I
time (days) 21
_ _
_ _
+0.3
+ 1.6
ataken from Ref. [I].
[I] [Z] [3] [4] [5] [6] [7] [S]
G. R. G. G. E. F. F. F.
H. GLADE and H. R. POST, Appl. Specrrosc. 24, 193 (1970). E. MICHAELIS and B. A. KILDAY, Adc. X-Ray Anal. 5, 405 (1962). ANDERMANN and J. D. ALLEN, Anal. C’hem.33, 1695 (1961). ANDERMANN and J. D. ALLEN, Adv. X-Ruy Anal. 4,414 (1961). L. GUNN. Adr. X-Roy Awl. 4. 382 (1961). CLAISSEand C. SAMSON, Adu. X-Ruy Anul. 5, 335 (1962). BERNSTEIN, Ado. X-Ray Anul. 5, 486 (1962). BERNSTEIN,Ado. X-Ray Anul. 6, 436 (1963). 835
250 +4.1 + 6.0 _
X36
B. VKEU~S and J. A. H~LSEN
the list being far from exhaustive. The growing availability of computer facilities and the application of excellent algorithms, with influence coeficients calculated by regression methods (e.g. LACHANCEand TRAILL[9], RASBERRYand HEINRICH[lo] or derived from fundamental parameters (e.g. JENKINS et al. [ll], CRISS[12]) provided considerable help in the analysis of complex matrices on a routine basis. The efforts of the analysts became focussed on computer methods and the interest in the physical state of the sample faded away, as can be seen from the decreasing number of papers on this subject. Only a markedly increased interest in the analysis of particulate samples (greatly stimulated by a growing interest in environmental studies), for the larger part done by energy dispersive spectrometers, can be noticed since the late sixties. LUBECKIet al. [13], HUNTERand RHODES[14], CRISS[~~], HOLYNSKAandMARKOWICZ[~~], VANESPEN~~~ADAMS[~~] and VANDYCK [18] discussed the particle size effect, arising in such particulate samples, but to the authors’ knowledge, no effort was made to apply these mathematical techniques for particulate samples to heterogeneous solids. In this paper, an introduction is presented to a fundamental approach of these problems. In a first approximation, the fluorescence processes in a binary solid matrix were simulated by a Monte Carlo method. The hypothetical sample consists of alternating lamellae, containing either element A or B. The analyzed surface is perpendicular to the plane of the lamellae, as shown in Fig. 1. This simple geometry has been adopted because the mathematical treatment is easy and a straightforward explanation of the results was to be expected. The majority of the results are obtained with photons impinging the surface along direction (a) in Fig. 1, a few others were obtained for photons following direction (b). All the photons from fluorescent radiation leaving the sample surface are counted, whatever their direction.
Fig. I. Geometry of the sample considered in the Monte Carlo Simulation.
[9] G. R. LACHANCEand R. J. TRAILL, Can. Spectrosc. 11,43 (1966). [lo] S. D. RASBERRYand K. F. J. HEINRICH, Anal. Chem. 46, 81 (1974). [l l] R. JENKINS, J. F. CROKE, R. L. NIEMANN and R. G. WESTBERG, Adu. X-Ray Anal. 18, 372 (1975). [12] J. W. CRISS, NRLXRF: a Fortran Program/or X-Ray Fluorescence Analysis, University of Athens, Georgia (1977). [13] A. LUBECKI, B. HOLYNSKAand M. WASILEWSKA,Spectrochim. Acta 238,465 (1968). [14] C. B. HUNTER and J. R. RHODES, X-Ray Spectrom. 1, 107 (1972). [lS] J. W. CRISS, Anal. Chem. 48, 179 (1978). [16] B. HOLYNSKAand A. MARKOWICZ,X-Ray Specrrom. 11, 117 (1982). [17] P. VAN ESPEN and F. ADAMS, X-Ray Spectrom. 10, 64 (1981). [I81 P. VAN DYCK, PhD. Thesis, Antwerp (1982).
