Introduction

This is a discussion of how to calculate a best estimate of specimen composition, consistent with the measured fluorescence intensities from the unknown specimen and standard reference materials, by indirect use of a physical-mathematical model of emitted intensities. Theoretical intensity calculations are discussed in a companion article, "Modeling the Physics …" For specificity, we will occasionally refer to the particular model described there, but the ways suggested here for using an intensity model can apply to whatever more elaborate and detailed models might become available. In general, a model is a way to calculate predicted fluorescence intensity, in terms of specimen composition, fundamental parameters of the elements, and experimental conditions. The main restriction in using a model is to be sure that "measured intensities" and "predicted intensities" refer to the same kind of "intensities". For example, if the model does not attempt to include the contributions from scattered background or line overlap in the spectrometer or detector, then one should make the appropriate corrections to the data before attempting to relate measured intensities and predicted intensities.

A Remarkably Simple, Atypical Example

Before getting into the general situation, consider a special case that is not only historically important, but also valuable for providing insight into quantifying the effect of each element on the measurement of another element. As presented in "Modeling the Physics …", predicted directly-fluoresced intensity for a particular line l, from element i, relative to the intensity from the pure element i, can be written as

. (Eqn 1)

If the incident spectrum, , is essentially monochromatic, as from an isotope source, a secondary target, or specially filtered tube output, or if we otherwise choose to assume that there is only a single value of , then there is no integration, and we can rearrange the equation to become

. (Eqn 2)

At this point, let us repeat, from the other article, our reason for using the subscript l. Most discussions, for convenience, just use the subscript i instead, referring to the emitting element. Strictly, the list of emitted lines, whether measured or not, is very different from the list of elements: some elements might not be fluoresced at all, while other elements might emit several different characteristic lines that can be useful in measurement and/or important in calculating secondary fluorescence. So, even though we usually speak of the "intensity from element i", those who write the computer programs and their user interfaces need to make the proper distinctions between lines and elements.

Using the expression for the mass absorption coefficient of a multielement specimen,

, (Eqn 3)

we can expand the and in Equation 2 to obtain

. (Eqn 4)

Noting that the sum of the mass fractions (C's) equals 1, we can express the denominator more compactly:

, (Eqn 5)

where

. (Eqn 6)

This kind of intensity equation often is written in the form

, (Eqn 7)

where the expression in parentheses is called the "matrix correction". This kind of equation has been written in many different ways over the years, with different definitions of the influence coefficients, whether written as alphas, a's, or other symbols. The important thing to observe is that each alpha expresses the effect of a single matrix element on the measured line, and so the relative effects of the various elements can be understood by looking at the values of the alphas. In particular, for this formulation, the alpha of the emitting element itself is zero, from Equation 6 (with j=i).

If the experimental situation really did involve monochromatic radiation, then Equation 7 could be used for analysis. For example, a value for each could be obtained by writing Equation 7 for a measured standard reference material, with the measured intensity substituted for , and the appropriate values substituted for the C's and alphas. Then, with an Equation 7 written for the each measured line from the unknown, one could substitute the measured values of intensities, the previously determined values for each , and appropriate values for the alphas. The unknown quantities, the C's, appear on both sides of the equations, but there are many techniques for solving such a set of equations that are linear in the unknowns.

Using General Influence Coefficients

If the experiment did not involve strictly monochromatic radiation, an expression like Equation 7 could still be useful, but there would be two important differences in procedure and concept. First, the alphas could not be calculated from Equation 6, which applies to a single wavelength. For a long time, analysts obtained the alphas (or other influence coefficients) mainly empirically - e.g., by writing a set of Equations 7 for each measured line from a suite of specially prepared binaries and/or multicomponent reference materials, and then solving for the influence coefficicents for each such line. Then, Lachance showed in 1970 that alphas can be calculated from a fundamental-parameters model in a way that considers the full range of wavelengths in the incident radiation, as well as secondary fluorescence. The other big difference in using alphas for broad-spectrum excitation is that each alpha no longer represents just the effect of one matrix element, but depends on the overall composition of the specimen. Fortunately, each alpha represents mainly the effect of the element it purports to describe, and varies relatively little over narrow ranges of specimen composition. Therefore, the alphas calculated for broad-spectrum excitation are still useful, not only for understanding the relative effects of different elements, but even in calculating composition.

