2541
Anal. Chem. 1985, 57, 2541-2544
Graphical Comparison of Mathematical Models for Quantitative X-ray Fluorescence Analysis of Cobalt Alloys GBrard Platbrood Laborelec, Laboratoire Belge de l’lndustrie Electrique, B.P. 1 1 , B-1640 Rhode St. Genese, Belgium
X-ray fluorescence has been Implemented for the analysls of cobalt alloys wlth high Cr and W contents. The developed program has enabled a rough evaiuatlon of matrlx effects and a comparlson of mathematical models common In Ilterature. The graphs of the resldual standard devlatlon of the fit, calculated from the measured and calculated concentratlons as a function of Interfering elements, have been lnvestlgated for some matrix effects, secondary absorption and secondary fluorescence, and three mathematlcal models. This method has enabled a determlnatlon of the best correction condltlons for X-ray fluorescence Intensities. The analysis deals wlth such major elements as Co, Cr, and W and such minor elements as Mo, Nb, l a , Ni, Fe, Mn, SI, and AI.
Available literature on corrections of X-ray fluorescence interelemental effects regarding cobalt alloys is rather limited. The matrix correction problems raised by the cobalt alloys are nevertheless similar to those of stainless steel. In 1971 Trojan et al. ( 1 ) issued the Ni, Co, Mn, and Si calibration curves, together with measuring conditions. Owing to the simple relations between intensities/concentrations for these elements and to the restricted investigation range of concentration variations, correction aspects dealing with matrix effects were not considered. On the contrary, analysis of cobalt alloys investigated in the present paper has necessitated a correction program for matrix effects. The concentration values of the major and minor elements used in the implemented series of cobalt standards will characterize the alloy investigated in the present paper: Mo (0.0-6%), Nb (0.0-2.48%), W (0.8-12.2%), Ta (0.0-0.12%), Ni (0.3-2.65%), Co (59.53-67.48%), Fe (0.24-1.78%), Mn (0.0-0.63%), Cr (19.28-30.08%), Si (0.0-0.89%), A1 (0.0-1.12%). This alloy is particularly used for the manufacture of turbine blades. In the 19709, several X-ray fluorescence correction models were available in literature (2), providing the analytical chemist with a whole series of correction formulas founded either on intensities or on concentrations. Laborelec (1978) has suggested a general approach to aid in the selection of an appropriate correction scheme. This approach can be used as a guideline for analysts when the adequate model and interfering elements have to be chosen for a given matrix type. Several practical bases have been studied within Laborelec and two of these have been dealt with in the literature ( 3 , 4 ) . As shown by these studies, the smallest analysis error for each element is achievable through the automatic comparison of the residual standard deviation graphs (basing on the comparison of calculated and measured concentrations of standards), as a function of the increasing number of interfering elements and taking into account each matrix effect. The graphic display is a tremendous tool as efficiency of the correction introduced by a series of elements can be appreciated nearly instantaneously, whereas the comparison of listed calculation results does not often give a comprehensive view. Nevertheless, our approach of the problems raised by corrections of matrix effects differs from Schreiner and Jen0003-2700/85/0357-2541$01.50/0
kins’ approach (5). The investigation of a practical case dealt with in our paper did not require more than 1 h of graphical analysis. This enabled us to propose a valid analytical solution for the matrix effect correction.
THEORY Within this computer program, four mathematical models have been investigated thoroughly: the straight line, the Lucas-Tooth and Price model (LTP) (6) and the models derived from the Lachance-Trail1 (LT) (7) and RasberryHeinrich (RH)(8)expressions. The overlapping X-ray lines have also been taken into account in the algorithms implementing the LTP, LT, and RH models (3). Further comments on the use of these interelemental correction models are included in the paper developing the theoretical foundations of the program (3). The essential foundation for the present study is the comparison of three graphs for each element, dealing respectively with secondary absorption, secondary excitation, and both effects investigated simultaneously. Each graph illustrates the evolution of the residual standard deviation of the fit r (on the y axis) as a function of the elements influencing the matrix effect (on the x axis), r being expressed as weight fraction
\
1
where r is usually referred to as “goodness-of-fit”or residual standard deviation of the fit, ns is the number of standards, p is the number of coefficients used in the correction formula, C, is the concentration of the standard i , and C’Lis the calculated concentration of the standard i . For an increasingly smaller r value, correction of matrix effects by the model will become more important. Consequently,the analysis error will be reduced proportionally. The first point (LR) on the x axis of the graphs will determine the value of r in the simple case of a linear model. The elements located on the x axis will be classified in descending order with respect to their estimated matrix effect (secondary absorption, secondary fluorescence). Regarding secondary absorption this classification is founded upon the use of a so-called y y magnitude defined as ~
t
=] ~y(C](rnax) - C,(rnln)) = PLIBCI
(2)
where ptl is the mass absorption coefficient, i is the measured element, j is the element that absorbs the X-ray fluorescence lines of element i , Cl(max)is the maximum concentration of element j , Cl(rn,n)is the minimum concentration of element j , and SCJ = C](max)- C](mln)* After classification of the elements in the descending order of their y V ,they will be implemented for corrective calculations. A similar approach will be adopted in dealing with the effect of secondary excitation by also defining a xLlcoefficient x.. = pCL.6C. (3) EJ 11 I As several rays of an element j can enhance an element i , 0 1985 American Chemical Society
2542
ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985
m-
100
LUCAS-TOOTH-PFflCE MOOEL - RASBERRY-HEINRICH MODEL
0
I
L
090 080
om
070
050
050
050
0 50
0LO 0 30
ow'l
1 1 " 1 1 1 1 I LR W MO NE TA SECONDARY FLUORESCENCE
SECDWARY ABSORPTION
Flgure 1.