Ab inirio calculations of XRF intensities in non-homogeneous matrices
837
2. NON-HOMOGENEOUSSOLIDSAMPLES In a perfect solid solution, every volume V has the same composition, independent of its position or its size. Consequently, for any X-ray photon of a given energy travelling through the material, the mass absorption coefficient (mat) is constant irrespective of its direction. For a given composition and a given photon energy, the mat is given by:
(1) where (p/~)~.~is the mat of element i for photons with wavelength I, y the weight fraction of element i in the sample and I: w = 1. Since there is no direct evidence that the fluorescence yield or photon absorption is dependent on crystallographic structure, perfect homogeneity implies that the intensity of fluorescent radiation is determined only by sample composition, under the assumption of constant geometry and experimental conditions. It is fundamental to the quantitative analysis of a homogeneous material that the intensity of any spectra1 line is a unique function of composition. As a consequence, an algorithm correcting for absorption and secondary fluorescence, as proposed by many authors, will give as a result a unique concentration. As we see in practice in our and in other laboratories (e.g. [ 19]), however, other factors are to be accounted for. Segregation into different phases is one of them and its importance with respect to intensity will be dealt with in the present paper. If a sample is completely homogeneous, the mat is constant throughout the sample and for a given photon energy defined by Eqn (1). The probability that a photon leaves the sample, may be estimated by Lambert’s law: P x exp
[
- ; pl 1 0
where 1 is the distance covered in the sample. Combining Eqn (1) and (2) yields:
If, however, the sample consists of element B, dispersed in a matrix of element A (Fig. 2), the probability to cross the sample boundary for a photon starting from point 2, is the combined probability of passing through the different path segments in the matrix or the dispersed phase without being absorbed:
Fig. 2. Second Phase B dispersed in a matrix A.
[19]
B. W. BUDENNSKY, X-Ray
Spectrom. 4, 166 (1975).
B. VREBOS and J. A. HELSEN
838
P ovcru,,= P, P, . . . P. = exp
[
(5)
f Ip,ZI, 0
It can be shown that
(6’4 and, since (7) Eqn (5) reduces to Eqn (3). The approximations in Eqns (6a, 6b) and Eqn (7) become more accurate with increasing number of terms in the summations, so that the thinner the lamellae, the more the system will approach the homogeneous case. But, as can be understood for photons starting from point Z in Fig. 1 and passing always through the same phase, the limit for this geometry can never be the homogeneous case. Equation (5) can easily be re-written for phases consisting of more than one element. Calculations on systems with complex geometry e.g. a random dispersion of spherical particles with a given size distribution, are in progress. 3. MONTECARLOSIMULATION Rather than to start the experimental verification of the effect of non-homogeneity by the preparation of binary samples with controlled segregation, it was preferred to simulate the fluorescence processes by a Monte Carlo method. The feasibility of this approach was tested on samples with the structure presented in Fig. 1 for the reasons mentioned earlier. The simulation method is extremely flexible with respect to all experimental parameters. The sample is a pile of parallel lamellae of which the thickness can be changed at will, while the composition of the sample as a whole remains constant. A change in composition is obtained by modifying the relative thickness. The volume fraction isdetermined by the ratios:
v, = ‘,/(‘A+ IS)
‘, = ‘B/tlA+I,)
If the density is known, the volume fraction can be transformed into weight fraction. Simulations have been performed for five different compositions of a binary “alloy”. Iron and chromium were taken in order to have absorption as well an enhancement effects. The weight fractions are tabulated in Table 2. The incident photons were given an energy of 10 keV. In each run the sample was supposed to be irradiated by 200 000 photons at an incident angle of 45” and “leaving” photons were
Table 2. Results obtained
with the Monte Carlo (MC) and NRLXRF
[12]
programs Relative intensity
Composition Weight fraction
Fe
Cr Cr
Fe
MC
0.18 0.25 0.50 0.647 0.786
0.82 0.75 0.50 0.353 0.214
0.229 0.294 0.537 0.652 0.779
NRLXRF 0.240 0.314 0.549 0.679 0.802
MC 0.591 0.486 0.253 0.157 0.090
NRLXRF 0.633 0.531 0.275 0.171 0.093
Ab initio calculations of XRF intensities in non-homogeneous matrices
839
assumed to be detected by a (hypothetical) hemispherical detector. This was done in order to improve the counting statistics by increasing the number of detected fluorescent photons. The algorithm of the computer program is presented schematically in Fig. 3. For every incident X-ray photon, the place of the initial interaction is randomly chosen (relative to the volume fractions of the two phases and according to Lambert’s law). Next, the interacting element is identified (Fe or Cr) and the involved phenomenon; in 867; of all cases, photo-electric absorption by K electrons is assumed (this is an approximation of the factor 1 - l/ri, in which ri is the jump ratio). Photo-electric absorption by a K electron can be followed by either an Auger transition or the generation of a fluorescent X-ray photon. The fluorescence yield determines the probability that the latter effect occurs. Of the generated Xray photons, a certain fraction is Ka, the rest being K/3. The relative amounts of Ka and KP are given in [20]. If no characteristic radiation is generated, the next incident photon is considered. If, on the contrary, a Ka or K/l photon is generated, its direction is randomly chosen and the place of the next interaction is determined. If phase boundaries are crossed, the appropriate changes in phase composition and mat are taken into account. If the photon escapes from the sample, it is counted disregarding its direction. We are aware that a lot of computing time is wasted by this direct approach, and that the calculation time can be decreased drastically by adopting
INPUT geometry fundsmental parameters
I INCIDENT PHOTON
I ABSORPTION select position of intersction
1
select directlon
I
generation 01 Ka
YES
NEXT PHOTON
t END I
1
Fig. 3. Algorithm of the Monte Carlo program.
