Demystification of Algorithms and Influence Coefficients in Quantitative XRF Analysis

Transcription

1 718 Demystification of Algorithms and Influence Coefficients in Quantitative XRF Analysis Gerald R. Lachance 1100 chemin St-Felix Hammond, ON, Canada KOA 2A0 Abstract The evolution ofthe influence coefficients concept and the multitude of models that have been proposed in which they are used has had a somewhat stormy life, This is not surprising considering the wide range of analytical contexts encountered in practice. It should also not be surprising that analysts have approached the task by initially addressing the simpler context {monochromatic excitation, absorption effect in binary systems) and gradually addressing the more common and comprehensive context (polychromatic excitation, absorption and enhancement effects defined explicitly from fundamental parameters in multi-component systems]. It is shown that algorithms are independent of analytical contexts, while influence coefficients, on the other hand, are defined specifically for a given analytical context. It is demonstrated that all validly derived algorithms are equivalent as to the physical reality of XRF emissions, but that some are expressed in more convenient models. This leads to the following conclusions: - any advancement made in extending the theoretical x-ray principles underlying influence coefficients in one algorithm can be extended to all; - the stigmas sometimes attached to algorithms due to the fact that they were derived for a restricted context are unwarranted - theoretically derived algorithms and theoretically defined influence coefficients can be derived from one another. The assignment of a preeminent role to any given one is untenable.

2 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website ICDD Website -

3 719 Introduction The concept of influence coefficient algorithms for the correction for matrix effects is a topic that has generated quite different visualizations and interpretations of the process of absorption and enhancement. In the case of algorithms that are derived from time-honored principles of x-ray fluorescence emission and influence coefficients that are defined explicitly in terms of fundamental parameters, the conflict is only apparent. The aim of this paper is to examine the correction terms for matrix effects within the framework [ aem + C { aa* )ij Cj ] where brackets [ ] and braces ( ) are used to demarcate the algorithm and influence coefficient terms, respectively. For clarity and ease of comparison, identical symbols are used to designate indentical entities in the various expressions rather than the myriad of different symbols used in the original sources. For space considerations, the following abbreviations will be used to delineate the analytical context under consideration: - mono and poly refer to monochromatic and polychromatic excitation sources, respectively - abs, enh and matrix refer to absorption, enhancement and matrix (abs and enh combined), respectively - bin and multi refer to binary and multi-element systems, respectively - theo, expr and regr refer to theoretical, experimental and regression procedures, respectively. Influence coefficients obtained by regression methods are also labelled empirical. In XRF, the combination of algorithms/influence coefficients can be visualized as mathematical expressions that accurately describe a large number of observations based on reliable fluorescence emission data measured in the laboratory. Thus, experimentally measured intensities are related to theoretical intensity expressions by the simple relation Ii 7 experimental, c/s units = gi cpi + si Ti + l l l )theoretical 7 arbitrary units

4 720 where g is a proportionality constant Pi, Si, and Ti refer to Primary, Secondary and Tertiary fluorescence emissions, respectively. The following treatment is limited to primary and secondary fluorescence emissions. Restricted analytical context The following expression, proposed by v. Hamos, was one of the first to define emitted intensity as a function of the emitted intensity emitted by the pure analyte, the analyte concentration in the specimen, and the ratio of their respective absorption coefficients pil = p(i)h ci (Pi*/ Ps*>k where pi A refers to the theoretically calculated primary fluorescence intensity P(i)k as above for a specimen of pure analyte i ci concentration (weight fraction) of element i l-4 total effective mass absorption coeffkient for pure analyte i Lb* total effective mass absorption coefficient for specimen s Expanding the denominator in the above expression for a three-element system leads to P = ih P (j)jy ci 4 'i VT + cj p; + Ck 11; (1) which is retained for generating expressions developed by the following authors:

5 721 (A) Sherman2T3 derived expressions that are equivalent to dividing the numerator and denominator by ci Pi* and expressing intensity ti as the time taken to register a fixed number of counts, which leads to t ih = t(i)h+ xj L 1 t(i)kijlh(- 5 C) i J (2) and t (i,hvtih) i + [c? { t(i)kij}~cj] = (3) where tg) refers to the pure analyte and Kij / cli*) i.e., (mono, abs, bin, theo) (B) Beattie and Brissey4 proposed expressions of a similar type - (Q+, Ci+& Kjj,bincj = 0 (4) where Qi = 10) / Ii and the value of &j,bin is obtained from the expression K ij,bin = (Qi-l)I c; J (5) i.e., for the context fpoly, matrix, bin, expr).