Analysis of Co, secondary absorptlon.
Figure 2.
evaluation of the secondary excitation is definitely more complicated than evaluation of the secondary absorption. Excitation conditions of the matrix may generate photon intensity variations of an element that causes secondary enhancement. Thus, only the main ray (for example Kar lines) of the element generating the secondary excitation will be used. Tertiary fluorescence is not considered in this rather simple approach. Owing to the complexity of the secondary fluorescence expression (2),we will content ourselves with this simplified approach that has already proved satisfactory in several cases. The coefficients calculation in the nonlinear expression of Lachance-Trail1 and Rasberry-Heinrich has been carried out with an optimalization technique based on the Marquardt algorithm (9). This one has been modified to accelerate the convergence process (10, 11).
~
RESULTS AND DISCUSSION
All matrix effects regarding each element will be considered successively,but the Co and Cr elements characterized by the highest matrix effect will be commented on more particularly. Analysis of the Element Co. First, secondary absorption will be treated. pco, is calculated through interpolation of the values pv given by Bertin (12) and is slightly different from that derived by the NRLXRF program (13) using the fundamental parameters. As a result of the rather small differences between the NRLXRF pcoJ and pcoJ, classification of the decreasing y', values, calculated from the pco, values, should not be modified. The j elements classified in decreasing order of ycoJare listed in Table I, under the Co column. When the
~~
LUCAS-TOOTH-PRICE MODEL __ LACHANCE-TRAILL MODEL L
EXPERIMENTAL SECTION Standards. The cobalt base alloys are certified reference
materials from MBH analytical LTD manufactured by Willan Metals, Ltd., Rotherham (Standards reference series 14936F and series 12667). Measurement Conditions. The X-ray fluorescence spectrometer SRSOl conditions are voltage 50 kV and current 40 mA. The crystal chosen for the lines Mo Kal, Nb K q , W Lo,, Ta Lol, Ni Kal, Co Ka,, Fe Kal, Mn K q , and Cr Kal is the LiF. The lines Si Kcq and A1 Kal have been measured on a PET crystal. The collimator 0.15O has been selected. The background has been measured for the elements Nb, Ta, Co, Fe, and Cr and the bruto intensities directly used for the elements Mo, W, Ni, Mn, Si, and Al. Software and Hardware Organization. A Tektronix 4051 (32K bytes) controls the Siemens spectrometer SRSOl via a logical controller. The intensities are transmitted to the host computer SEL 32 (Systems Engineering Laboratories, 1Megabyte RAM, Fortran 77'). Five computation programs are used, SN:ATLAG, SN:ULISS, SN:LECCO, SN.PTBO, and SN.DRAW. Different files using the fluorescence wavelengths, mass attenuation coefficients, edges, and data are required to perform the calculations.
Analysis of Co, secondary fluorescence.
i
1
000liR
Flgure 3.
\1
i
1 Mo
,
l
[
w do
l
'
i
l
,
Ai
,
l
h
CR C I AN SECONDARY ABSORPTION
Analysis of Cr, secondary absorption.