[ZO] W. BAMBYNEK, B. CRASEMANN, R. W. FINK, H. U. FREUND, H. MARK, C. D. SWIFT, R. E. PRICE, P. VENUGOPALA RAO, Rev. Mod. Phys. 44, 716 (1972).
B. VREBOS andJ. A.
840
HELSEN
variance reductance, as did GARDNERand HAWTHORNE[21]. However, in this exploratory phase, the program fitted a PDP-1 l-based minicomputer and ran overnight. For the more complex geometries large computer facilities are being used. 4. RESULTSAND DISCUSSION The intensity is expected to tend toward the pure element intensity on increasing lamellae thickness. Decreasing thickness will make the intensity approach the homogeneous case, without, however, reaching this limit for the reasons explained in section 2. In the present binary system, iron radiation is attenuated by absorption only. Figure 4 indicates that this anticipated trend of the function of intensity versus lamellae thickness is verified for all thicknesses and concentrations concerned. Extrapolation to zero thickness yields an intensity somewhat lower than calculated for a solid solution of both elements: this has been calculated by the FORTRAN program NRLXRF [12], using the theoretical formulas. The results are shown in Table 2, the absolute intensities, obtained with the Monte Carlo program have been transformed to relative intensities by dividing them by the intensity of the respective pure element. In the case of chromium, absorption and secondary fluorescence are the competing phenomena, and the shape of the relationship intensity vs thickness is less clear (Fig. 5). In Fig. 5 some peculiar aspects can be seen for the higher Cr-concentration, apparently a maximum is found, the intensity at zero thickness is about equal to the pure element intensity, and the scatter of the points is much higher, although a number of photons is counted comparable to iron. As the thickness of the lamellae decreases, the probability of Cr Ka absorption in iron increases ( = decrease of the Cr-characteristic radiation) but, on the other
Fig. 4. Intensity (in kilocounts) of Fe Ka YSthickness of the lamellae. [21].
R. P. GARDNER
and A. R.
HAWTHORNE, X-Ray
Spectrom.
4, 138 (1975).
Ab initio calculations of XRF intensities in non-homogeneous
841
matrices
I
11’
I
0
5
I IO
THICKNESS
I
I
16
20
1
[t.ml
Fig. 5. Instensity (in kilocounts) of Cr Ka vs thickness of the lamellae.
