Critical Evaluation of Correction Methods for lnterelement Effects in X-Ray Fluorescence Analysis Applied to Binary Mixtures Cesia Shenberg and Saadia Arniel Soreq N u c i e a r R e s e a r c h C e n t r e . Y a v n e , l s r a e i
Different methods of correction for interelement effects in X-ray fluorescence analysis, as applied to two-element samples, were examined and evaluated. These correction methods were applied to the determination of Cu in Cu-Fe solutions and solids. The methods of calculation based on the a correction factor were checked over a wide range of Cu and Fe concentrations with the result that the cr factor was found to vary considerably. Introducing peak ratios instead of intensity into the N correction method formula extended this method over a wider range of concentrations, but it is limited to samples of the same chemical composition as the standards. The original peak ratio method of correction was independent of both the concentration range and chemical composition: accuracies of 3.5% were obtained with samples containing two main elements. This was compared with 4.0’10 error obtained when the cy modified correction method was used for samples of the same chemical composition as standards. The error was about 50% for samples of different chemical composition than standards. The original N method gave even much larger errors and therefore cannot be applied to the concentration range covered by the peak ratio method.
By using X-ray fluorescence analysis for the determination of a single predominant element in a sample, the corresponding intensity is essentially a linear function of its Concentration. a t least over a limited range. When the concentration of a n element reaches several percents, variations in the self-absorption and the backscattering effects become significant. The determination becomes further complicated when dealing with samples containing more than one major elemental constituent. Since the measured intensity for each element can be affected by the absorption or fluorescence of its X-rays by the other elements present, it becomes necessary to take into account both matrix and interelement effects. A number of papers have been published describing the use of correction procedures for the compensation of interelement effects in X-ray spectrometry. Sherman (I, 2) attempted to derive a general mathematical relationship between the measured intensity and concentration of an element in multielement samples. First, he derived the intensity-concentration relationship (I-C) for the element having the highest atomic number. The I-C relationships for elements of successively lower atomic number were then corrected for enhancement by the spectra of elements of higher atomic number. However, the equations were successful only for limited concentration ranges in two or three component systems (3). (1) J Sherman Arner Soc Test Mater Spec Tech Pub1 157. 27-33 (19541 ( 2 ) J Sherman Spectroch/m Acta 7. 283-306 (1955) ( 3 ) E P Bertin Principles and Practice of X-Ray Spectrometric Analysis Plenum Press New York N Y 1970
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Beattie and Brissey ( 4 ) analyzed three and four component alloys by introducing so called cy factors for interelement correction into the linear simultaneous equations. For a binary system A-B these factors have the form
where cyAB is the correction factor for element A in the presence of element B. CYBA is the correction factor for element B in the presence of element A; W is the weight fraction and the respective R’s are defined by R.4 =
I.LAJI.Ls; Ru = IRRIIR.~
where I A A and I,, are the peak intensities of a sample of pure A and of A in the actual sample, respectively, and Z B ~and Z B ~are the same for element B. The following assumptions were made: 1) the sample is homogeneous and infinitely thick, 2 ) the primary X-ray radiation is monochromatic, and 3) enhancement by interelement secondary excitation within the sample has the same effect as low matrix absorption. They obtained only a n approximate analysis of chromium-iron-nickel-molybdenum alloys. The range studied was: 3 7 4 7 % Fe, 16-32% Cr, 13-9570 Xi, and 2-7% Mo. It is possible to evalute N coefficients with high accuracy when the major constituents have approximately the same concentration in all samples. Lucas-Tooth and Pyne ( 5 ) have applied the cy correction method to the determination of chromium in the range of 10-20’30 in high alloy steels containing between 50-8070 Fe. Their basic equation was derived from one additional assumption: when elements B and C are added to a sample containing only element A, the absorption of the spectral line of element A by element B or C is assumed to be proportional to the weight fraction of B or C. In this model the I-C relationship was expressed as:
where ZA, Zri, and I,, are the measured intensities of the elements A, B, and C in the actual sample. I A , is the expected intensity of element A if there were no interelement effects; CY = correction factors for element A in the presence of elements B and C. Variations of this expression have been reported by other authors (6-10). A method for converting measured X-ray intensities to mass concentrations has been described by Lachance ( 6 ) and applied by Lachance and Traill (7, 8) to the analysis of a multicomponent alloy system. The basic equation of the Lachance-Trail1 relationship was used by Hughes (9) to (41 H . J . Beattie and R M Brissey, A n a / Chem.. 26. 980 11954) ( 5 ) H J Lucas-Tooth and E . C Pyne, Advan X-Ray A n a l . . 7 , 523 ( 1964). (61 G R Lachance, Geol. Surv Can. Pap.. 64-50 (1964). (7) G R Lachance and R J Traill. Can Spectrosc , 11 ( 2 ) . 43 (1966) ( 8 ) R J Traill and G R Lachance Can. Spectrosc. 11 ( 3 ) . 63 (1966) (9) H Hughes Analyst ( L o n d o n ) ,97. No 1152. 161 (1972).