6 722 (C) A slightly different initial treatment, equivalent to dividing the numerator and denominator by CLi only, yields the expression (6) which leads to the algorithm C,=R, [ci+xj K. 13.c. 3 J (7) where & = Pi / Pfi, In the above equation, Criss and Birks5 proposed that the K coefficients be determined for the context (poly, matrix, multi, regr}. (D) Tertian, on the other hand, determined the K coefficients using the expression K = ij (1-K) Ri xl-ci ci (8) for the context { poly, matrix, bin, theo (& calculated)). Subsequently, Tertian and Vie le Sage7 proposed the following expression, where the K,j* coefficient relates to the absorption effect, while ~j relates to the enhancement effect, which are combined as the matrix effect in coefficient Kij, {mono, matrix, bin, theof K = K,Tj-Cihij ij 1+(1-C) hij (9)

7 723 (E) To eliminate the problems associated with having the term Ci on both sides of the equal sign, Lachance and Trail18 proposed the expression Ci = R, I l+cj aijhcj] (10) where d = ijh (11) thus, ~j h for the context (mono, abs, bin, theo}. Comprehensive analytical context The following treatment examines the five algorithms, equations (2), (3) (4), (7), (10) and other algorithms proposed subsequently in the light of a number of theoretical developments, namely, the contributions of Gillam and Heal who essentially retained the expression derived by v. Hamos, eqn (l), for primary fluorescence emission, and proposed an expression for secondary fluorescence emission (enhancement), i.e., a term to be added to primary emission. Sherman2 who derived integrals for primary, secondary and tertiary fluorescence emissions. Shiraiwa and Fujino who confirmed the expressions of Gillam and Heal and noted the missing 112 and l/4 terms in the expressions for secondary and tertiary emissions, respectively, as proposed by Sherman. Criss and Birks5 who retained the expressions proposed by Shiraiwa and Fujino for primary and

8 724 secondary emissions, but proposed : (a) that the integrals be replaced by a finite summation over the number of discrete effective wavelength intervals; (b) using experimentally measured x-ray tube spectra as excitation sources for theoretical calculations. This resulted in the well known and still widely used Fundamental Parameters Approach which has the general formulation Pi + si = gici CA ob@zrn2 (enh term) (12) Following further developments in the concept of theoretical influence coefficients by de Jongh1 y12, Tertian and Claisse 3, Broll and Tertian14, Rousseau*, Lachance16, and Lachance and Claisse17, the following expression was proposed Pi + Si = Pt,,Ci - cj AijCj + cj EijCj (13) as an alternative formulation equivalent to the Criss and Birks algorithm.. Transposition leads to Pci,Ci = Pi + Si + xja..c - c 13 j j E..C. 17 _7 (14) an intermediate expression that is easily amenable to defining theoretical influence coefficients for the practical, i.e., comprehensive analytical context: { polychromatic excitation source, absorption, enhancement and their combined (matrix) effects, multi-element systems, coefficients defined explicitly). For example, influence coefficients aijh {mono, abs, bin, theo) in the Lachance-Trail1 algorithm can be replaced by coefficients mij (poly, matrix, multi, theo} Ci = Ri l'+xj mijcj J (15)

9 4 725 where R = (Pi + Si) / Phi) = Ii / I,,, and m = ij A ij -Eij pi+5 1 (16) It therefore follows that any of the Kij coefficients in previously shown Kij expressions, where defined for restricted analytical contexts, can now be defined by the usual expression, & = ~j + 1, and is now applicable for the comprehensive analytical contexts. The ~j coefficients can also be defined in terms of absorption and enhancement effects defined separately, namely m ij = A ij p,+s,- E ij pi+si = a ij -eij (17) The intermediate expression is also amenable to defining the absorption and enhancement intluence coefficients aij and Q. Defined as and A;; a., = -2 1.J 'i (18) E ij E - ij 'i (19) the coefficients are applicable for the Rousseau algorithm

10 726 'i I l+cj cxij cj 1 = Ri.1+cj (20) and for the Broll-Tertian algorithm Ci = Ri ~+~jpl-.ij~}cj] (21) The intermediate expression is also amenable to defining influence coefficients for the de Jongh algorithm12 ci = Ki(Pi+si) L'+xj cxijecjj (22) or Ci = R,(l+m,,) L'+& cxij,cj] (23) where & = (1 + q,) / P,, Note: in the above, the summation over j includes 5 but not e, the eliminated element. The values q, and aij,, are given by a = iie i m..-mie E-mie (24)