hand, the secondary fluorescence of chromium by iron radiation increases. The absorption of Fe Ka in chromium is very important, but for each absorbed Fe photon, the probability that subsequently a Cr Ka photon leaves the sample is rather small. Of all Fe Ka photons absorbed in chromium, only 86% excite K electrons; of these K-electron-vacancies, only 25.3 “/;, (fluorescence yield for chromium) are decaying radiatively and only 88 “/;, of the photons emitted in this way are Ka. These fundamental parameters are taken from BAMBYNEK et al. [20]. Therefore, the probability that the absorption of a Ka photon by chromium yields a Cr Ka photon, moving towards the detector is about 0.096 ( = 0.86 x 0.253 x 0.88 x 0.5) (the factor 0.5 is introduced to account for photons moving away from the detector). This means that for every 1000 Fe-photons (Ka and K/?) absorbed in chromium, there are only 96 Cr Ka photons moving towards the detector. Due to absorption, only a fraction of these is actually detected. One can easily calculate that the intensity of Cr Ka radiation is attenuated by 6 7, for a path length of 1 pm in pure chromium and 8 2, in 1 pm of pure iron. This means that the extra absorption of Cr Ka photons by iron is about 2 7; (with respect to chromium). On the other hand, about 28 7;; of Fe Ka radiation is absorbed in 1 pm chromium, but, as mentioned earlier, less than 9 ‘;oof the absorbed photons give rise to a Cr Ka photon, so that in 1 pm chromium only about 2.5 ‘;, ( = 0.28 x 0.09) of the incident Fe Ka and K/l radiation is “transformed” into Cr Ka photons: this is the same order of magnitude as the extra absorption of Cr Ka photons in iron (with respect to chromium). It must be emphasized that these percentages indicate only an order of magnitude. They depend of course on the flux of Fe photons entering the chromium phase and of Cr Ka photons which enter the iron phase. Thus, these percentages depend on the geometry and concentration. Both effects are rather small and oppose each other, accounting for the larger scatter of the chromium results. Obviously, these percentages depend upon the numbers of Cr K and Fe K photons crossing the phase boundaries. This is the reason why in some cases, the Cr Ka radiation is
X42
B. VRELIOSand J. A. HELSEN
increased, as the thickness decreases, and for other compositions, the inverse tendency is noticed (Fig. 5). Figure 4 also clearly illustrates that for the geometry involved, it is impossible, to derive the iron content from the Fe Ka intensity, without knowing the thickness of the lamella. The intensity of fluorescent radiation is not only a function of concentration, but depends also on how iron is distributed in the sample. The observed effect diminishes if the difference in composition between the two phases decreases. In Fig. 4 curve f represents a sample consisting of a homogeneous phase (25 ‘J,,,Fe and 75 ‘i;;Cr by weight) and pure iron phase with an overall iron content of 50”,,. The net effect is smaller. 5. CONCLUSION The influence of the non-homogeneity of a solid sample on the intensity of the characteristic radiation is simulated by a Monte Carlo Method and proved to be real for one particular geometry. The effect is extremely large (for iron up to 200%) and for chromium up to 20% in the considered binary alloy but will be attenuated when segregation is less pronounced (compound phases) or when the dispersed phase is very finely divided (e.g. with sizes much smaller than the critical thickness). The results of the simulations have not yet been checked experimentally. The described approach of absorption and enhancement effects on intensity of X-ray lines will be expanded to more common types of segregation (e.g. spherical particles randomly dispersed in the matrix). From the described results, however, it is obvious that segregation is a parameter not to be neglected in XRF analysis and should be integrated in the current correction algorithms. This is the final aim of the present study. Acknowledgments-IWONL, the institute for stimulation of scientific research in industry and agronomy, is gratefully acknowledged by B. VREBOSfor grantinghim a fellowship. We wish to thank DR. W. AERTSand Prof. L. DELAEYfor many fruitful discussions.
Arm,
Vol.
188.
No.
516. pp. 835-842.
0584-8547/83
1983. 0
Prinlcd in Great Britain.
Ab initio calculations
53.00 + .oO
1983. Pergamon Press Ltd.
of XRF intensities in non-homogeneous
matrices
B. VREBOSand J. A. HELSEN Katholieke Univesiteit Leuven (KUL), Departement Metaalkunde. G. de Croylaan (Receiued
2, B-3030Heverlee,
Belgium
30 November 1982)
Abstract-The physical state of a multiphase solid material affects the intensity of spectral lines in quantitative X-ray fluorescence analysis. The current correction algorithms for absorption and secondary fluorescence do not account for this effect. In the present paper, the relative influence ofa second phase in a solid on the XRF intensity of Ka-lines with respect to the homogeneous case is demonstrated by a Monte Carlo simulation of X-ray fluorescence in a (hypothetical) binary, fully segregated alloy of a particular geometry.
