Improving accuracy and capabilities of X-ray fluorescence method using intensity ratios

Nuclear Instruments and Methods in Physics Research B 397 (2017) 67–74

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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Improving accuracy and capabilities of X-ray fluorescence method using intensity ratios Andrey V. Garmay ⇑, Kirill V. Oskolok Department of Chemistry, Moscow State University, Moscow 119991, Russia

a r t i c l e

i n f o

Article history: Received 20 October 2016 Received in revised form 1 February 2017 Accepted 25 February 2017 Available online 8 March 2017 Keywords: X-ray fluorescence Intensity ratio Scattered radiation Steel Oxides Iron ore

a b s t r a c t An X-ray fluorescence analysis algorithm is proposed which is based on a use of ratios of X-ray fluorescence lines intensities. Such an analytical signal is more stable and leads to improved accuracy. Novel calibration equations are proposed which are suitable for analysis in a broad range of matrix compositions. To apply the algorithm to analysis of samples containing significant amount of undetectable elements a use of a dependence of a Rayleigh-to-Compton intensity ratio on a total content of these elements is suggested. The technique’s validity is shown by analysis of standard steel samples, model metal oxides mixture and iron ore samples. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction X-ray fluorescence (XRF) method allows fast and precise determination of elemental composition of samples of various genesis. Different techniques and calibration methods have been proposed since early 1950-s in order to reach the best precision and to broad an application field of the method. A comprehensive review of these techniques including fundamental parameters method (FPM) is given by R.M. Rousseau [1]. However, all the techniques were originally designed for analysis of homogeneous samples with flat polished surface, e.g. massive steel samples, fused borate disks, etc. Therefore, none of the techniques (except FPM) can be applied for the analysis of non-conventional samples (finished technical products, samples with unpolished surface, metal filings, various powders with different grain size, etc.) without appropriate calibration and often a great number of reference samples, but adequate standard samples are often unavailable [2]. Besides, when a specimen contains significant amounts of elements that cannot be detected (undetectable elements (UE)), most of these techniques fail to give adequate results. Some other methods using primary radiation scattered by a sample were suggested to solve this problem Abbreviations: XRF, X-ray fluorescence; FPM, fundamental parameters method; UE, undetectable elements; FA, fundamental algorithm; LSM, least squares method; BFP, backscatter fundamental parameters; DE, detectable elements; NRMSE, normalized root-mean-square error; APE, absolute percentage error. ⇑ Corresponding author. E-mail address: [email protected] (A.V. Garmay). http://dx.doi.org/10.1016/j.nimb.2017.02.072 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.

[3–8], but their application is usually limited to a narrow matrix composition range restricted by calibration samples and often requires a large number of reference standards. Il’in showed that if a ratio of XRF lines intensities is applied as an analytical signal, then accuracy of analysis improves and the influence of the sample shape and surface quality decreases significantly [2]. However, matrix effects influence is still strong and should be taken into account. In a current work, we present a novel equations with theoretical coefficients based on intensity ratios. These equations allow one to improve the accuracy of XRF analysis, to broad the matrix composition range, in which the analysis can be performed with the same accuracy with the same calibration samples, and to analyze nonconventional samples with a reduced number of standard samples. 2. Theory 2.1. Algorithm for samples not containing UE A dependence of XRF intensity ratio on a concentration ratio occurred to be linear in a broad range of concentration ratios [2]. However, it is still influenced by other elements. Let us consider Ni-Fe-Cr tertiary system under monochromatic approximation. To take into account that the primary radiation is polychromatic and heterogeneous due to its characteristic constituents, it is better to use different effective wavelengths kpr i of the primary radiation for each element i [9,10]. These wavelengths are chosen so that

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their influence is similar to the averaged influence of the whole primary radiation spectrum. Then nickel-to-iron Ka-lines intensity ratio is given by the following equation:

INiK a g Ni C Ni kNi pNiK a ¼ IFeK a g Fe C Fe kFe pFeK a

l

pr Fe ðkFe Þ



pr pr Ni ðkNi Þ  Ipr ðkNi Þ=M Ni pr Ipr ðkFe Þ=M Fe þ 1=2  dFe;NiK a

þ 1=2  dFe;NiKb

; ð1Þ

where IiKa is an intensity of Ka-line of an element i, gi is a proportionality constant dependent upon the instrument used, ci is a mass fraction of the element i, Ipr(k) is an intensity of primary radiation at a k wavelength,

ki ¼

Si  1 xi ; Si

Si is a jump ratio of element i for K-edge of considered line, xi is a fluorescence yield for considered line of element i, pi is a fraction of element’s i Ka-line in the total intensity of K-series, li(k) is a mass attenuation coefficient of the element i at the k wavelength,