One should keep in mind that alphas do depend on overall composition, and so the values of the alphas need to be recalculated as a computer program proceeds through successively better estimates of composition. For many years, a major impetus for using theoretically calculated influence coefficients, of various forms, was the desire to make the calculation for each unknown specimen as quick as possible by using a single set of coefficients for a range of specimen compositions. With the faster computers of recent years, it is an unnecessary limitation on accuracy to use a fixed set of alphas for different unknown specimens, or even for different steps in the calculation for a single unknown.

Many workers over the years have written influence-coefficient expressions in different forms, and for a while there were intense discussions about which form was best. However, Lachance has shown in recent years that the different common forms of influence coefficients are simply algebraic transformations of each other, so that once the coefficients of any particular kind are calculated, they can be readily converted to another kind, mathematically equivalent. For details and interpretations for the wide range of formalisms that have been used, consult the book

Gerald R. Lachance and Fernand Claisse, Quantitative X-Ray Fluorescence Analysis, Theory and Application, John Wiley & Sons, Chichester, New York, etc. (1995)

Using a General Intensity Model

In a more general implementation of fluorescence intensity models, there is no need at all to use any kind of alphas or other influence coefficients within the calculation. In fact, since alphas need to be recalculated for each successively better estimate of specimen composition, the use of alphas would actually make the calculation slightly (but not significantly) slower than the general approach described below. An example of a general use of an intensity model follows. Later, it will be generalized further.

First, we need a way to put theoretical intensities and measured intensities on the same scale. One approach is to deal in terms of relative x-ray intensity,

, (Eqn 8)

the ratio of intensity from a specimen to the intensity of the same line from a specimen of the pure emitting element. Instead of actually measuring the pure element, one could measure any standard reference material containing the element. For the known composition, the corresponding theoretical R would be calculated, from Equation 8 and the intensity model. Then the effective pure-element intensity would be found by rewriting Equation 8 as

(Eqn 9)

and substituting the theoretical R and the actual measured I for that line. Now, any measured I from a multicomponent specimen can be put on the same scale as the theoretical intensities by dividing it by the P that was found as just described.

Now that measured and predicted intensities are on the same scale, the analysis problem is a matter of finding the specimen composition whose predicted intensities best match the measured intensities from that specimen. We should not expect there to be any composition that would give a perfect match, partly because of inevitable measurement errors, but mainly because the model should not be expected to match the experimental reality perfectly, no matter how detailed the model. For example, we might arrive at a specimen composition for which all the predicted intensities were higher than the measured intensities by some factor, say 1.004. That kind of situation would be one way to define a "best" match: equal relative errors in intensities. More detailed mathematical definitions of "best estimate of composition" might include considerations of estimated random errors in the measurements, concentration levels of the individual elements (e.g., requiring smaller relative mismatches for higher concentrations), etc.

The remaining computational problem is how to proceed from one estimate of composition for the unknown to a better estimate of composition, preferably in a way that moves toward the desired "best" estimate. This should be an iteration method that gets essentially the same result no matter what the first estimate of composition is. There are many ways to iterate, but we will just mention one particular approach that is easy to describe, and that applies to specimens for which a line is measured for every element.

We can express the intensity model in a way that is analogous to Equation 1:

, (Eqn 10)

where the function f represents the ratio of integrals in Equation 1, or the corresponding part of any theoretical intensity model, no matter how complicated. Then, we can re-write the equation as

. (Eqn 11)

Now, for any current estimate of composition, the measured R and the assumed C's are substituted on the right side of the equation, and the next estimate of is calculated. This numerical process is repeated for successive estimates of composition until there is no significant change. The set of next estimates of mass fractions must be rescaled on each iteration to add to 1, because that is what mass fractions means, and that is what is needed to calculate mass absorption coefficients, etc.

In designing a computer program to do the iteration, it will be necessary to use an iteration method that considers the particular definition of "best estimate" that the user specifies, and which allows for unmeasured elements, more than one measured line for an element, fixed combinations of elements (e.g., oxides), fixed amounts of some components (diluents, fluxes, binders, etc.), and other variations in experimental situations.