correction coefficientsare to be calculated, these elements have to be entered in the same order. As noticeable in Figure 1, the r value no longer decreases significantly as soon as the first three elements, Cr, W, and Mo, are entered (see the curve corresponding to the Lachance-Trail1 model). In order to achieve the lowest r value, practically the same results are obtained when the Rasberry-Heinrich or LucasTooth models are implemented. With respect to the standards, the Co element shows an important absolute analysis error (&0.5%) due to the balanced calculation of the concentrations. Consequently, the r value beneath this analysis error cannot be greatly improved by means of the corrections. The secondary fluorescence excitation initiated by the elements W, Mo, Nb, and T a is rather small and does not improve the r factor in any case (Figure 2). Thus, the Lachance-Traill formula has been adopted, and the matrix effect shall only take into account the three elements Cr, W, and
Mo. Analysis of the Element Cr. The same procedure has been adopted for this analysis of the Cr element in order to achieve the highest accuracy. Gamma calculations ycrjhave enabled a classification of the interfering elements in descending order of their absorption effects. Referring to Figure 3, the different models, Lucas-Tooth Price and LachanceTrail1 can be compared. In the initial correction phase, the r values derived from both models are nearly equivalent at the input of the first elements. But when the number of elements entered in the correction is increased, the difference between the r values, as analyzed by both selected models,
ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985
2543
Table I. List of Interfering Elements j on i for Secondary Absorption and of Elements j Taken into Account in the Secondary Fluorescence Matrix Effect of Element P
i
interfering elements j on i for secondary absorptionb W Ni co Fe Cr
1 2 3 4 6 6 7 8 9
W Mo Nb co Cr Ni Fe Ta Mn
10
A1
11
Si
Cr W Mo Fe Nb co Mn Ni Al Si Ta
Cr W
Mo Nb co Mn Ni Fe A1 Si Ta
Cr W Mo
Nb co Ni Fe A1 Si Mn Ta
W Mo Nb co Cr Ni Fe A1
elements j taken into account in secondary fluorescence matrix effect of ic Ni co Fe Cr W Mo Nb Ta
W Ni Mo Ni Ta
W Mo
Nb Ta
co W Fe Ni Mo
Nb Ta
Si
Mn Ta
-
"For X-ray fluorescence measurements, the elements stated from the left to the right in the table (Mo Al) will be introduced in the analysis order. In this table, only the elements i where absorption and/or enhancement effects are appreciable, have been considered. *Set in decreasing order of their yU. Set in decreasing order of their xv. ' L9
- LUCAS-TOOTH-PRICE W
LUCAS-TOOTH PRICE MODEL - RASBERRY-HEINRICH MODEL
L
- RASEERRY- HEINRICH W E L
n -
L R c 0 W F E N I w ) F B T A
~ ~ W Q E S C E N C E
LR CO W
Flgure 4. Analysis of Cr, secondary fluorescence.
increases. The expectable analysis accuracy amounts to nearly 0.1% absolute. When the excitation effect is considered (Table I, Figure 4),it is observed that the input of Co causes a significant decrease of the r value, and the elements W, Fe, and Ni also cause a smaller, although significant, decrease of the r value. Considering practical aspects, which involve the use of a minimum of elements for correction purposes, the selected solution will be decided as follows: utilization of the Rasberry-Heinrich formula as the corrective expression and correction based on four elements (Co, W, Fe, Ni) for the secondary fluorescence. The third type of graph (Figure 5) that can be drawn with this program will only be considered for information purposes. On Figure 5, the elements are introduced by pairs in the correction formula: one for the secondary absorption effect and one for the secondary fluorescence effect. Proceeding as in the Lucas-Tooth Price expression, no distinction is made between both effects. The correction will only take into account the different elements. On the contrary, in the correction formula where both effects are considered, two terms will be introduced simultaneously in the Rasberry-Heinrich expression: one under for the secondary absorption and another under @,Cj/(l+ C,) for the secondary fluorescence. Regarding Figure 5, the excellent result achieved by this correction involving eight elements is misleading; as a matter of fact, by use of the nine standards, a system of determined equations is approached. Consequently we will no longer refer to this correction that uses a mixed expression, although in
SECONDARY ABSORPTION FE N I Mo NB TA SECWDARY FLUORESCENCE
Flgure 5. Analysis of Cr, both effects (secondary absorption and
secondary fluorescence). Table 11. Correction Conditions and the Chosen Models for the Studied Cobalt Alloys
chosen matrix elements introduced element model" effectsb in the correction Mo Nb W Ta Ni co Fe Mn Cr Si A1
LTP LR LTP LR
SA
W, Co, Cr
SA
W, Mo, Nb, Co, Cr
LTP LTP
SA SA
Cr,W,Mo Cr, W, Mo, Nb, Co
SF
Co, W, Fe, Ni
SA
Co, Cr, W, Mo
oc
oc RH oc LT
remarks
Co overlapping Cr overlapping Co overlapping
" Key: LT, Lachance-Traill; LTP, Lucas-Tooth and Price model; RH, Rasberry and Heinrich model; LR, linear regression (two coefficients); OC, overlapping correction (three coefficients). Key: SA, secondary absorption; SF, secondary fluorescence. stainless steel analysis, it has provided the best results regarding chroming analysis (3). This approach is coefficients consuming. Other Elements. Owing to the extremely limited concentration ranges, the linear model will be implemented for Nb and Ta since the tested matrix corrections do not improve sufficiently analysis errors. Ni, Mn, and Si (see formula OC)
2544
ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985
T a b l e 111. A n a l y s i s
of t h e U n k n o w n S a m p l e 1
element
effective concn, %
Mo
0.0
Nb
1.44 10.34 0.12 0.63 64.06 1.24 0.54 20.79 0.63
W Ta
Ni co Fe
Mn
Cr Si A1
0.00
calcd concn, %
a
1.51 10.33 0.12 0.61
63.46 1.24 0.52 20.97 0.55 b
" T h e concentration was less t h a n t h e limit o f detection, 0.066. * T h e concentration was less t h a n t h e limit of detection, 0.008. Table
IV. A n a l y s i s of the U n k n o w n S a m p l e 2 element
effective concn, %
calcd concn, %
Mo Nb W
4.97
4.96
0.00
a
1.36
1.34
Ta
0.00
Ni co
0.60 60.50 0.81 0.62 30.08 0.89 0.26
Fe
Mn
Cr Si A1
b
0.62 60.47 0.81
0.63 29.90 0.82 0.26
a T h e concentration was less t h a n t h e limit o f detection, 0.06. * T h e concentration was less t h a n t h e limit o f detection, 0.01.