Mi ¼

n X C j  ðlj ðkpr i Þ  cosec/ þ lj ðkiK a Þ  cosecwÞ; j¼1

di,jKx is an enhancement correction factor: pr dij ¼ li ðkjKx Þkj C j lj ðkpr j Þ  Ipr ðkj Þ , ! n X i pr C k  ðlk ðkj Þ  cosec/ þ lk ðkiK a Þ  cosecwÞ ;  LjKx

sin /

C l ðkpr Þ k k k j

X ln 1 þ X

sin / 

!

pr k ðkj Þ

l

Ck kX

C l ðk Þ k k k jKx !

þX

sin w

C l ðk Þ k k k iK a

C k lk ðkiK a Þ kX  ln 1 þ ; C l ðk Þ sin w  k k k jKx

MFe pr lFe ðkpr I k Fe Þ pr ð Fe Þ pr M Ni lNi ðkpr I k Ni Þ pr ð Ni Þ

þ

M Fe MNi ½1=2dFe;NiK a þ1=2dFe;NiKb 

ð Þ

pr Ni kNi

l

;

ð2Þ

ð Þ

pr Ipr kNi

where kNi=Fe ¼ g Ni kNi pNiK a =g Fe kFe pFeK a . Let us then consider Mi/Mj  const (which is, of course, rather rough approximation). Then, taking into account that CNi = (CNi + CFe)RNi/Fe/(1 + RNi/Fe) and CFe = (CNi + CFe)/(1 + RNi/Fe), where RNi/Fe = CNi/CFe, and that CNi + CFe = 1  CCr in the considered case, we can rearrange (2) as follows:

INiK a a0 þ aCr  C Cr ¼ kNi=Fe RNi=Fe ; IFeK a b0 þ bCr  C Cr

ð3Þ

where     pr  a0 ¼ RNi=Fe  lNi kpr ð1 þ RNi=Fe Þ; Fe ; kFeK a þ lFe kFe ; kFeK a

aCr ¼ lCr ðkpr Fe ; kFeK a Þ  a0 ; ;k ÞþlFe ðkpr ;k Þ l ðkpr ÞIpr ðkpr Þ RNi=Fe lNi ðkpr Ni NiK a Ni NiK a

Fe b0 ¼ lFe ðkFe pr pr Ni Ni ÞIpr ðkNi Þ

þ 12

RNi=Fe 1þRNi=Fe

P kNi MNiprMFe

ck lNi ðkNi ;kFeK a Þ



1þRNi=Fe



Fe lFe ðkNiK a ÞpNiK a LFe NiK a þ lFe ðkNiKb ÞpNiKb LNiKb ;

l ðkpr ÞIpr ðkpr Þ

pr Fe bCr ¼ lFe ðkFe pr pr lCr ðkNi ; kNiK a Þ  b0 ; Ni Ni ÞIpr ðkNi Þ

li ðkj ; kk Þ ¼ li ðkj Þ  cosec/ þ li ðkk Þ  cosecw:

P P a00 þ k–i;j a01k C k a10 þ k–i;j a1k C k P P ; a1 ¼ : 1 þ k–i;j b01k C k 1 þ k–i;j b1k C k

However, it was found empirically that often second-order terms are necessary to describe a dependence of a0 on concentrations Ck more accurately. Therefore

a0 ¼

INiK a C Ni ¼ kNi=Fe IFeK a C Fe

ð5Þ

Nevertheless, both the coefficients a0 and a1 were found to be best fit by an expression similar to that of the coefficient a in (4), so after dividing both numerator and denominator of (4) by b0

a0 ¼

u is an angle between the sample and an incident x-rays, w is an angle at which XRF radiation is detected (‘‘take-off angle”) [11]. Rearrangement of (1) gives