A N A L Y T I C A L C H E M I S T R Y , VOL. 4 6 , NO. 1 1 , S E P T E M B E R 1 9 7 4
Table I. C u and Fe Values Obtained with Standards of Different Concentration Ranges Concn found, '/o
Actual concn, %
cu
Fe
Std No. 1
0.62
0.507
Std No. 2
2.48
6.09
Std No. 1
0.62
0.507
Std No. 3
1.24
Sample
1.24
+ Std No. 2.
}
0.357
0.543
3161
8535
... ...
...
0.268
-- 1 8 . 7 3
- 25098
8210
... ...
.. ..
.. ..
...
10.1
..
2.03
...
...
Using Std No. 1
an
Using Std NO.1
..
Elemental concentrations
a
0.260 0.268
~~
0,275
$Xu = 0.93kc,b~ekbek'1"r
0,291 0.313 0,357
describe a' practical method for determination of correction factors by using synthetic standards. Jenkins and Campbell-Whitelaw (10) employed the Lachance-Trail1 method for the graphical calculation of correction factors. They clearly emphasize the dependence of the a factor on the following expression:
C" I'
=
[+'C.;]:'['C,X
-
2.13a 4.94b
Another method of correction, the peak ratio method recently described (12), permits accurate analysis in mixtures and ores. This method was applied to the determination of Cu in Cu-Fe solutions and solids using backscattered As X-rays as reference. The method is based on the prior estimation of the concentration of the interfering element. This permits correction of the Cu K X-rays intensity in Cu-Fe solutions and in Cu ores. The following formula was derived:
~~
+ 0.507% Fe + 2.03% Fe 0 . 6 2 % Cu + 0.507% F e 1 . 2 4 % Cu + 1 0 . 1 % F e 0 . 6 2 % C u + 0.507% F e 1.24% Cu + 20.3% Fe 0 . 6 2 % C u + 0.507% F e 1.86% C u + 2 0 . 3 % F e 0.62% Cu + 0.507% Fe 4 . 9 6 % C u + 20.3% F e 0 . 6 2 % C u + 0.507% F e 2 . 4 8 % Cu + 6 . 0 9 % F e
1.24a 1.70b
+ Std No. 3.
Table 11. (Y Correction Factor Calculated for Different Fe and Cu Concentrations 0.62% Cu 1.24% Cu
...