1. I Nra0oucr10~ CAREFULANALYSIS of analytical data obtained from XRF-spectra in our laboratory or published in the current literature shows that, in spite of the application of correction algorithms and the use of a sufficient number of standards, the precision is not always as good as counting statistics or instrument quality leads us to expect. When the same sample of a rolled copper-zinc-aluminium alloy is spectrometrically analyzed after different heat treatments, the compositioncalculated by the Lachane-Trail1 algorithm (or by the Philips’ a-program) is apparently changing. Wet chemical analysis, however, showed the composition to be constant; the surface finish (grinding on Sic-paper nr. 800) was the same after each treatment (Table 1). The application of any other correction algorithm is not able to account for the apparent change in composition. The aging of lead-tin alloy standards is another problem that has long been recognized in spectrochemical analysis (Table l), ref. [ 11. By digging into the literature, more examples of this kind have been found, especially in the days of the beginning of the generalized application of XRF-analysis. MICHAELIS and KILDAY[2] found in 1962 that surface finish could minimize but not eliminate apparent differences in concentration of silicon in hypereutectic aluminium alloys, obtained by different casting techniques, but with the same nominal chemical composition. Particle size effects of particulate samples such as soils, cement, dusts and minerals were recognized by different authors. Some examples are described by ANDERMANN and ALLEN [3,4] and GUNN[5] in 1961. CLAISSEand SAMSON [6] and BERNSTEIN [7,8] in 1962 and 1963; Table I. Apparent change in composition by aging of alloys at IOWC Alloy
Change in I!,,,
63 7” tin in soldera 10% tin in soldera (1) 69 ‘;,, copper in Cu-Zn-Al
I
time (days) 21
_ _
_ _
+0.3
+ 1.6
ataken from Ref. [I].
[I] [Z] [3] [4] [5] [6] [7] [S]
G. R. G. G. E. F. F. F.
H. GLADE and H. R. POST, Appl. Specrrosc. 24, 193 (1970). E. MICHAELIS and B. A. KILDAY, Adc. X-Ray Anal. 5, 405 (1962). ANDERMANN and J. D. ALLEN, Anal. C’hem.33, 1695 (1961). ANDERMANN and J. D. ALLEN, Adv. X-Ruy Anal. 4,414 (1961). L. GUNN. Adr. X-Roy Awl. 4. 382 (1961). CLAISSEand C. SAMSON, Adu. X-Ruy Anul. 5, 335 (1962). BERNSTEIN, Ado. X-Ray Anul. 5, 486 (1962). BERNSTEIN,Ado. X-Ray Anul. 6, 436 (1963). 835
250 +4.1 + 6.0 _
X36
B. VKEU~S and J. A. H~LSEN
the list being far from exhaustive. The growing availability of computer facilities and the application of excellent algorithms, with influence coeficients calculated by regression methods (e.g. LACHANCEand TRAILL[9], RASBERRYand HEINRICH[lo] or derived from fundamental parameters (e.g. JENKINS et al. [ll], CRISS[12]) provided considerable help in the analysis of complex matrices on a routine basis. The efforts of the analysts became focussed on computer methods and the interest in the physical state of the sample faded away, as can be seen from the decreasing number of papers on this subject. Only a markedly increased interest in the analysis of particulate samples (greatly stimulated by a growing interest in environmental studies), for the larger part done by energy dispersive spectrometers, can be noticed since the late sixties. LUBECKIet al. [13], HUNTERand RHODES[14], CRISS[~~], HOLYNSKAandMARKOWICZ[~~], VANESPEN~~~ADAMS[~~] and VANDYCK [18] discussed the particle size effect, arising in such particulate samples, but to the authors’ knowledge, no effort was made to apply these mathematical techniques for particulate samples to heterogeneous solids. In this paper, an introduction is presented to a fundamental approach of these problems. In a first approximation, the fluorescence processes in a binary solid matrix were simulated by a Monte Carlo method. The hypothetical sample consists of alternating lamellae, containing either element A or B. The analyzed surface is perpendicular to the plane of the lamellae, as shown in Fig. 1. This simple geometry has been adopted because the mathematical treatment is easy and a straightforward explanation of the results was to be expected. The majority of the results are obtained with photons impinging the surface along direction (a) in Fig. 1, a few others were obtained for photons following direction (b). All the photons from fluorescent radiation leaving the sample surface are counted, whatever their direction.
Fig. I. Geometry of the sample considered in the Monte Carlo Simulation.