ð4Þ

However, beside approximations used, there is a problem of coefficients ai, bi being dependent on the concentration ratio Ri/j of the chosen pair of elements i and j. Thus, the Eq. (4) will require a very close first approximation of the sample composition and will allow one to obtain accurate results only in narrow ranges of concentrations of all elements, i.e. in narrow range of the matrix composition. Such a situation took place with Rousseau’s fundamental algorithm (FA), which required a use of a Claisse-Quintin algorithm with separate calibration samples set for obtaining the first approximation [1]. An alternative approach was suggested by Pavlinsky and Vladimirova [12]. They first calculated XRF lines intensities for several tens of samples and then used this calculated intensities to find the values of the parameters of their model by the least squares method (LSM). This allows one to broad an operating range of matrix compositions and to reduce bias caused by the approximations used. Therefore, we also apply this approach to calculate the parameters of our model. However, when the concentration ratios Ri/j are relatively large, some of the assumptions made do not work, which gives rise to a significant constant term, though the dependence is still linear:

IiKx ¼ a0 þ a1 Ri=j : IjKy

k¼1

LijKx ¼ X

X a0 þ ak C k IiKx Xk–i;j ¼ ki=j Ri=j ¼ aRi=j : IjKy b0 þ bk C k k–i;j

l



For multielement sample, the Eq. (3) will take the following form:

P P a00 þ k–i;j a01k C k þ k–i;j a02k C 2k : P P 1 þ k–i;j b01k C k þ k–i;j b02k C 2k

2.2. Algorithm for samples containing UE Almost all algorithms of XRF analysis of samples with significant content of UE are models based on the use of a scattered radiation. Often they are more or less adequate regression models [13]. Nevertheless, some models have theoretical basis. Among them Bakhtiarov’s technique [6], so-called backscatter fundamental parameters (BFP) method [3,4] and Szaloki’s algorithm [5] are notable. The first one is suitable for determination of heavy metals in any concentration and in any light matrix. However, empirical regression models are necessary for analysis of samples containing high amounts of other heavy elements, especially heavier than the element being determined [6]. Thus, matrix composition range will be restricted by calibration samples, and errors caused by difference in samples’ shape, particle sizes, surface quality are possible. BFP is an attempt to expand FPM to samples containing UE, but it requires using absolute intensities of scattered radiation, that in fact again leads to a dependence of the results on samples’ shape, surface quality and other experimental factors difficult to control. Szaloki’s algorithm is also an attempt of expanding FPM to samples containing UE. It utilizes using Rayleigh-to-Compton intensity ratio IR/C to determine an average atomic number of the sample ZS

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[5]. Then, if a close estimation of concentrations of detectable elements (DE) is available, one can calculate an average atomic number of UE ZUE, use it for calculating the corresponding mass attenuation coefficients and apply FPM. However, such an approach faces several difficulties. First, several standards are necessary to find parameters of the dependence of IR/C on ZS by LSM. Second, even relatively small discrepancy between calculated and real ZS can cause rather large errors in final concentration. Third, the best method of calculating ZS is questionable, and most of them result in calibrations appropriate only for narrow matrix composition ranges, i.e. for very close objects [14]. Finally, close estimation of DE concentrations is crucial; otherwise adequate results cannot be achieved. Indeed, ZS is a function of ZUE, DE average atomic number ZDE and their total content in the sample CDE:

P P P Zi Ci Z UE;i C UE;i þ Z DE;i C DE;i ZS ¼ P ¼ Ci 1 P P Z UE;i C UE;i Z DE;i C DE;i ¼ C UE P þ C DE P ¼ C UE Z UE þ C DE Z DE C UE;i C DE;i ¼ ð1  C DE ÞZ UE þ C DE Z DE ¼ Z UE þ C DE ðZ DE  Z UE Þ:

Here CUE is a total content of UE. In fact, in Szaloki’s algorithm, we fix CDE and ZDE and find the most optimal ZUE for this fixed pair of parameters. Then we fix this ZUE and calculate new DE concentrations by FPM, which fit measured intensities in a better way, but these new DE concentrations will strongly depend on ZUE. As a result, final concentrations will differ from the first estimation hardly more than 10%, and if the first estimation is far from the correct sample composition, then results can still be far from adequate, with a relative deviation about several tens of percent. We propose another approach to the problem, namely to fix ZUE and to find CUE. For these purpose a dependence of Rayleigh-toCompton intensity ratio on CUE can be used:

IR=C ¼ a0 þ a1 C UE :

ð6Þ

Indeed, if we divide an equation describing a relation between Rayleigh intensity and a sample composition by a corresponding equation for Compton intensity [15], we will obtain the following expression:

IR=C ¼

k R þ Dk kR

P

P C l ðk ; k þ DkÞ ðdrR =dXÞi C i Pi i R R P ; C i li ðkR ; kR Þ ðdrC =dXÞi C i

ð7Þ

where kR is a wavelength of a Rayleigh peak maximum, Dk is a shift of a Compton peak maximum from kR, (drR/dX)i and (drC/dX)i are mass differential Rayleigh and Compton scattering cross-sections of the element i. A dependence of (drC/dX)i on atomic number is relatively P P The C i li ðkR ; kR þ DkÞ= weak, so ðdrC =dXÞi ci  const. P C i li ðkR ; kR Þ ratio is almost constant for all elements except those having absorption edges between Rayleigh and Compton peaks. So the Eq. (7) can be rearranged to give a following expression:

IR=C ¼ KððdrR =dXÞUE  ðdrR =dXÞDE ÞC UE þ KðdrR =dXÞDE ; where K ¼ kRkþRDk

P P ci li ðkR ;kR þDkÞ ci P P ci li ðkR ;kR Þ

ðdrC =dXÞi ci

ð8Þ

 const, ðdrR =dXÞUE and

ðdrR =dXÞDE are average mass differential Rayleigh scattering cross-sections for UE and DE, P ðdrR =dXÞDE ¼ ¼

C DE;i ðdrR =dXÞDE;i P C DE;i !1 ! X C DE;i X C DE;i : 1þ ðdrR =dXÞDE;k þ ðdrR =dXÞDE;i C C DE;k i–k DE;k i–k ð9Þ

Here CDE,k is a weight fraction of any DE. Combination of (8) and (9), followed by some algebraic manipulations, gives an equation

X   a00 þ a0i C DE;i =C DE;k IR=C ¼

X

i–k

1þ

i–k

C DE;i =C DE;k



X   a10 þ a1i C DE;i =C DE;k þ

X ðC DE;i =C DE;k Þ

i–k

1þ

C UE ;

ð10Þ

i–k

where

a00 ¼ KðdrR =dXÞDE;k ; a0i ¼ K ðdrR =dXÞDE;i ; h i a10 ¼ K ðdrR =dXÞUE  ðdrR =dXÞDE;k ; h i a1i ¼ K ðdrR =dXÞUE  ðdrR =dXÞDE;i : The Eq. (10) is obviously similar to (6). However, if we want to use it in a broad range of matrix compositions, we again should better calculate parameters of (10) by LSM using preliminarily calculated Rayleigh and Compton peaks intensities. In this way, the Eq. (10) is applicable in a broad range of UE total concentration. The only limitation is that ZUE should be known. In general, it can be determined for a reference sample and fixed, but in this case only samples with the same ZUE could be analyzed with sufficient accuracy. However, almost all modern spectrometers can detect XRF radiation of elements starting with sodium or magnesium, so their content or at least their contribution to ZUE can be estimated more or less precisely. On the other hand, for the most widespread elements of the second period, i.e. for carbon and oxygen, the differences in the parameters of (10) are not critical (fig. 1). So an error will not raise dramatically if ZUE = 6 is used to represent oxygen or carbonate matrix, or vice versa. Besides, indeed ZUE is often known, for example, if an analyst investigates rocks or soils of known genesis or if he has reasons for expecting some certain light components in the samples. For instance, Perrett et al. expect only water and oxygen to be major undetectable components in Martian geochemical samples [8]. For analysis one should solve a system of Eqs. (5) and (10) with P a condition C i ¼ 1; which is necessary to calculate absolute concentrations from their ratios. On the other hand, such a normalization can be a source of error [1]. Instead, one can find one concentration (e.g. the concentration of one of the major components of the sample) by another XRF technique and use it to derive other concentrations from the corresponding ratios. Since intensity ratios allow one to reduce the influence of the samples’ shape and other factors, it seems better to use a technique where a ratio of the XRF radiation intensity to a scattered radiation intensity is used [6,16]. However, in this case, an algorithm will be more difficult and time-consuming because of the necessity of calibration of the other technique, and the matrix composition range will be narrower. Still our algorithm can be accurate enough even with a conP dition C i ¼ 1. 3. Calculations XRF, Rayleigh and Compton peaks intensities were calculated via a program on VisualBasic.NET. Fundamental parameters and formulae were taken from [15,17–20]. In particular, the intensity distributions of X-ray tubes were calculated as described by Finkelshtein and Pavlova [20]. In the scattering spectrum of a Mo-anode X-ray tube of a ‘‘Spectroscan Max-G” spectrometer characteristic lines of copper were found. Therefore, the intensity distribution of this X-ray tube was calculated as a sum of Mo- and Cu-anode X-ray tubes spectra, so that Mo/Cu Ka-lines calculated intensity ratio was equal to that of the coherent scattering intensities, measured with a polytetrafluoroethylene sample. Parameters of (5) and (10) were calculated using Microsoft Excel 2013. For analysis of five-component steel samples parameters of (5) were calculated in a following way. Three sets of hypothetical

