AADDVVAANNCCEESS IINN XX--RRAAYY AANNAALLYYSSIISS VVoolluummee 3355BB A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. ADVANCES IN X-RA Y ANAL YSIS Volume 35B Edited by Charles S. Barrett John V. GiIfrich University of Denver SachslFreeman Associates Denver, Colorado Washington, D. C. Ting C. Huang Ron Jenkins IBM Almaden Research Center ICPDS-ICDD San Iose, Califomia Swarthmore, Pennsylvania Gregory J. McCarthy Paul K. Predecki North Dakota State University University of Denver Fargo, North Dakota Denver, Colorado Richard Ryon Deane K. Smith Lawrence Livermore National Laboratory Pennsylvania State University Livermore, California University Park, Pennsylvania Sponsored by University of Denver Department of Engineering and JCPDS - International Centre for Diffraction Data Springer Science+Business Media, LLC The Library of Congress cataloged the first volume of this titIe as folIows: Conference on Application of X-ray Analysis. Proceedings 6th- 1957- [Denver] v. iIIus. 24-28 Cßl. annual. No proceedings published for the first 5 conferences. Vals. for 1958- caIIed also: Advances in X-ray analysis, v. 2- Proceedings for 1957 issued by the conference under an earlier name: Conference on Industrial Applications of X-ray Analysis. Other slight variations in name of conference. Vol. for 1957 published by the University of Denver, Denver Research Institute, Metallurgy Division. VoIs. for 1958- distributed by Plenum Press, New York. Conferences sponsored by University of Denver , Denver Research Institute. 1. X-rays-Industrial applications-Congresses. I. Denver University. Denver Research Institute H. TitIe: Advances in X-ray analysis. TA406.5.C6 58-35928 ISBN 978-1-4613-6532-7 ISBN 978-1-4615-3460-0 (eBook) DOI 10.1007/978-1-4615-3460-0 Proceedings of combined First Pacific-International Congress on X-Ray Analytical Methods (pICXAM) and Fortieth Annual Conference on Applications of X-Ray Analysis, held August 7-16, 1991, in Hilo and Honolulu, Hawaii © Springer Science+Business Media New York 1992 Originally published by Plenum Press in 1992 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, e1ectronic, mechanical, photocopying, microfIlming, recording, or otherwise, without written permission from the Publisher MATHEMATICAL CORRECTION PROCEDURES IN XRF - THE LONG AND THE SHORT Gerald R. Lachance Nepean, Ontario, Canada The processes of absorption and enhancement in x-ray fluorescence spectrometry preclude the direct conversion of measured intensities to concentrations unless the specimens being analyzed are practically identical to the reference materials that were used as standards for calibration. While well known techniques of spectrometry such as the use of internal standards or techniques specific to x-ray fluorescence such as Compton scatter may be used in some instances, the analysis of alloys for example is not amenable to these techniques. The goal of mathematical correction procedures is to compensate adequately for matrix effects and therefore to extend the concentration range within which analysis may be accomplished. The evolution of the concepts and expressions that have been proposed for this purpose spans some 50 years, is somewhat fascinating and examined in outline after a brief nostalgic look at where it all began. Having set out to "examine spectra of a few elements in greater detail," Moseley! describes how he ". .. used as targets a number of substances mounted on a truck inside an exhausted tube ... " with provision for ". .. each target to be brought in turn into the line of fire." Of the elements chosen as forming a continuous series (Z 20-30) ". .. calcium alone gave any trouble. In this case ... the layer of lime which covered the surface of the metal gave off such a quantity of gas that the x-ray could only be excited for a second or two at-a-time." Prophetically, Moseley concluded that "the prevalence of lines due to impurities suggest that this may prove a powerful method of chemical analysis." In a second paper2, the instrumentation is described: ". .. the aluminium trolley that carries the targets can be drawn to and fro by means of silk fishing lines wound on brass bobbins. The slit (which defines the x-ray beam) should be fixed exactly opposite the focus-spot of the cathode stream, though a slight error can be remedied by deflecting the cathode rays with a magnet." Consider some of Moseley's comments: ". .. the examination of keltium would be of exceptional interest, as no place ... , ... lines due to impurities were frequently present but caused little trouble except in the rare-earth group ... The x-ray spectrum of praesodymia showed that it consisted roughly 50% La, 35% Ce and 15% Pr ... " Advances in X-Ray Analysis. Vol. 35 Edited by C.S. Barrett et al. • Plenum Press. New York. 1992 693 694 X. MATHEMATICAL METHODS IN XRS Some twenty years later, von Hevest determined lead in a number of alloys. The procedure may be summarized as follows: - one sample of zinc with 0.087% Pb used as standard -equal exposures made with other samples -Pb concentration calculated by direct ratio with the comment, "Results for the alloys of unknown lead content ranged from 0.064 to 0.008 percent and, although no analysis by another method was carried out, these values were in good