[9] G. R. LACHANCEand R. J. TRAILL, Can. Spectrosc. 11,43 (1966). [lo] S. D. RASBERRYand K. F. J. HEINRICH, Anal. Chem. 46, 81 (1974). [l l] R. JENKINS, J. F. CROKE, R. L. NIEMANN and R. G. WESTBERG, Adu. X-Ray Anal. 18, 372 (1975). [12] J. W. CRISS, NRLXRF: a Fortran Program/or X-Ray Fluorescence Analysis, University of Athens, Georgia (1977). [13] A. LUBECKI, B. HOLYNSKAand M. WASILEWSKA,Spectrochim. Acta 238,465 (1968). [14] C. B. HUNTER and J. R. RHODES, X-Ray Spectrom. 1, 107 (1972). [lS] J. W. CRISS, Anal. Chem. 48, 179 (1978). [16] B. HOLYNSKAand A. MARKOWICZ,X-Ray Specrrom. 11, 117 (1982). [17] P. VAN ESPEN and F. ADAMS, X-Ray Spectrom. 10, 64 (1981). [I81 P. VAN DYCK, PhD. Thesis, Antwerp (1982).
Ab inirio calculations of XRF intensities in non-homogeneous matrices
837
2. NON-HOMOGENEOUSSOLIDSAMPLES In a perfect solid solution, every volume V has the same composition, independent of its position or its size. Consequently, for any X-ray photon of a given energy travelling through the material, the mass absorption coefficient (mat) is constant irrespective of its direction. For a given composition and a given photon energy, the mat is given by:
(1) where (p/~)~.~is the mat of element i for photons with wavelength I, y the weight fraction of element i in the sample and I: w = 1. Since there is no direct evidence that the fluorescence yield or photon absorption is dependent on crystallographic structure, perfect homogeneity implies that the intensity of fluorescent radiation is determined only by sample composition, under the assumption of constant geometry and experimental conditions. It is fundamental to the quantitative analysis of a homogeneous material that the intensity of any spectra1 line is a unique function of composition. As a consequence, an algorithm correcting for absorption and secondary fluorescence, as proposed by many authors, will give as a result a unique concentration. As we see in practice in our and in other laboratories (e.g. [ 19]), however, other factors are to be accounted for. Segregation into different phases is one of them and its importance with respect to intensity will be dealt with in the present paper. If a sample is completely homogeneous, the mat is constant throughout the sample and for a given photon energy defined by Eqn (1). The probability that a photon leaves the sample, may be estimated by Lambert’s law: P x exp
[
- ; pl 1 0
where 1 is the distance covered in the sample. Combining Eqn (1) and (2) yields:
If, however, the sample consists of element B, dispersed in a matrix of element A (Fig. 2), the probability to cross the sample boundary for a photon starting from point 2, is the combined probability of passing through the different path segments in the matrix or the dispersed phase without being absorbed:
Fig. 2. Second Phase B dispersed in a matrix A.
[19]
B. W. BUDENNSKY, X-Ray
Spectrom. 4, 166 (1975).
B. VREBOS and J. A. HELSEN
838
P ovcru,,= P, P, . . . P. = exp
[
(5)
f Ip,ZI, 0
It can be shown that
(6’4 and, since (7) Eqn (5) reduces to Eqn (3). The approximations in Eqns (6a, 6b) and Eqn (7) become more accurate with increasing number of terms in the summations, so that the thinner the lamellae, the more the system will approach the homogeneous case. But, as can be understood for photons starting from point Z in Fig. 1 and passing always through the same phase, the limit for this geometry can never be the homogeneous case. Equation (5) can easily be re-written for phases consisting of more than one element. Calculations on systems with complex geometry e.g. a random dispersion of spherical particles with a given size distribution, are in progress. 3. MONTECARLOSIMULATION Rather than to start the experimental verification of the effect of non-homogeneity by the preparation of binary samples with controlled segregation, it was preferred to simulate the fluorescence processes by a Monte Carlo method. The feasibility of this approach was tested on samples with the structure presented in Fig. 1 for the reasons mentioned earlier. The simulation method is extremely flexible with respect to all experimental parameters. The sample is a pile of parallel lamellae of which the thickness can be changed at will, while the composition of the sample as a whole remains constant. A change in composition is obtained by modifying the relative thickness. The volume fraction isdetermined by the ratios:
v, = ‘,/(‘A+ IS)
‘, = ‘B/tlA+I,)