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Fig. 1. Rayleigh-to-Compton intensity ratio dependence on UE concentration for carbon and oxygen.

ternary calibration samples were used, consisting of two elements being determined and one disturbing element. In every set two elements concentration ratios varied (we used five values in a range of 0,1 – 1 for Cr/Fe and Fe/Ni, four values in ranges of 0,001 – 0,0075 and 0,005 – 0,025 for Ti/Fe and six values in a range of 0,0025 – 0,05 for Mn/Fe) and a concentration of the disturbing element was fixed (10%, 30%, 50%, 70%, 90%). Besides, intensities were calculated for one set of binary samples with different concentration ratios and for one set of five-component sample. Then the parameters of (5) were calculated by non-linear LSM using calculated intensities. To reduce an error caused by various assumptions and approximations, the Eq. (5) was calibrated to fit relative intensities, calculated via the fundamental Sherman equation [11,17], using 10–15 values. For analysis, a system of the Eq. (5) is solved iteratively. Usually 3–4 iterations are sufficient for convergence. For analysis of model metal oxide mixture parameters of (5) were determined in the same way, using 12 concentration ratio values in a range of 0,5 – 2,2 for both Cr/Ni and Cr/Co with disturbing metal concentration varying from 0 to 90% and oxygen concentration varying from 0 to 50%. The same calculated intensities were used to determine parameters of (10) by LSM. For analysis of iron ore materials 11 Ca/Fe concentration ratio values were used, with UE total content varying from 10 to 90 wt.%. The same calculated intensities were used to determine parameters of (10) by LSM. Analysis by FPM was performed via a program on VisualBasic. NET. For analysis by FA, macros for Microsoft Excel 2013 on Visual Basic for Applications 7.1 was used. Calculations for analysis by

BFP, Szaloki’s and Bakhtiarov’s algorithms were made as described in [1,3–6,17] with a use of Microsoft Excel 2013 and the program on VisualBasic.NET for calculation of the XRF intensities. 4. Materials and methods For approbation of the developed algorithm standard steel samples, model metal oxides mixture and iron ore agglomerates and concentrates were analyzed. Steel contained iron, chromium, nickel and small amounts of titan and manganese. Oxides mixture consisted of NiO, CoO and Cr2O3. Major components of iron ore materials were iron, calcium, oxygen and silicon. Steel and oxide mixture spectra were measured with a wavelength dispersive spectrometer ‘‘Spectroscan Max-G” (Spectron, Russian Federation) with a molybdenum anode X-ray tube (voltage is 40 kV, incidence and take-off angles are 80° and 30°, respectively). Scanning step was 2 mÅ, spectra were collected for 10 s at each wavelength. Iron ore agglomerates and concentrates spectra were measured with a wavelength dispersive spectrometer ‘‘S8 Tiger” (Bruker, Germany) with a rhodium anode X-ray tube (60 kV, 63° incidence angle, 45° take-off angle). Spectra were measured in a ‘‘Best detection” mode, wavelength range varying from 119 to 25,000 mÅ and total measurement time being equal to 17 min. Samples were preliminarily grinded in a ball mill, mixed with boric acid (9:1) and pressed to form tablets. All the intensities were corrected for background. Compton peak intensity was corrected for double scattering. For this purpose, the fraction of the doubly scattered radiation in the total Compton peak intensity for each element was taken from the paper

