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ANALYTICAL CHEMISTRY, VOL. 51, NO. 7, JUNE 1979
Differential Delta-Coefficient Method for the Correction of Matrix Effects in X-ray Fluorescence Analysis Fernand Claisse Department of Mining and Metallurgy, Universitl Laval, Qulbec, Canada, G 7K 7P4
Tran Phuc Thinh” Laboratoire Central, Ministkre des Transports, Gouvernement du Qulbec, Qulbec, Canada, G I P 3 W8
Unknown samples are compared to only one standard of similar type; their compositions are calculated from ratios of X-ray line intensities and equations obtained by derivation of the well known Lachance and Traill, Rasberry and Heinrich, and Claisse and Quintin relations. Corrections for matrix effects are small, to the extent that several correction terms can often be omitted. The method probably leads to the ultimate accuracy that can be reached In XRF analysis.
RA = I A ( ~ ~ ~ ~ element) I ~ ) / I A ( ~ (2) ~ ~ ~ The composition of the sample is referred to that of the standard (*) by: c, = sc, (3)
e,*+
Writing CA* as a function of the C,*’s as in Equation 1,taking the ratio of CAICA*and combining with Equations 2 and 3,
Numerical methods of matrix effect corrections in X-ray fluorescence analysis are numerous, and the analyst must usually make a compromise between high accuracy and simplicity of calculations. The Lachance and Traill (L-T) method ( 1 ) is by far the simplest, and being theoretically sound, leads to acceptable results most of the time; the Rasberry and Heinrich (R-H) relation (2) is also simple and is used for certain combinations of elements; the equation of Claisse and Quintin (Q-C) ( 3 ) is very accurate but the large number of influence coefficients required is a disadvantage; other important ones are the fundamental parameter method of Criss and Birks (4) and the variable coefficient method of Tertian ( 5 ) ,both of which are difficult to apply on a routine basis. Except for the last two methods, all the others are approximations of the theoretical equation of fluorescence intensities as a function of composition and, as a consequence, they do not have the same accuracy a t all levels of concentrations. However, they contain parameters that can be adjusted for better application to smaller ranges of compositions. Consequently, if any of these relations is made to fit exactly a given standard, it should apply accurately to other samples of the same type with similar compositions. The unknown samples can be considered as the standard with a slight change in composition. Therefore, the differences of X-ray fluorescence line intensities between the sample and the reference standard are related to small differences in concentrations and to small differences in the matrix effects. Since these corrections for matrix effects are small, it should be easier to make them accurately than when larger corrections must be made as when other methods of calculation are used. Such corrections will be expressed as differential equations developed from the relations of Claisse and Quintin, of Lachance and Traill and of Rasberry and Heinrich. A similar approach has been made by others (6, 7) but their theoretical work was limited to only one relation between fluorescence intensity and concentration (Equation 8). The present paper is more general and it includes all types of relations, more explanations are given on the manipulation of the numerical coefficients, and more emphasis is made on the great advantages of the method. THEORY The Claisse-Quintin Differential Relation. The C-Q relation ( 3 ) for element A in a sample is: 0003-2700/79/0351-0954$01.00/0
{[(l+ Za,C,* + ,za,,c,*2...I + ( Z a , A C , + 2&Y,,C,*AC, + ...) I / ( 1 + Z”,C,* + Z”,,C,*2 + ...)) (4)
C A = C**(Z,/Z*,)