Selection and Use of Standard Reference Materials

The preceding mentioned how a single standard reference material (or just "standard" for brevity) could be used to determine the effective pure-element intensity, for the purpose of scaling the theoretical calculation to match the experimental reality. In that case, it would be desirable to choose a standard whose composition is reasonably similar to the unknown's, so that there would be less of a burden on the theory to relate specimens with widely different compositions. Unfortunately, we must often do the best we can with standards that are very different from the unknowns. For many types of specimens, such as alloys and glasses, the fundamental-parameters methods have been very successful when using just pure-element standards or pure oxides, if those were all that were available. In general, a well-designed computer program should allow the analyst to use whatever standards are available and have well known compositions. It has been found that one can get better results using a few very well characterized standards than a larger number of more dubious materials.

Global vs. Local Use of Standards

Before the development of the programs NRLXRF and XRF11 in the late 1970's, calculation procedures generally handled standards in one of two extreme ways -- totally "global", or totally "local", as explained below.

Regression methods generally calculated calibration coefficients once and for all, by a global, compromise fit to all the available standards. Unfortunately, unless one used only the bare minimum number of standards, the fitted parameters could not possibly match all the standards exactly, and often missed some of the data points by more than the experimental error. That traditional situation has the disturbing effect of violating the concept of a "standard" -- if the measured x-ray intensities from an unknown happened to be the same as those from one of the standards, then one should reasonably expect a calculation procedure to produce a composition equal to the composition of that standard (within experimental uncertainty).

The other extreme way of handling standards was by a strictly local rescaling of the calibration curve, by pinning the calibration to a single "favorite" standard in the composition range of interest. That procedure forced the analyst to choose one favorite standard for each x-ray line, and ignore other standards that might be equally valid. If one chose different standards for different composition ranges, the calibration curve could change abruptly, and by more than the experimental uncertainty.

No theory, no matter how elaborate, can model all instrumental responses exactly, and no empirical equation can fit all the standards all the time. However, this fact does not lead inevitably to the two undesirable situations mentioned above (violation of standards and abrupt switches in curves), as many of us once supposed. It is possible -- in fact, it should be required of any analysis software, for XRF or other methods -- that (1) the analyst may input data for as many different standards as desired beyond the required minimum, (2) the computer program may not violate the analyst's assertion that they are indeed standards, and (3) there may not be any abrupt changes in results for slight changes in data.

Adaptive Regression

Starting in the late 1970's, programs like NRLXRF and XRF11, unlike other programs for quantitative analysis, handled standards by a method that may be called "adaptive regression". If more than one standard has been measured for some lines, then for each unknown, the program essentially recalibrates, and places greatest emphasis on the standards whose intensities are most similar to the intensities from the unknown being processed. The emphasis weights vary smoothly as intensities change. The weighting can also include the experimental uncertainties in the data from the different standards. Besides being a smooth, automatic recalibration, the results cannot "violate" any standard by more than the experimental uncertainty.

The process of adaptive regression is described in the documentation for NRLXRF in terms of a smooth adjustment of the effective pure-element intensity, since the scaling factor is in those units. The procedure in XRF11 is basically the same, but it might be thought of also as a repeated, re-weighted regression - i.e., "adaptive" regression. For other applications, similar procedures have been called "metric interpolation" and "moving least-squares approximation", either "interpolating" (exact matching at each data point) or "non-interpolating" (allowing errors consistent with errors in the data). Obviously, there are many details involved in the calculation. This brief discussion was provided to support the following section.

Using Standards with Versatile Analysis Software

It is always very important to consider carefully the selection, preparation, and use of standards (i.e., certified or standard reference materials, or specimens prepared from certified materials). This is even more critical when using fundamental-parameters calculations for analysis, because the calibration will be pinned to selected standards, and will suffer from any biases in the standards.

The subject of specimen preparation will be mentioned only briefly here. As everyone knows, specimen preparation must be sufficiently reproducible that different specimens prepared from the same original material will lead to essentially the same measured x-ray intensities. Besides simple reproducibility, it is just as important to consider homogeneity within each specimen. If standard and unknown specimens are not all homogeneous, then the accuracy of analysis will be limited. It is best, of course, to assure that all specimens are homogeneous - for example, by using solution and fusion techniques.

Certain precautions must be taken if it is necessary to measure specimens that are irregular, rough, or particulate down to the very fine scale that matters in x-ray fluorescence (sometimes 1 micron or smaller). All specimens should have the same degree of heterogeneity, considering the size and composition distributions of particles, voids, and/or binders. That means, in effect, that standards and unknowns should be prepared in exactly the same way, and should have approximately the same compositions on a very fine scale.

Besides homogeneity, perhaps the next most important consideration is to be sure that so-called "standards" are really standards - that is, their complete composition is known with an accuracy that is the same as or better than what is desired in analysis.