Table 1 Results of the analysis of one of the standard steel samples (wt.%). Element

Certified

FA

FPM

Fe Cr Ni Mn Ti

69.42 21.23 6.71 2.01 0.11

74.04 20.91 7.05 2.0 0.11

70.15 21.29 6.59 1.88 0.09

Our algorithm 1 standard

2 standards

70.08 21.27 6.66 1.91 0.09

69.79 21.35 6.73 2.02 0.11

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[21]. This fraction was multiplied by elements concentrations and subtracted from the total Compton peak intensity. Proportionality constants gi from the Eq. (1) were determined as ratios of the measured intensities to the calculated ones with the use of one standard steel sample (for analysis with ‘‘Spectroscan Max-G”) or one iron ore agglomerate sample (for analysis with ‘‘S8 Tiger”). Proportionality constants for Rayleigh and Compton peaks were determined in the same way with the use of sodium sulfate sample (for ‘‘Spectroscan Max-G”) and the agglomerate sample (for ‘‘S8 Tiger”).

for calibration. Higher NRMSE values for titan and manganese are caused by a weak signal due to small concentrations and spectral interferences of MnKa and CrKb lines. Accuracy can be further improved if more than one calibration samples are used. For example, we analyzed the same steel samples with two calibration samples. The results are represented in the Table 1 and in the Fig. 2. They are more accurate than the results, obtained by FA and FPM. Thus, XRF intensity ratios being used, analysis accuracy improves.

5. Results and discussion

5.2. Iron ore materials analysis

5.1. Standard steel samples analysis

For approbation of the developed technique, we analyzed iron ore agglomerates and concentrates of known composition. For comparison, we also analyzed the samples by BFP, Szaloki’s and Bakhtiarov’s algorithms. Some of the results are represented in a Table 2 and in a Fig. 3. One reference sample was used for calibration of our algorithm, BFP and Szaloki’s algorithm, and six reference samples were necessary for Bakhtiarov’s algorithm. As follows from the Table 2 and the Fig. 3, all the algorithms provide almost the same accuracy, when adequate reference samples are used. However, Szaloki’s algorithm fail to give accurate results for iron in iron ore concentrates, because the first approximation of its concentrations, obtained by proportion with one agglomerate sample, was too far from the certified values. That is why the corresponding NRMSE values are quite high. Other algorithms gave adequate results for iron ore concentrates.

Steel samples analysis results are given in a table 1. Normalized root-mean-square errors (NRMSE)

NRMSE; % ¼ 100% 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 1 ðC i  C i;0 Þ2 n C0

;

where Ci,0 are certified concentrations, C 0 is a mean of the certified concentrations, are represented in a Fig. 2. For comparison, we also analyzed the samples by FPM and FA. FA requires a calibration with a large amount of standard samples, but only four samples were used. This is probably a reason of relatively high NRMSE values for FA. In contrast, results obtained by our algorithm and FPM are both accurate enough, though only one standard sample was used

Fig. 2. (Color online) NRMSE values (%) for steel analysis results, obtained by FA, FPM and our algorithm with one and two standard samples used for calibration.

Table 2 Results of the analysis of the iron ore materials (wt.%). Sample

Agglomerate Concentrate

Element

Fe Ca Fe Ca

Certified

49.80 10.10 69.35 0.12

Algorithm Our

BFP

Szaloki

Bakhtiarov

49.05 10.05 69.76 0.11

48.92 9.72 69.78 0.12

48.18 10.01 77.85 0.10

49.18 10.15 69.99 0.12

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5.3. Metal oxides mixture analysis To test if our algorithm is applicable without adequate reference samples we used it for analysis of model Cr-Co-Ni oxides mixture. These oxides powders were put together in a special cuvette and mixed by a glass stick, i.e. it was not homogenized in a proper way and pressed. Thus, it was a powder sample with irregular surface, and there were no adequate reference sample for it. For comparison, we also analyzed the mixture with BFP, Szaloki’s and Bakhtiarov’s algorithms. The results are given in a Table 3. Absolute percentage errors (APE) of the results



