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.

In the last decade, the Fundamental Parameter (FT) method has been increasingly used in quantitative XRF analysis as an effective means for interelement effect corrections. Many authors have contributed to its continuous development both in theory and practice [1 - 26]. Among various implementations, the FP-alpha method has been used widely in bulk-sample analysis because of its unique advantage of speed [1 - 6]. In recent years, however, with the advent of faster computers, a rigorous full FP approach has become feasible and practical. This is especially true in the field of multilayer thin-film, analysis, where the FP-alpha approach has not yet been applied.

Qualitative inorganic analysis is required for the identification of unknowns, the classification of type, and sometimes to decide what subsequent quantitative analysis is needed. The traditional way of performing qualitative XRF analysis on unknown materials is by subjecting the sample to a full spectral scan. This takes time and an experienced operator to interpret the spectra. Classifying the elements detected as major, minor or trace can also be person dependent. Round robin tests have confirmed this by showing considerable variation in results between laboratories.

A fundamental parameter algorithm for quantitative XFA with characteristic Kα- and Lα-radiations of the specimens is presented. This allows an application to programmable pocket calculators or a fast evaluation of measured countrates for the unknown composition of the specimen by means of PCs. For the practical utilization we need the incidence angle and the takeoff angle of the x-rays in the instrument, the high voltage V in kV which is supplied to the x-ray tube (E0) and reference measurements on either pure elements or a single reference specimen. The quality of the analytical results depends on the concentration differences of the unknown specimen and the reference specimen.

Kramers described the cross section of electron interaction with target atoms of atomic number Z by

where Eo is the kinetic energy of impinging electrons, and E o S) the energy of x-ray photons of the continuum, Smith et al modified this equation, introducing an exponent x, so that

We applied the cross-section σS, E to the evaluation of experimental results. The evaluation of the measured spectral responses of the x-ray signals nE was performed by

where f(deff) describes the absorption of x-rays of energy E in the target, RE accounts for backscattering of electrons, DE quantifies the efficiency of x-ray detection within the solid angle Ω.

Modern powder diffraction employing computer-controlled diffractometers now allows quantitative analytical methods to use the whole diffraction trace rather then only individual peaks. Two such methods are in common use the Rietveld method which refines the crystal structures of the component phases as part of the matching calculation, and the patternfitting method which uses reference patterns from a database. Potential accuracies of these methods seems to be around 1 % absolute. The most severe limitation on the potential accuracy of these methods is particle statistics, which has been reviewed in considerable detail.

Calculated patterns play an essential role in X-ray powder diffraction analysis. This paper gives examples of their use in qualitative analysis for evaluating and supplementing reference patterns in the ICDD Powder Diffraction File (PDF), in quantitative analysis for calculating Reference Intensity Ratios (RIRs), in ceil parameter refinements for indexing of low-symmetry/large unit cell diffractograms, in powder pattern determination for validating intensities and recognizing preferred orientation, in new materials synthesis for verification of structure type and phase purity, and for modeling the effects of solid solution substitution.

EDXRF analysis is conveniently split into two parts: (1) the determination of X-ray intensities and (2) the determination of elemental amounts from X-ray intensities. For the first, most EDXRF analysis has been done by some method of integrating the essentially Gaussian distribution of observed full energy pulse heights. This might be done, for example, by least-square fitting of Gaussian distributions superimposed on a straight line or a quadratic background. Recently more elaborate shapes of the energy peaks also have been considered (Kennedy, 1990). After the X-ray intensities have been determined, interelement effects between the analyte element and other elements must be corrected for in order to obtain the elemental amounts from X-ray intensities. This correction can be done either by an empirical correction procedure as in the influence coefficient method which requires measurements on a number of standard samples to determine the required coefficients, or by theoretical calculation as in the fundamental parameters method which does not require standard samples.

Some x-ray fluorescence - concentration relationships in the framework of XRF were researched. Fundamental calculation approaches for primary fluorescence and incoherent scattering were realized for evaluation of matrix influence. A new binary approach was produced for the cases considered, and its unification was related to the empirical and regression types of the “standard background” technique, widely used in the analytical practice of XRF. It is confirmed that application of the calculations by fundamental parameters (FP) in combination with the empirical approach allows the reduction of the set of standards (to as few as one) in the analysis procedure with wide variations in matrices and concentrations, without loss of accuracy.

The utilization and optimization of the properties of materials follows most effectively from a detailed knowledge and understanding of the positions and energetics of their constituent atoms, generally obtained from scattering/diffraction experiments involving electrons, neutrons or electromagnetic radiation. For the most part, these experiments are undertaken on individual crystals of the material, thereby preserving the resolution (and information content) of the three-dimensional reciprocal lattice. However, many of the substances of greatest academic and technical importance either do not crystallize with dimensions large enough for single-crystal studies, or display the properties of maximal interest only when present in finely-divided (powdered) form. In a diffraction experiment, the reciprocal lattice is then collapsed on to the single dimension of the 2θ scale.