Noticing that the function inside the first bracket of the numerator is the same as the denominator and expressing the “apparent concentration” of A as
one obtains:
with
Since the values of (ACJ2and aijare small, the linear terms only are retained in Equation 6 and the terms containing mixed indices like a,, in the denominator of Equation 7 are neglected. Note that in the expression of dB (Equation 7), the numerator contains terms for element B only but in the denominator, the summation extends over all the elements in the standard except element A. The Lachance-Trail1 Differential Relation. The L - T relation (1) differs from the C-Q relation (Equation 1) by the absence of second-order terms: CA = RA (1 + Ba,C,) (8) Then, Equation 5 and 6 remain the same and Equation 7 becomes 6B
=
“B
1+
c aici*
ifA
The Rasberry-Heinrich Differential Relation. The R-H relation ( 2 ) for the sample is:
C
1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 7, JUNE 1979
where the terms with subscript i are for absorbing elements and those with subscript j are for enhancing elements. Combining Equations 3 and 10:
As a first approximation (Taylor series), we can write
DETERMINATION O F T H E a COEFFICIENTS N coefficients can be calculated from fundamental parameters ( 5 , 7-9) or obtained experimentally. The latter method, which we will describe, is usually simpler. The experimental determination of CY parameters is easier with binary specimens of different compositions covering all the elements in the samples. Obtaining these specimens is the major difficulty encountered in natural samples, but when the samples are prepared as borate fusion disks, such binary specimens are easily made from pure oxides. G r a p h i c a l D e t e r m i n a t i o n . For a binary system, Equations 1, 8, and 10 become C-Q:
Replacing Equation 12 in Equation 11 and proceeding as for the C-Q relation:
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+ ap,CB + a B B C & ) CA = R A (1 + CUBC B )
CA = RA (1
L-T:
(15) (16)
R-H: Equation 16 when absorption dominates. with
CA=RA
(
I + - TBCACB) when enhancement dominates
APPLICATION O F T H E DIFFERENTIAL RELATIONS T h e calculation of any of the three differential relations is the same but since the C-Q differential relation (Equation 6) is more accurate, yet as simple as the others, the calculation will be referred to that relation only. The procedure is similar to the iteration procedure described by Lachance and Trail1 ( I ) with a coefficients and is detailed below. The 6 coefficients and the composition of the reference standard are known. Step 1. Measure the net line intensities of the elements in the sample and in the standard. Step 2. Calculate the apparent concentrations in the sample using Equation 5. Step 3. Calculate the ACLvalues using Equation 3. T h e first time this equation is applied, use the apparent concentrations for the concentrations C,. Step 4. Calculate the concentrations using Equation 6. Use the AC values obtained in step 3. Step 5. Repeat steps 3 and 4 as many times as necessary until the C values become stable. This composition is the one looked for. In each iteration, always use the same values for Capp (Equation 6). CALCULATION O F T H E 6 COEFFICIENTS Equations 7 , 9, and 14 indicate that the 6 coefficients depend on t h e cy coefficients; the latter can be obtained as described in the next section. T h e 6 coefficients also depend on the composition of the standard. For routine analysis, it is obvious that one reliable standard should be chosen and all the samples referred to it. Then a set of 6 coefficients is calculated once for all. However if a new standard is used, it is not necessary to determine the a coefficients again; it is necessary only to calculate a new set of 6 coefficients using the same N values and the composition of the new standard.