C i  C i;0

APEi ; % ¼ 100% 

C i;0

are represented in a Fig. 4. For Bakhtiarov’s algorithm, reference samples necessary for determination of parameters were replaced by calculated intensities, and one reference sample was used to determine proportionality constants between calculated and measured intensities. For Szaloki’s algorithm, two ways of determining the first approximation of concentrations were used (Szaloki I and Szaloki II in the Table 3 and in the Fig. 4). In the first case, concentrations were taken equal to the ratios of XRF lines intensities to calculated XRF intensities of pure elements’ radiation. In the second case, concentrations were estimated using very close reference sample. As follows from the Table 3 and the Fig. 4, only two algorithms can provide adequate results, namely Szaloki’s and ours. However, the former requires a close first estimation of concentrations. Otherwise, it also fails to give accurate results. Besides, Szaloki’s algorithm requires representing the only UE (oxygen) as an element with ZUE  12.5 (a sum of manganese and aluminum), which is quite incorrect.

However, the accuracy in this case is lower than that for iron ore samples. The reason is a lower resolution of the spectrometer and overlaps of Rayleigh and Compton peaks and a false peak at 750 mÅ corresponding to NiKb line (1500 mÅ). 5.4. Application of our algorithm to samples containing unidentified UE For iron ore agglomerates and concentrates and for model metal oxides mixture, approximate ZUE was known to be close to that of oxygen. We wanted to check out if our algorithm can be applied with satisfactory correctness, when UE are not known and inadequate ZUE is used in the model. For this purpose, we analyzed the same samples with fixed ZUE = 6. For iron ore samples, NRMSE values for iron and calcium rose from 1.7% and 3.1% to 6.9% and 6.6%, respectively. Though these values are relatively high, the results are still satisfactory. Besides, if some reference samples are available, relative error can be reduced to 1–4%. For oxides mixture, APE for chrome, cobalt and nickel are 5.7%, 11% and 4.3%. They are thus comparable with the results obtained with ZUE = 8 (6.0%, 8.5%, 2.4%). 5.5. Broadening matrix composition range Parameters of the Eq. (5) occurred to be almost constant for 0.5 – 2 orders of concentration ratios and for all significant concentrations of disturbing elements. Parameters of the Eq. (10) are almost constant for the whole range of UE concentration and for at least 1–2 orders of DE concentration ratios. That is another reason for the quantity of reference samples to be reduced, and that is why

Fig. 3. (Color online) NRMSE values (%) for iron ore materials analysis results, obtained by BFP, Szaloki’s, Bakhtiarov’s and our algorithms.

Table 3 Metal oxides mixture analysis results (wt.%). Element

Cr Co Ni

Added

22.95 26.17 26.08

Algorithm Our

BFP

Szaloki I

Szaloki II

Bakhtiarov

24.33 28.39 26.70

2.3 4.5 4.7

21.63 14.38 13.47

24.12 25.84 24.71

24.48 55.26 13.40

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Fig. 4. (Color online) APE values (%) for the results of analysis of model metal oxides mixture.

a very close first approximation is not required. However, after a pair of iterations, if necessary, the parameters can be refined. 6. Conclusions We proposed a novel algorithm of XRF analysis based on using calibration equations with theoretical coefficients, utilizing using XRF lines intensity ratios and Rayleigh-to-Compton intensity ratio. These relative analytical signals reduce the influence of different experimental factors hard to control or take into account, such as primary radiation fluctuations and sample shape variations. The algorithm provide sufficient accuracy of analysis, comparable with the most precise XRF techniques. It does not require large sets of reference samples and can be used for analysis of various nonconventional samples, powders. Samples containing large amounts of UE can be analyzed with sufficient accuracy without adequate reference samples, while other techniques like Szaloki’s algorithm do require adequate reference samples and often very close first estimation of concentrations. Analysis without adequate reference samples with sufficient accuracy is also possible. 7. Funding This work was supported by the Russian Scientific Foundation (Grant No. 14-23-00012) Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.nimb.2017.02. 072. References [1] R.M. Rousseau, Corrections for matrix effects in X-ray fluorescence analysis – A tutorial, Spectrochim. Acta B 61 (2006) 759–777, http://dx.doi.org/10.1016/j. sab.2006.06.014. [2] N.P. Il’in, An alternative version of X-ray fluorescence analysis, J. Anal. Chem. 66 (2011) 894–917, http://dx.doi.org/10.1134/S1061934811100054.

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