will be submitted to an overlap correction. Table I1 includes the best correction conditions for all the elements and particularly for the elements Mo, W, Fe, Al. Matrix effects and elements intervening in the corrections have been selected in the same way as the previously investigated Co and Cr. Unknown Analysis. The achievable accuracy level regarding the analysis of unknown samples will be illustrated by two practical examples of sample analysis with composition of the samples being known. The fundamentals (3) describe the adopted procedure for analysis of unknowns (Tables 111 and IV). Calculated concentrations are derived from the matrix effects and from the elements determined through stand-
ardization, as described earlier. The error calculation method worked out by Plesch (14) is referred to in order to determine the calculated error and the detection limit. The analysis error being a function of the measured concentration, it will grow smaller if the measured concentration is approaching one of the standards implement for calibration purposes. The chosen value meets the stated confidence level of 90%. The measurements show a satisfactory accuracy. The aim of the adopted simultaneous analysis of four standards and the unknown samples is 2-fold: enabling an excellent recalibration of intensities and ensuring an efficient verification of analysis accuracy. By use of the graphical display of r as a function of the interfering elements, it becomes easy to evaluate the effect of any additional element introduced in the correction. Owing to the modular structure of the program, correction formulas other than those mentioned in the paper are also implementable. However, considering the satisfactory results of the used correction formulas alternative models have not been tested. Providing of the organigram of different programs and of the file structure has been neglected intentionally in this paper. Detailed information on this topic will be provided directly by the author on request. Registry No. Co, 7440-48-4;Cr, 7440-47-3;W, 7440-33-7;Mo, 7439-98-7; Nb, 7440-03-1; Ta, 7440-25-7; Ni, 7440-02-0; Fe, 7439-89-6; Mn, 7439-96-5; Si, 7440-21-3; Al, 7429-90-5.
LITERATURE CITED Trovan, W.; Witkowska, S.; Stankiewicz, W. Rudy Met. Niezelaz. 1971, 16, 229. Tertian, R.; Claisse, F. "Princlples of Quantitative X-Ray Fluorescence Analysis"; Heyden: London, 1982; Chapter 10. Platbrood, G.; Simon, S. X-Ray Spectrom. 1982, 1 7 , 121. Platbrood, 0. Vortrage vom 3. Colloquium "Anwendungsmogiichkeiten der Rontgenfluoreszenzanalyse" am 2 1J22.1.1982; Landwirtschaftliche Untersuchungs- and Forechungsanstalt, Joseph-Konig Institut, Monster; p 127. Schreiner, W. N.; Jenkins, R. X-Ray Spectrom. 1979, 8, 33. Lucas-Tooth, H. J.; Price, B. J. Metallurgica 1961, 64, 149. Lachance, G. R.; Traill, R. J. Can. Specfrosc. 1966, 11, 43. Rasberry, S. D.; Heinrich, K. F. Y. Anal. Chem. 1974, 46, 81. Marquardt, D. W. J. SOC. Ind. Appl. Math. 1963, 11, 431. Janssens, N. Sous-Routine MARQ10, MARQl1, MARQIP, LINREG. Description et Mode d'emploi; LABORELEC, September 1980. Platbrood, G. J. Appl. Crystallogr. 1985, 18, 114-119. Bertin, E. P. "Principles and Practice of X-ray Spectrometry Analysis"; Plenum: New York, 1975; Appendix 7A. Criss, J. W. "NRLXRF, a Fortran Program for X-ray Fluorescence Analysis"; Cosmic Program DOD-00065, 1977. Plesch, R. X-Ray Spectrom. 1976, 7, 156.
RECEIVED for review March 15,1985. Accepted June 24,1985.