(17)
In the case of Equation 16, the N coefficient can be obtained by plotting (CAIRA) - 1 as a function of CB;the CY coefficient is equal to the slope of the best straight line through the points. T h e procedure is the same for Equation 17 except that ((CAIRA) - 1)(1+ CA) is plotted as a function of CB. The two coefficients of Equation 15 are obtained by plotting (1/ CB)(CA/RA))- 1) as a function of CB; N B is equal to the ordinate at the origin and N B B is equal to the slope of the best straight line. Regression Analysis. The determination of a coefficients by regression analysis from a set of specimens containing several elements is questionable because small errors in X-ray intensities sometimes yield large errors in calculated coefficients. However, regression analysis is an excellent method to determine a coefficients when it is applied to a binary system. Regression analysis is specially useful when the samples with high concentrations of element A cannot be fused; then, IA(pure) in Equation 2 must be considered as an additional unknown parameter. DISCUSSION The potential of the differential 6-influence-coefficient method as compared to the a-influence-coefficient methods is very promising. 1. One reference standard only is measured, as compared to one pure element sample for each element when the a methods are used. It is true that several binary specimens are also required to determine the a coefficients but such measurements are made only once. It will be shown later that in some cases it is not even necessary to determine the a coefficients. 2. The corrections for matrix effects are much smaller. The cy and 6 coefficients appear in equations of the same form (compare Equations 6 and 8 ) , and Equations 7 , 9, and 14 indicate that they have about the same value and the same accuracy. However, the correction terms are given by aC in one case and by 6AC in the other. The values of AC will usually not be allowed to be higher than 0.1 while C can be as high as 1. As a result of the smaller corrections, errors on coefficients or on concentrations will yield smaller errors on CA. Examples of this are reported by Heinrich and Rasberry ( I O ) and by Frechette et al. (11) This situation can be explained very simply by means of Figure 1 which represents the fluorescence intensity I as a function of composition C for a binary system: a binary system is chosen because it is easier t o understand, but a more
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 7, JUNE 1979
CONCENTRATION
Flgure 1. Comparison between the a and 6 methods. A sample with
composition C, will be observed as having composition C, (a method) or C3 (6 method) if an inexact a (or corresponding 6) value is used. S represents the reference standard complex system would be explained in the same way. Let us take I A as the measured intensity for a given element A. Using the a method with the exact a value (curve A), the exact concentration C1 is obtained; if an inexact a value (curve B) is used, the inexact concentration Cz is obtained. In the 6 method, the calculations are based on the curvature of the I vs. C curve in the vicinity of the point corresponding to the reference standard (represented by Cs and Is). If the exact a coefficient is used in the calculation of the 6 coefficient, the variation of I with C about the standard S is obviously the same as in curve A; consequently, the value of IAwill again give the exact concentration C,. However, if the inexact a value is used in the calculation of 6, the curvature will be about the same as the curvature observed on the curve B (thicker section) but it will be used about the S point. For the intensity I*, one finds the concentration C3 which is very close to C1. Indeed Figure 1 shows that very large errors on the a coefficients are required to induce significant errors on concentrations when the 6 method is used. 3. A most interesting consequence of the discussion in the preceding paragraph is that 6 parameters with relatively low accuracy can be used without decreasing significantly the accuracy of the calculated concentrations. This means that sometimes it is not necessary to determine the a coefficients; they can be obtained from the literature, even for spectrometers with different geometries. For example Lachance (7) has calculated a few a coefficients for three combinations of incidence and emergence angles which cover .most spectrometers on the market. The largest difference from the median was f0.04 for a values between -0.6 to + 1.8. In the worst situations, with a ACB of 0.1, this represents a relative error of about 0.04 X 0.1 = 0.004, or 0.4% on CA as calculated from Equation 6. 4. Because the matrix corrections 6AC are very small when AC is small, corrections for minor elements can be omitted more often than in the a methods. Even a major element can be ignored in the calculations if its concentration is always constant and equal to that in the standard. 5 . The fluorescence intensities of the pure elements are not required in the &coefficient method. Since these intensities are always higher and sometimes considerably higher than those of the same elements in the sample, errors on dead-time corrections are less significant in the 6 method. 6. Further to the above advantages, the calculation of 6 coefficients from Claisse-Quintin a coefficients (Equation 7)