agreement with other evidence as to the composition of the samples." Thus was the stage set for x-ray fluorescence to emerge as a technique for quantitative elemental determination and the derivation of mathematical expressions for converting intensities to concentration. For easier comparison, a common set of symbols (see Appendix) is used rather than those in the original expressions. One of the first theoretically derived expressions relating emitted intensity to specimen composition is due to von Hamos.4 For the restricted analytical context: binary specimen irradiated by a monochromatic incident source and secondary fluorescence emission (enhancement) is not present, the emitted intensity is given by (1) with the variable K; defined as a function of Cj, J1.j' J1.j and instrument geometry only. Current practice for influence coefficient models tends towards the calculation of the RECIPROCAL of K; as a function of all the elements EXCEPT Cj. Some nine years later, Sherman5 derived the following expression for a monochromatic incident source wavelength A IllI' C. g. l. = I I (2) I • • Cj Ili + Cj Ilj + ... where gj is a proportionality constant IliI = Ili (A) esc V' Il *i = Ili (A) esc V' + Ili (Ai) esc V" * Il j = Ilj (A) esc V' + Ilj (Ai) esc V" thereby extending Eq (1) to multi-element systems. The terms for absorption of the incident and emergent x-radiations were combined for each element individually, the asterisk indicating that the path lengths of each having been taken into consideration. Sherman6 and Shiraiwa and Fujin07 further developed the theory of emitted intensity as a function of specimen composition and fundamental parameters for a polychromatic incident beam but their treatment is beyond the scope of this presentation. It incorporates an expression for enhancement previously reported, without proof, by Gillam and Heal.8 However it is interesting to note that in a discussion of the problem of extending G. R. LACHANCE 695 monochromatic to polychromatic formulations, Sherman concluded "Instead of measuring intensity. .. as counts per second, it is advantageous to represent the intensity by the reciprocal of the counting rate, i.e., time required for the counter to reach a predetermined fIxed count." This led to the expression * * {k) + (Cj) IJ. j + IJ. k + (3) C. * C. * IJ. i IJ. i I I It is noted that tj / t(i) = I(i) / Ij , the reciprocal of the generally current practice of expressing relative intensity as (4) Beattie and Brissey9 retained Sherman's defInition of relative intensity and reformulated Eq (3) yielding * IJ.. - I) C. + (-1.*. ) C. + ... = 0 (5) I J lJ.i Faced with the prospect of working with a polychromatic incident source and lacking the practical means of dealing with integrals over the wavelength range of interest, Beattie and Brissey opted to calculate the values of the influence coefficients from experimental intensity data using the expression Kij = CC.j (1I(iJ. - I) (6) J I which are then introduced in Eq (5) - (I(IiiJ -I)C. + Kij Cj +... = 0 (7) I For the determination of tantalum in niobium oxides, Campbell and CarllO proposed the following expression to correct for the matrix effect due to the presence of impurities (8) where Cj.expr is the value determined experimentally from a calibration of tantalum oxide in niobium oxide and C is given by j l. (jor specimen) CJ. = JI j(for 1 % C) , (or Cj determined chemically). It was observed that the experimental data agreed qualitatively with those expected from calculations of the linear absorption coefficients for the various matrices. Mitchellll working with the same oxide system also generated influence coefficient values experimentally (termed absorption-enhancement indices) and examined their pattern for a given analyte by observing plots of indices versus matrix element atomic number. 696 X. MATHEMATICAL METHODS IN XRS A quite different concept was proposed by Lucas-Tooth and Price12 for compensating mathematically for matrix effects. Here the individual corrections are a function of intensities of the matrix elements rather than their concentrations. In a subsequent work, Lucas-Tooth and Pyne13 proposed a model in which apparent concentrations are substituted for the intensities of the matrix elements. In either case the influence coefficients are obtained by regression of experimental data. The coefficients are empirical, i.e., have little or no physical meaning, and it is generally acknowledged that this approach is only applicable over very limited concentration ranges. In the mid 60's Lachance and Traill14 proposed a correction model that circumvents the over-definition problem of the Beattie-Brissey model Eq (5), namely (9) with the stipulation that the values of the influence coefficients may be calculated provided the assumption can be made that a single wavelength may represent (i.e., be equivalent to) a polychromatic incident beam. This was an oversimplification that was subsequently abandoned in favor of the concept that coefficients may be Q related to 'fundamental influence coefficients,' thus retaining a close link to the basic principles of x-ray physics. From 1968 on, the emergence of the fundamental parameters approach proposed by Criss and Birks15 provided a method for the correction of matrix