If the complete composition of a standard is not specified, then accurate results will depend on two assumptions: the unspecified portions of all standards are the same material, and the corresponding part of the unknowns is that material. Keep in mind that all elements present in an unknown should be identified when using fundamental-parameters calculations (or when expecting high accuracy from other approaches), even though some elements might not be of interest in the final analysis report.

As described in the section on "adaptive regression", a program using that approach will believe the user when a "standard" is specified, and will assure that the calibration conforms to every standard, no matter how much contortion might be required. Unlike the global, compromise regression used in the earlier empirical methods, adaptive regression cannot be expected to smooth out irregularities in the standards data and ignore extreme outliers. Whether or not this is desirable depends on the accuracy expected. The adaptive approach is appropriate for accurate results based on accurate standards. The older, global approach is more appropriate when high accuracy cannot be expected from the available standards. In such situations, the adaptive approach can still be used, if arbitrarily large uncertainties are assigned to the measured intensities, so that all standards will be weighted equally.

The conclusion to be reached from this discussion of adaptive regression is that one should measure and input only the best standards, if the best results are desired. It has been observed in the analysis of stainless steels that it can be better to base the results on pure-element standards (which are completely and accurately known) than to rely on a type-standard whose certified components add up to only 95 per cent or so. Of course, it is even better to use one or more type-standards that are certified up to 99 per cent or more.

Checking the Consistency of Standards Used

The preceding discussed how a program can rescale theoretical calculations to agree with whatever standards are available, automatically emphasizing those standards whose intensities are most similar to the intensities from the unknown being analyzed. It was pointed out that this approach is desirable for accurate analysis based on accurate standards, since the standards are not "averaged" as is done in most other procedures. However, to achieve accurate results in this way, it is important to use only standards that are accurately and completely known.

If the available standards are not accurate and self-consistent, it would be better to suppress the localized adjustment (as described later), in order to prevent irregularities in the analysis results. When more than one standard is used for a particular measured line, it is advisable to check the consistency of the standards.

To test standards against each other, you can input intensities for an "unknown" equal to the intensities from one of the standards. If you use that standard also as a standard for the analysis, you should obtain essentially the correct analysis for the "unknown" (considering the assumed random error in the intensities). If you omit that standard (as a standard), the results for the "unknown" will be based only on the remaining standard(s), and will tend to show up inconsistencies among the standards. If the remaining standard(s) are very different from the standard being tested, the results will also reflect the program's ability to produce accurate answers based on dissimilar standards. Thus, when testing a standard, you are also testing the calculations, and it is not obvious what should be blamed for any observed errors. However, errors from inconsistent standards tend to look different from errors in calculation when several standards are tested against each other. Other testing strategies sometimes can make the distinction.

Another way to check the consistency of standards is to take any specific unknown and analyze it on the basis of just one standard at a time. For example, start out using all standards for all lines. Then, for each measured line being tested, analyze the data for the unknown using just one standard at a time for that line. If the results for the measured element (or its compound) are related smoothly to the intensities for that element in the standards, then there is no reason to suspect any of the standards. In that case, any errors will be smooth, and will reflect the inability of the theory to model all aspects of the experiment (such as residual background and dead-time effects). Also, in that case, the program's automatic emphasis of the most similar standards will be valuable. On the other hand, if results for the unknown are not related smoothly to the intensities of the measured line in the standard, then the irregularity is most likely a result of irregularities (inconsistencies) in the standards.

If the standards are not consistent to the degree of accuracy desired, there are two main courses of action. If only a few of the standards are doubtful, those few can be omitted (with regard to the particular measured line being tested), so long as there remain enough standards for analysis of unknowns. To omit a standard with regard to some lines, but not all lines, just omit the intensities for those lines from that standard. In the other situation, where there are no obvious outliers, and the standards appear to be inconsistent on a random basis, then an overall averaging of the standards would be advisable. To accomplish this, input very large assumed random errors in the data from the standards. The relative emphasis on the different standards will tend to be the same if the values provided are large enough (e.g., 200 per cent). If smaller, and more reasonably-looking, errors are assigned, the program will still be able to distort the calibration curve to match each standard more closely that might be desired.

The analyst who wants the most accuracy out of borderline standards, including standards whose certified composition adds to much less than 100%, will always have to use careful judgment in the selection and weighting of standards.