has the additional advantage that corrections are accurate for a larger interval of composition about that of the reference standard. A solution to the problem of matrix corrections by comparing a sample with a standard of the same type is not new and several attempts have been made. Tertian ( 5 ) arrived at Equation 6 earlier but he did not exploit his finding further because he thought that the “the validity of the coefficients was relatively narrow in terms of composition”. Instead he (6) developed his “Self-consistent Calibration Method” in which a coefficients for use with the Lachance-Trail1 expression (Equation 8) are obtained from a reference standard and from samples made by mixing the standard with elements in known proportions. The essential idea in that approach is to obtain corrections that apply in a region of composition next to the composition of a reference standard in such a way that the corrections are zero when they are applied to the standard itself. This is exactly what the 6 method does; as a result, the Tertian’s Self-consistent Calibration Method and the 6 coefficient method are theoretically equivalent. The main difference is in the magnitude of the corrections which are several times smaller in Equation 6 than in Equation 8. As a consequence, when Equation 8 is used, all the elements must be measured except those for which a C is very small. On the contrary, the occurrence of small 6AC corrections in Equation 6 is frequent and only the elements whose concentrations vary significantly must be measured. It appears to the authors of this paper that Tertian’s method (6) to obtain the a coefficients is a good alternative to the use of binary samples. A combination of that method with our method for the calculation of 6 coefficients (Equation 7 ) and of final concentrations (Equation 6) could perhaps yield a method which would have most of the advantages of the two. Anderson, Mander, and Leitner (12) also use a n addition technique to obtain 6 coefficients instead of a coefficients, for use with Equations 5 and 6; however their Equation 3 for the calculation of 6 coefficients is not correct when used in combination with their correction for dilution. Lachance (13) has applied the 6 coefficient method for several years in an indirect way: the X-ray intensities from a standard are measured and used in Equations 8 and 2 with the known concentrations and known a coefficients to calculate intensities for each element. The same equations are used again with the calculated IA(pure) intensities to determine the unknown composition of the specimens. T h e mathematical calculations are less straightforward and more iterations are needed to obtain the final composition. Apart from that, the Lachance approach is identical to the 6 method based on Equations 8 and 9. Amoury and Pryor (14) use Equations 5 and 6 and obtain 6 coefficients by regression analysis on known samples. Instead of using the actual values of cA*/IA* in Equation 5 they consider these terms as unknown parameters. In their procedure a large number of standards is required; the questionable applicability of regression analysis to samples with similar compositions is reflected as deviations in their analytical results. Heinrich and Rasberry (10) have proposed their DeltaAdaptation Method based on a relation in the form A(CA/RA) = 1 ~ ( Y B A C B (18)
+
Theoretically i t is equivalent to our 6 method based on Equation 11 but practically i t has some disadvantages. (a) RA must be known which means that the intensity of the pure elements must be measured. (b) The calculation of ACA from A(CA/RA) is more complex than in Equation 3 and is undesirable when iterations are calculated by hand or with a pocket calculator.
ANALYTICAL CHEMISTRY, VOL. 51, NO. 7, JUNE 1979
CONCLUSION T h e 8-correction approach has great advantages: it takes the best from the numerical correction methods, and leaves behind most of their inconveniences; it is simple to apply and very accurate; it is particularly well suited to the analysis of industrial products where composition variations are usually not very large. The authors believe that the 8 method will be in common use in the near future. A paper on the analysis of cements appears in this Journal (11);although that paper is not intended to be a typical example of application of the 6 method, it nevertheless shows that the method offers definite advantages in relation with accuracy and simplicity.
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R . Tertian, X-Ray Spectrom., 4, 52 (1975). W. K . Dejongh, X-Ray Spectrom., 2 , 151 (1973). G. R. Lachance, Can. Spectrosc., 15, 64 (1970). R. Rousseau and F. Claisse, X-Ray Spectrom., 3, 3 1 (1974). K . F. J. Heinrich and S. D. Rasberry, Adv. X-Ray Anal., 17, 309 (1974). G.Frichette, J. C. Gbert, T. P. Thinh, R. Rousseau, and F. Claisse, Anal. Chem., 5 0 , following paper in this issue. (12) C. H. Anderson, J. E. Mander and J. W. Leitner. Adv. X-Ray Anal., 17, (6) (7) (8) (9) (10) ( 11)
217 (1974). (13) G. R. Lachance, Energy, Mines and Resources Canada, Ottawa, Canada, personal communication, 1978. (14) M. A. Amoury and K. S. Pryor, Denver X-Ray Conference, 1973.