If the density is known, the volume fraction can be transformed into weight fraction. Simulations have been performed for five different compositions of a binary “alloy”. Iron and chromium were taken in order to have absorption as well an enhancement effects. The weight fractions are tabulated in Table 2. The incident photons were given an energy of 10 keV. In each run the sample was supposed to be irradiated by 200 000 photons at an incident angle of 45” and “leaving” photons were
Table 2. Results obtained
with the Monte Carlo (MC) and NRLXRF
[12]
programs Relative intensity
Composition Weight fraction
Fe
Cr Cr
Fe
MC
0.18 0.25 0.50 0.647 0.786
0.82 0.75 0.50 0.353 0.214
0.229 0.294 0.537 0.652 0.779
NRLXRF 0.240 0.314 0.549 0.679 0.802
MC 0.591 0.486 0.253 0.157 0.090
NRLXRF 0.633 0.531 0.275 0.171 0.093
Ab initio calculations of XRF intensities in non-homogeneous matrices
839
assumed to be detected by a (hypothetical) hemispherical detector. This was done in order to improve the counting statistics by increasing the number of detected fluorescent photons. The algorithm of the computer program is presented schematically in Fig. 3. For every incident X-ray photon, the place of the initial interaction is randomly chosen (relative to the volume fractions of the two phases and according to Lambert’s law). Next, the interacting element is identified (Fe or Cr) and the involved phenomenon; in 867; of all cases, photo-electric absorption by K electrons is assumed (this is an approximation of the factor 1 - l/ri, in which ri is the jump ratio). Photo-electric absorption by a K electron can be followed by either an Auger transition or the generation of a fluorescent X-ray photon. The fluorescence yield determines the probability that the latter effect occurs. Of the generated Xray photons, a certain fraction is Ka, the rest being K/3. The relative amounts of Ka and KP are given in [20]. If no characteristic radiation is generated, the next incident photon is considered. If, on the contrary, a Ka or K/l photon is generated, its direction is randomly chosen and the place of the next interaction is determined. If phase boundaries are crossed, the appropriate changes in phase composition and mat are taken into account. If the photon escapes from the sample, it is counted disregarding its direction. We are aware that a lot of computing time is wasted by this direct approach, and that the calculation time can be decreased drastically by adopting
INPUT geometry fundsmental parameters
I INCIDENT PHOTON
I ABSORPTION select position of intersction
1
select directlon
I
generation 01 Ka
YES
NEXT PHOTON
t END I
1
Fig. 3. Algorithm of the Monte Carlo program.
[ZO] W. BAMBYNEK, B. CRASEMANN, R. W. FINK, H. U. FREUND, H. MARK, C. D. SWIFT, R. E. PRICE, P. VENUGOPALA RAO, Rev. Mod. Phys. 44, 716 (1972).
B. VREBOS andJ. A.
840
HELSEN
variance reductance, as did GARDNERand HAWTHORNE[21]. However, in this exploratory phase, the program fitted a PDP-1 l-based minicomputer and ran overnight. For the more complex geometries large computer facilities are being used. 4. RESULTSAND DISCUSSION The intensity is expected to tend toward the pure element intensity on increasing lamellae thickness. Decreasing thickness will make the intensity approach the homogeneous case, without, however, reaching this limit for the reasons explained in section 2. In the present binary system, iron radiation is attenuated by absorption only. Figure 4 indicates that this anticipated trend of the function of intensity versus lamellae thickness is verified for all thicknesses and concentrations concerned. Extrapolation to zero thickness yields an intensity somewhat lower than calculated for a solid solution of both elements: this has been calculated by the FORTRAN program NRLXRF [12], using the theoretical formulas. The results are shown in Table 2, the absolute intensities, obtained with the Monte Carlo program have been transformed to relative intensities by dividing them by the intensity of the respective pure element. In the case of chromium, absorption and secondary fluorescence are the competing phenomena, and the shape of the relationship intensity vs thickness is less clear (Fig. 5). In Fig. 5 some peculiar aspects can be seen for the higher Cr-concentration, apparently a maximum is found, the intensity at zero thickness is about equal to the pure element intensity, and the scatter of the points is much higher, although a number of photons is counted comparable to iron. As the thickness of the lamellae decreases, the probability of Cr Ka absorption in iron increases ( = decrease of the Cr-characteristic radiation) but, on the other
Fig. 4. Intensity (in kilocounts) of Fe Ka YSthickness of the lamellae. [21].