...
c,-)+ 11
where Pc and P are intensities of the element in sample and standard, and CX and CB are the respective concentrations, i represents the main and j the interfering element. The intensity varies in a linear manner with concentration only in a narrow concentration range and it is obvious t h a t matrix and interelement effects will limit the cy correction method. It has been reported (11) t h a t the influence of self-absorption can be minimized by a comparative method using target X-rays backscattered from the sample, as a reference for calculating the exact elemental concentration. Therefore, a modification of the original a method, which introduces ratios of intensities of the X-ray fluorescent line to the target backscattered line, instead of absolute intensities, would give improved results over a greater concentration range. (10) R Jenkins and A Campbell-Whltelaw, Can Spectrosc (1970) (11) C Shenberg and S Amlel Anal Chem 43, 1025 (1971)
where a, = (Fe/As) for x% of Cu; b, = (Cu/As) for y% of Fe; kFe and kcu are the proportionality coefficients obtained from pure Fe and pure Cu standards, respectively. ksl is the slope obtained from the relation between the corrected Cu/As peak ratio and the estimated Fe concentration. High accuracy was obtained over a wide range of Cu and Fe concentration even when standards were of different chemical composition than the samples. This method, by using corrected intensity ratios based on pure one element standards, seems a priori to be advantageous and insensitive to the factors affecting the 01 correction method. The present work is an evaluation of the 01 correction methods (4-IO), their possible modifications and the peak ratio method (12). The methods were compared as applied to the accurate determination of Cu in the presence of Fe in liquid and solid samples.
EXPERIMENTAL AND RESULTS The experimental procedures and the equipment used in the present work were described previously (12). cy Correction Method. A sample containing Fe and Cu was chosen to check the a correction method. Fe and Cu concentrations were calculated from the previously published data (12) by the method of Jenkins e t al ( I O ) , using two different pairs of standards. The results are shown in Table I. As can be seen, two different values for Fe and Cu concentrations were obtained when using standards of different concentration ranges. Table I1 shows the a values calculated for various Cu and Fe concentrations. The cy factor when checked over a wide range of concentrations was found not to be constant. The a factor in the range studied is strongly influenced by different Cu concentrations. This effect is clearly expressed in Table I11
1 5 ( 2 ) 32 (12) C Shenberg, A Ben Haim. and S Amiel, A n a / Chem ( 1 973)
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
4 5 , 1804
1513
~
~
T a b l e 111. CY Coefficients a s a Function of Varying C u C o n c e n t r a t i o n s at C o n s t a n t F e C o n c e n t r a t i o n a
'hFe
5% c u Sample
C,
Standard
1.24 1.24 7.44 1.24 1.86 4.96
1 2
3 4 5 "
Intensity of Cu K X-rays (counts/lO min) 11
I,
2749 6015 20004 1553 2231 5645
11400 4022 2737 15162 14676 13893
3'
10.1 2.03 2.03 20.3 20.3 20.3
Intensity of Fe K X-rays (counts/lO min) A(C3'
C,' I,a __ I,Y c,a
- C,*)
a
0
1.00
...
-8.07 -8.07 10.2 10.2 10.2
0,457 0.825 1.770 1.848 1.948
0.0673 0.0217 0.0755 0.0831 0.0929
Calculated according to R. Jenkins e t a l . (IO)(x indicates samples and s, standards).
T a b l e IV. I n t e r e l e m e n t Effects in t h e Modified CY M e t h o d f o r Different Fe and C u C o n c e n t r a t i o n s c i x R .I a
No.of sample
% CU, Ci
% Fe, C j
Cu/As, R i
Fe/As, R j
Standard
1.24 24.8 12.4 7.44 1.24 1.24 1.86 4.96 2.48 1.24 0.62 0.95 0.95 0.95 0.95 0.95 0.95 0.95
10.14 3.04 2.54 2.03 0.507 20.3 20.3 20.3 6.09 2.03 0.507 22.8 15.21 10.14 5.07 2.03 1.01 0.507
2.280 55.617 27.862 18.568 3.175 1.559 2.384 6.824 5.031 2.988 1.540 1.253 1.491 1.772 2.077 2.325 2.410 2.502
9.457 4.354 3,244 2.541 0.443 15.222 15.680 16.791 6.186 1.999 0.412 18.119 13,671 9.777 5.111 1.981 0.881 0.426
1
2 3
4 5 6 7 8
9 10 11
12 13
14 15 16 17 a
C
=
. I
1
-10 -9 -8
12 I1
I
I
-7 -6 -5
-4
1
I
1
I
-3 -2
-
-
1'0 -I
09 r
I
I
I
I
2
3
I
I
I
I
I
4
5
6
7
8
-
I 9
1 IO
(10) 0
__
R i X Ci'
0
1.000
-7.10 -7.60
0 I818
-8.11
-9.633 10.16 10.16 10.16 -4.05 -8.11 -9.633 12.66 5.07 0.