RECEIVED for review January 4, 1978. Accepted February 22, 1979. A part of this paper has been presented (no. 431) a t the Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland, Ohio, March 1-5, 1976. Grateful acknowledgment is made for the support from the Department of Transportation, Government of Quebec, and the Canadian National Research Council (Grant No. A-931 to F.C.).
LITERATURE CITED G. R. Lachance and R. J. Traill, Can. Spectrosc., 1 1 , 43 (1966). S. D. Rasberry and K. F. J. Heinrich. Anal. Chem., 46, 81 (1974). F. Claisse and M. Quintin. Can. Spectrosc., 12, 129 (1967). J. W. Criss and L. S. Birks, Anal. Chem., 40, 1080 (1968). (5) R. Tertian, X-Ray Spectrom., 2 , 95 (1973).
(1) (2) (3) (4)
X-ray Fluorescence Analysis of Cements Guy Frbchette, Jean-Claude HGbert, and Tran Phuc Thinh" Laboratoire Central, Ministbre des Transports, Gouvernement du Quebec, Qudbec, Canada, G 1P 3 W8
Richard Rousseau Institut National de la Recherche Scientifique. UniversitB du QuBbec, Qudbec
Fernand Claisse Dgpadement de mines et m6tallurgie, Universitd Laval, Qudbec, Canada G 7K 7P4
The combination of a borate fusion and "Delta-Coefficient'' corrections for matrix effects is shown to yield a method for the analysis of cements in which high accuracy, rapidity, and slmpllcity are compatible. The features of the method are: (a) Counting statistics is nearly the only source of errors. (b) Weighing of flux and sample are nearly the only manipulations. ( c ) One standard only is required. (d) When cement composition is rather constant, the method described can be shortened significantly. The analysis of the six major elements In NBS-633-639 cements was performed with an average accuracy of 0.07 YO.
X-ray Fluorescence is currently used in the analysis of cements because of its rapidity and reproducibility. A complete analytical procedure incorporates the following steps: (a) sample preparation: grinding, pelletizing or fusion; (b) X-ray measurements; (c) determination of sample composition by comparison with other cements, by calibration lines, or by calculations with influence coefficients. In addition, preparatory work must be done once only to work out t h e methods, for example, obtaining the required standards or determination of influence coefficients. Accuracy and rapidity are both desirable but as they are usually uncompatible one makes a compromise between the possible procedures. For example, pelletizing which is associated with some inaccuracies due to particle size effects is often adopted for sample preparation instead of the more accurate b u t sometimes slower fusion process. For similar 0003-2700/79/0351-0957$01 .OO/O
reasons, corrections for matrix effects are sometimes based on an empirical relation already in the computer instead of a more fundamental and more accurate relation. The ideal situation is one where each step of the procedure is at the same time the simplest, the fastest, and the most accurate of all the available ways to make it; the purpose of this paper is to present a procedure which is close to this ideal situation. T h a t procedure is a combination of a borate fusion and corrections for matrix effects by t h e 6 method described by Claisse and Thinh ( 1 ) . I t features rapid, simple, and reproducible sample preparation; complete freedom from particle size effects; simple and accurate corrections for matrix effects; calibration through one standard only; accuracy comparable to the better wet chemical methods. First, a procedure is described with the objective of obtaining maximum accuracy. Second, it is shown how the procedure can be shortened considerably with nearly no loss in accuracy.
EXPERIMENTAL Sample Preparation. The fusion method ( 2 )was adopted using mixtures of 40% calcined cement and 60% pure lithium tetraborate as flux (2.8 and 4.2 g, respectively); the reasons for this choice are: (1) Complete elimination of particle size effects. (2) High ratio of sample to flux results in strong X-ray lines with easier detection of light elements in low concentrations such as Mg, and lower statistical errors on counting. (3) Absence of a heavy absorber or other additives in the flux also results in stronger X-ray lines and minimizes the number G 1979 American Chemical Society