effects based on physical principles of x-ray fluorescence emission and led to the re evaluation of older influence coefficient models. Criss and Birks compared two mathematical approaches for corrected matrix effects: the empirical regression method which had been in use for many years, and a new method that accounts for matrix effects by means of measured spectral distributions of x-ray tube generated incident sources, mass attenuation coefficients, fluorescence yields, etc. The model used for the regression method was that of Beattie and Brissey [Eq (5)] but simplified, updated and optimized for computer evaluation. The coefficients are determined using experimental data by solving sets of simultaneous equations of the type (10) For a three component system, nine influence coefficients are required and unknowns are analyzed by rewriting the above equation in the following form and q, ... solving for C j, etc., with the stipulation that Cj + Cj + ~ = 1.0 It was observed that influence coefficients selected in this way do not represent the effect of one element on another but are the best set of numbers to describe the intensities measured. G. R. LACHANCE 697 The fundamental parameters approach for the correction of matrix effects proposed by Criss and Birks is based on the concept of using either measured or calculated spectral distributions of x-ray tubes and replacing integrals by summations over a discrete number of effective wavelength intervals. This led to the expression for primary fluorescence emission. ~ Ilj(A) leA) 11 A P. = g C L.." (11) , I I A 1l .. (A) esc lJr' + 1l .. (A) esc lJr" where I (A) is the incident intensity for the interval 11 A :E II. . = i C Il,; :E C, = 1.0 j and to the expression (abridged) that includes both primary and secondary fluorescence emission, i.e., enhancement present. 1 Pi + Si = Pi (1 + eij Cj) (12) An iteration procedure is used to determine the weight fractions in unknowns. The measured relative intensities are scaled to 1.0 and used as the first estimate of composition. A set of theoretical relative intensities are calculated and compared to the measured values. The differences are used to make a better estimate of composition. The process is repeated until for some set of assumed concentrations, the calculated relative intensities agree with the measured values. Subsequently it was shown, Lachance,16 that the expressions for Pi + Si and P(i) could be combined leading to the formalism -1 1 Pi + Si = p(i! Ci Xj,a Cj + Xj,. Cj (13) where ~ .. and ~.e are the sums of the prorated absorption and enhancement monochromatic terms, respectively. Thus, the primary and secondary fluorescence emitted intensity is equal to product P(I) C minus the sum of each individual 1 absorption effect plus the sum of each individual enhancement effect. Solving the P(i) C leads to i 1 -7 p(i! Cj = Pi + Si + Xj,a Cj Xj,. Cj (14) which can be interpreted as 'the intensity corrected for matrix effect is equal to the emitted intensity plus a correction term that compensates for the decrease in intensity due to absorption minus a correction term that compensates for the increased intensity due to secondary enhancement.' Defining ~.n = ~ .• -~.e and substituting in Eq (14) gives E PC' C. = P. + S. + x. c. (15) ',' , , j loll 1 which can also be expressed as P. + S, ~ X. C. = -'- - [1 + L.." __1, _"_ C.] (16) I Per) j Pi + Sj 1 698 X. MATHEMATICAL METHODS IN XRS or -7 Cj = Rj [1 + njj C) (17) i.e., Criss and Birks' fundamental parameters equation expressed in influence coefficient formalism. Two concepts that have been proposed in the domain of fundamental influence coefficients are examined next, namely the option to choose an element other than the analyte for elimination in the correction for matrix effect term, and the option to calculate influence coefficients specific to absorption and enhancement. Given that the experimental determination of influence coefficients requires a large number of standards and that these are not always available, de Jongh17,18 proposed a model wherein an element other than the analyte is arbitrarily chosen for elimination in contrast to Eq (17) which is based on the elimination of the analyte. The process is described as follows: q / R, is calculated using a fundamental parameters expression for an average composition ... q / R, is calculated with respect to the average composition due to 0.1% change of element j, leading to the computation of delta-coefficients. The elimination of a selected components yields beta-coefficient values, Betas are converted to 'ALPHAS' so that they may be used in expressions similar to Eq (17), i.e., . -7 Cj = Rj [1 + njjn C) (18) where n is retained as the symbol for the influence coefficients to indicate that the coefficients are in the domain of fundamental influence coefficients. The subscript ijn refers to the influence of an exchange of j and n on the value of C R, of the j / analyte i. It is noted that j n in the summation term Eq (18) whereas j i in the ¢ ¢ summation term Eq (17). The fact that Eq (14) may be broken down to p(.I ) C.I = P.I + Lj x.lo a CJ. (19a) and (19b) and transformed to Cj = R/ (1 +Lj Qjj,p C) (20a) and +L Rj = R/ (1 j ejj,p C) (20b) where P. X. X. R.I ' =P-('/). , a jj,P =~P..' e jj,P =-hP!.., . I