-5.07 -8.11 -9.13 -9.633
0.820 0.737 0.718 1.463 1.435 1.337 0.907 0.763 0,740 1.394 1.172 0.986 0.841 0,751 0,725 0.698
which shows different a values for various Cu concentrations a t a given Fe concentration. Therefore the influence of the main element concentration on the CY factor cannot be neglected (Figure 1). Accordingly it is necessary to know a priori the approximate concentrations of both the main and the interfering elements, since the factors are constant only in a limited range of the two elements and the standard concentration must be near the midpoint of this range. These results are in agreement with the conclusions previously reported (11) that intensities of the X-ray lines vary with elemental concentration in a nonlinear manner when checked over a wide concentration range. Modified CY Correction Method. The range of concentration for which accurate results can be obtained is increased to a great extent with a slight modification of Lucas-Tooth and Pyne's ( 5 ) or Lachance-Traill's (&8) correction procedures. Substitution of peak ratios of a n X-ray fluorescent line to a target backscattered line into their formula instead of intensity or concentration, gives:
Figure 1. Graphical presentation of the CY correction factor calculated according to R. Jenkins and A . Campbell-Whitelaw
1514
Cj')'
concentrations of the main (i) and the interfering ( j ) elements; Rj = Cu/As ratio; x indicates samples and s, standards.
.
I
-
A(Cjx
A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 11, S E P T E M B E R 1974
C,' =
R" C,'[l RIS
+ C Y ( R-, ~R,')]
(1)
05t
Graphical presentation of the cy correction factor calculated according to the modified method Figure 2.
h,
*omm
8
~ o ~ G o o d o N ~ m
t-
t-
o m * * r i m o o m m c o u .a o. m. o. N N W
where C represents concentrations of the main element (i); R, = Cu/As ratio; R, = Fe/As ratio; x indicates samples and s = standards. a is the correction factor. The results according to Equation 1 are shown in Table IV and graphically represented in Figure 2. A more nearly constant a was obtained over a wide range of concentration. This modified procedure was extended to Fe-containing ores, using liquids as standards for analyses of solid samples by introducing an additional constant factor into Equation 1. This factor expresses the difference in attenuation for solids and solutions and was calculated according to the conclusions reported previously (12). In order to evaluate different forms of the a correction method, a factors were calculated according to three different modifications, a1, a2, 013, based on the methods of LucasTooth and Pyne, Lachance and Traill, and the authors, respectively. The three coefficients are defined in Table V. In this table, the three coefficients are calculated for each of 10 samples. Eleven samples were used having known Cu and Fe concentrations over a wide range. In the equations in the table, the first sample, the standard, was designated as s and the other 10 samples as x. Thus, for each of the three variations of a , 10 values were obtained for a wide Cu-Fe concentration range. The largest discrepancies were obtained for a1 since the calculations were based on intensities alone. These discrepancies were smaller for a2 which corrects for concentrations of the interfering elements. Only a3 calculated according to the peak ratios, gave results comparable over the whole range of concentrations studied. Consequently, the a correction method in its original form cannot be applied to the determination of Cu and Fe in the wide concentration range covered by the peak ratio method (12). Peak Ratio Method. To make a comparison between two methods, it was necessary to use the peak ratio method and the a correction method modified according t o the peak ratios. The results are shown in Table VI. There is no difference between the results obtained by the peak ratio method and by the modified cy method when standards and samples were of the same chemical composition. Large errors (-50%) were obtained for samples of chemical compositions different than the standards using the modified a method.