R. P. GARDNER
and A. R.
HAWTHORNE, X-Ray
Spectrom.
4, 138 (1975).
Ab initio calculations of XRF intensities in non-homogeneous
841
matrices
I
11’
I
0
5
I IO
THICKNESS
I
I
16
20
1
[t.ml
Fig. 5. Instensity (in kilocounts) of Cr Ka vs thickness of the lamellae.
hand, the secondary fluorescence of chromium by iron radiation increases. The absorption of Fe Ka in chromium is very important, but for each absorbed Fe photon, the probability that subsequently a Cr Ka photon leaves the sample is rather small. Of all Fe Ka photons absorbed in chromium, only 86% excite K electrons; of these K-electron-vacancies, only 25.3 “/;, (fluorescence yield for chromium) are decaying radiatively and only 88 “/;, of the photons emitted in this way are Ka. These fundamental parameters are taken from BAMBYNEK et al. [20]. Therefore, the probability that the absorption of a Ka photon by chromium yields a Cr Ka photon, moving towards the detector is about 0.096 ( = 0.86 x 0.253 x 0.88 x 0.5) (the factor 0.5 is introduced to account for photons moving away from the detector). This means that for every 1000 Fe-photons (Ka and K/?) absorbed in chromium, there are only 96 Cr Ka photons moving towards the detector. Due to absorption, only a fraction of these is actually detected. One can easily calculate that the intensity of Cr Ka radiation is attenuated by 6 7, for a path length of 1 pm in pure chromium and 8 2, in 1 pm of pure iron. This means that the extra absorption of Cr Ka photons by iron is about 2 7; (with respect to chromium). On the other hand, about 28 7;; of Fe Ka radiation is absorbed in 1 pm chromium, but, as mentioned earlier, less than 9 ‘;oof the absorbed photons give rise to a Cr Ka photon, so that in 1 pm chromium only about 2.5 ‘;, ( = 0.28 x 0.09) of the incident Fe Ka and K/l radiation is “transformed” into Cr Ka photons: this is the same order of magnitude as the extra absorption of Cr Ka photons in iron (with respect to chromium). It must be emphasized that these percentages indicate only an order of magnitude. They depend of course on the flux of Fe photons entering the chromium phase and of Cr Ka photons which enter the iron phase. Thus, these percentages depend on the geometry and concentration. Both effects are rather small and oppose each other, accounting for the larger scatter of the chromium results. Obviously, these percentages depend upon the numbers of Cr K and Fe K photons crossing the phase boundaries. This is the reason why in some cases, the Cr Ka radiation is
X42
B. VRELIOSand J. A. HELSEN
increased, as the thickness decreases, and for other compositions, the inverse tendency is noticed (Fig. 5). Figure 4 also clearly illustrates that for the geometry involved, it is impossible, to derive the iron content from the Fe Ka intensity, without knowing the thickness of the lamella. The intensity of fluorescent radiation is not only a function of concentration, but depends also on how iron is distributed in the sample. The observed effect diminishes if the difference in composition between the two phases decreases. In Fig. 4 curve f represents a sample consisting of a homogeneous phase (25 ‘J,,,Fe and 75 ‘i;;Cr by weight) and pure iron phase with an overall iron content of 50”,,. The net effect is smaller. 5. CONCLUSION The influence of the non-homogeneity of a solid sample on the intensity of the characteristic radiation is simulated by a Monte Carlo Method and proved to be real for one particular geometry. The effect is extremely large (for iron up to 200%) and for chromium up to 20% in the considered binary alloy but will be attenuated when segregation is less pronounced (compound phases) or when the dispersed phase is very finely divided (e.g. with sizes much smaller than the critical thickness). The results of the simulations have not yet been checked experimentally. The described approach of absorption and enhancement effects on intensity of X-ray lines will be expanded to more common types of segregation (e.g. spherical particles randomly dispersed in the matrix). From the described results, however, it is obvious that segregation is a parameter not to be neglected in XRF analysis and should be integrated in the current correction algorithms. This is the final aim of the present study. Acknowledgments-IWONL, the institute for stimulation of scientific research in industry and agronomy, is gratefully acknowledged by B. VREBOSfor grantinghim a fellowship. We wish to thank DR. W. AERTSand Prof. L. DELAEYfor many fruitful discussions.