11 727 and 01.. = ve i m..-m. ;i,ie (25) Note: m, = 0 Numerical confirmation The analytical context chosen is that of Rasberry and Heinrich18 : Excitation source: W target tube operated at 45 kv (CP) Spectrometer geometry: incident angle = 63 ; emergent angle = 33 System: Cr-Fe-Ni; Analyte: Cr Specimen 5324: weight fractions C,, = C,, = C, = Experimental data: relative intensities &, = RFe = RNi = Given that the total of the three weight fractions is , it is preferable for the task at hand that the data refer to a specimen whose weight fractions equal unity. Therefore, the following numerical example is based on hypothetical specimen 5324N and calculated relative intensities f?om fimdamental parameters. The results show very close agreement between the measured relative intensities and their calculated counter part: Specimen 5324N: weight fractions C, = C,, = C, = Theoretical data: relative intensities Rpr = RFe = RNi = P, + S, = Pee,) = P, = t,,= s t,,,=63.10 s N = krre = E,, = kcrni = E,, =

12 728 Theoretical influence coefficients: acrfe = / = ccrfe = / = qrni = / = ecrni = / = aclrfe = / = ecrfe = / = foci = / = ecrni = / = W3Fe = = ~, = = GrFe = = I<crNi = = Substitution in: (A) Sherman: eqns (2) and (3) tcr= (0.6781) (0.533 / 0.272) f0.9663) (0.195 / 0.272) = ( ) (0.6781) (0.9663) = 0.00 (B) Beattie and Brissey: eqn (4) -( ) i.6781) (0.9663) = 0.00 where Qcr = ( / ) = = 1 + (0.6781) (0.533 / 0.272) + (0.9663) (0.195 / 0.272) =

13 729 (Cl ew (7) where C,, = [ ( } (0.9663) = Qr= = (D) Lachance-Traill: eqn (10) C, = [l + ( ) ( ) 0.195}] = (E) de Jongh: eqn (23) C, = ( (0.6781) [l + ((0.0 - ( )) / ) = (( ( ))/0.6781) j (F) Rousseau: eqn (20) C,= [(l + (0.1064) (0.2567) 0.195) = ,' (1 + (0.5399) to.3022) 0.195)] (G) Broll-Tertian: eqn (21)

14 n C, = [l + ( (0.272 / 0.331)) = ( (0.272 / )) (H) Lachance-Traill: eqn (17) C,, = [l + ( ) ( ) = Conclusions Theoretically derived algorithms to correct for matrix effects are not per se restricted to a given analytical context. On the other hand, the definition of intluence coefficients can reflect the processes, i.e., match the experimental operating conditions. Thus, any advancement made in extending the field of application is applicable to all. Since any given theoretical algorithm or influence coefficient can be defined/derived from any other, it therefore follows that: the stigmas sometimes attached to the fact that a given algorithm was derived originally for a restricted analytical context is unwarranted. For example, (Kij; monochromatic, absorption, binary) in a given algorithm can be replaced by (m,j + 1; polychromatic, absorption and enhancement combined, multi-element systems). the designation of a preeminent role or status to any one theoretical algorithm or to a specific theoretical influence coefficient definition is also questionable. References 1. L. v. Hamos, Ark. Mat. Astron. Fys., 3 la, 1, (1945). 2. J. Sherman, Spectrochim. Acta, 7, 283, (1955). 3. J. Sherman, ASTM Spec. Tech. Publ. No. 157,27, (1954). 4. H. J. Beattie and R. M. Brissey, Anal. Chem., 26, 980, (1954).

15 J. W. Criss and L. S. Birks, Anal. Chem., 40, 1080, (1968). 6. R. Tertian, X-Ray Spectrom., 2,95, (1973). 7. R. Tertian and R. Vie le Sage, X-Ray Spectrom., 6, 123, (1977). 8. G. R. Lachance and R. J. Traill, Can. Spectrosc., 11,43, (1966). 9. E. Gillam and H. T. Heal, Br. J. Appl. Phys., 3, 353, (1952). 10. T. Shiraiwa and N. Fujino, Jpn. J. Appl. Phys., 5, 886, (1966). 11. W. K. de Jongh, X-Ray Spectrom., 2, 151, (1973). 12. W. K. de Jongh, X-Ray Spectrom., 8,52, (1979). 13. R. Tertian and F. Claisse, Principles of Quantitative &Ray Fluorescence Analysis Heyden, London, (1982). 14. N. Broil and R. Tertian, X-Ray Spectrom., 12, 30, (1983). 15. R. M. Rousseau, X-Ray Spectrom., 13, 115, (1984). 16. G. R. Lachance, Adv. X-Ray Anal., 3 1,471, (1988). 17. G. R. Lachance and F. Claisse, Quantitative X-Ray Flzcorescence Analysis Theov and Application, Wiley, Chichester, (1995). 18. S. D. Rasberry and K. F. J. Heir&h, Anal. Chem., 46, 81, (1974).