CONCLUSIONS Correction methods based on a coefficient can be used with a n accuracy of a few percent in industrial processing control. This is useful when the analyzed samples have approximately known concentrations. Prior knowledge of the chemical composition of the analyzed samples and the concentration range allows preparation of standards that will cover this range. A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 11, S E P T E M B E R 1974
1515
Table VI. Calculation of Cu Content i n Liquid and Solid Samples by Two Different Methods
Standardc Liquid samples 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Solid samples 18d 19d 20d 21d 22d 23d
70
cu
Error, %
Yo Cu calcd
Actual concn Sample no.
b
b
Fe
...
1.24
10.14
0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.62 1.24 2.48 4.96 1.86 1.24 1.24 7.44 12.40 24.8
0.507 1.01 2.03 5.07 10.14 15.21 22.8 0,507 2.03 6.09 20.3 20.3 20.3 0,507 2.03 2.54 3.04
0.96 0.95 0.96 0.97 0.97 0.92 0.87 0.59 1.23 2.45 4.59 1.56 1.01 1.22 7.84 12.12 25.3
0.92 0.93 0.94 0.97 0.95 0.95 0.58 1.20 2.38 4.93 1.65 1.06 1.20 7.60 11.74 24.5
+1.1 0.0 +1.1 +2.1 +2.1 -3.2 -7.4 -4.8 -0.8 -1.2 -7.5 -16.1 -18.5 -1.6 +5.4 -2.3 +2.0
-3.2 -2.1 $1.0 +2.1 0.0 0.0 -6.5 -3.2 -4.0 -0.6 -11.3 -14.5 -3.2 +2.2 -5.3 -1.2
0.79 0.99 1.40 2.26 2.64 4.85
1.02 1.30 0.92 0.73 0.75 1.27
0.76 1.02 1.41 2.20 2.65 5.04
0.76 1.02 1.39 2.17 2.61 5.02
-3.8 +3.0 $0.7 -2.7 +0.4 +3.9
-3.8 +3.0 -0.7 -4.0 -1.1 $3.9
51.43 35.51 78.3
... ... ...
74.07 53.74 116.02
51.00 37.00 79.80
...
Mean error, 1 4 . 0 24 (carbonate) 25 (silicate) 26 (oxide)
...
Mean error, *3.5
+44.0 +51.3 +48.2
-0.8 +4.2 +1.9
Mean error, $ 4 7 . 8
Mean error, 1 2 . 3
Using the modified a correction method. Using the peak ratio method. The standard was used only for the modified a correction method; for the peak ratio method pure Cu and pure Fe solutions were used as standards. Cu ores from Timna Copper Mines containing different amounts of Fe.
In general, the a correction method is limited to a narrow concentration range because intensities of the X-ray lines of the interfering element vary with the concentration in a nonlinear manner. The variation in concentration of the main element cannot be neglected, a fact which was not considered in the a method. The introduction of peak ratios in the cr method instead of intensity, gives improved results over a much greater concentration range. However, the modified method is limited to samples whose chemical compositions are similar to those of the standards used. The peak ratio method of correction is independent of both the concentration range studied and the chemical composition. The correction formula derived combines constants obtained from pure Cu and pure Fe solutions and a compensatory factor for the influence of lower 2 ele-
1516
ments. This last factor can be neglected when Fe is absent-e.g. in copper oxide, carbonate, silicate samples, etc. This allows use of the same one-element standards for analyzing samples containing only one main element and two-element mixtures of solutions and solids. Satisfactory analytical results were obtained for solutions of 0.6-25 wt/vol % Cu in the presence of 0.4-23 wt/vol % Fe. The procedure was extended to solid mixtures and ores containing up to 80% Cu. The accuracy of the method is about 3.5%.
ACKNOWLEDGMENT We wish to express our appreciation to H. L. Finston for his valuable remarks. RECEIVED for review June 20, 1974 Accepted February 6, 1974.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