It is well-known that in EDXRF, using high-resolution semiconductor detectors, evaluation of x-ray fluorescence radiation line intensities from multiplex spectrometric information represents definite difficulties, especially in automation of measurement-calculation procedures.

A common spectrum decomposition problem is to get the following parameters: a number of spectral lines and their centroids, intensities from measured experimental data and their errors.

We have developed special software for solving this problem using personal computers and high-level programming language C. It uses profiles of real-form lines of pure chemical elements produced by semiconductor detector spectrometers and these techniques: digital filters with parameters for suppression of background, multiplex structure analysis, and stable linear least-squares fit to get peak intensities. Also it established special criteria for reliability of the results.

We compared our investigation with software “EDXRF” (ver.1.32) arid spectrum decomposition with Gaussian peaks.

The magnitudes of the peak intensities in a powder diffraction pattern are not uniformly distributed as a function of diffraction angle. The Lorentz-polarization factor, X-ray form-factor and the thermal vibration parameters all conspire to progressively decrease the intensities of the peaks as sinθ/λ increases. In spite of this, diffraction data for Rietveld analysis is universally collected using the same counting time for each step in the pattern. As a result, peaks at high angles are collected with lower counting precision than those at low angles, despite the fact that the high-angle region has a higher density of peaks and therefore contains more information than the low-angle part of the pattern. Indeed, the intensity, position and profile of reflections at medium and high sinθ/λ are often more easily modelled since they are subject to less interference from systematic effects such as extinction, specimen transparency and α1/α2 overlap errors. In the present work a novel variable step-counting-time regime has been devised that increases the counting times with diffraction angle in a manner inversely proportional to the intensity fall-off calculated from the Lorentz-polarization, form-factor and thermal parameters. The effect of this new data collection regime on the results of Rietveld refinement of a number of materials of varying structural complexity is discussed.

The fundamental parameters method (FPM) enables one to determine with high accuracy the chemical composition of homogeneous samples, having only one reference sample. However, the reference sample composition should be similar to that of the samples analyzed.

The x-ray fluorescence analysis of multicomponent heterogeneous materials (ores, minerals, their mixtures, welding electrode coating mixtures, fluxes, etc.) is made by the Δμ-correction method based on the combined use of the fundamental and empirical correlations maintaining all the advantages of the FPM. Sample composition is calculated on the basis of the element intensities measured in the sample and in the reference specimen and is corrected for the disturbing effect of excitation conditions and heterogeneity as well as the calculated values of one of the fundamental parameters (μ1). At the preliminary stage of calibration, the coefficients are determined using regression and the absolute fundamental expression for the element fluorescence intensity.

SIROQUANT is a suite of personal ccmputer programs which runs on an IBM-compatible 386 or 486 personal computer, and gives phase quantification of mineral samples hy Rietveld refinement using the full XRD profile. Eight phases can be quantified in a mineral with the 640 K RAM version and 11 in the EM version.

New features have been added since the original description was written (J.C. Taylor, Powder Diffraction, 6, 2, 1991) including calculations of amorphous percentages using spiked samples, particle size calculations, use of “measured” as well as calculated patterns as standards, pre-programmed refinement “Stages” with different variables in each stage, and improved menus.

Some advanced applications of the system using these new features are described.

The Kulenkampff-Kramers formula giving the spectral distribution of the continuum as a function of wavelength allows a correct calculation of background under the peak. It is only necessary to measure two backgrounds, one on each side of the peak, The true background under the peak is given by multiplying them by adequate coefficients and adding them up. This method has been applied to the determination of low amounts of Ce, La, Ba and Cs in geological samples.

The effect of sample granularity on diffracted x-ray intensity was evaluated by measuring the 2θ dependence of x-ray fluorescence from various samples. Measurements were made in the symmetric geometry on samples ranging from single crystals to highly absorbing coarse powders. A characteristic shape for the absorption correction was observed. A demonstration of the sensitivity of Rietveld refined site occupation parameters is made on CuAu and Cu50Au44Ni6 alloys refined with and without granularity corrections. These alloys provide a good example of the effect of granularity due to their large linear x-ray absorption coefficients. Sample granularity and refined thermal parameters obtained from the Rietveld analysis were found to be correlated. Without a granularity correction, the refined thermal parameters are too low and can actually become negative in an attempt to compensate for granularity, A general shape for granularity correction can be included in refinement procedures. If no granularity correction is included, data should be restricted to above 30° 2θ, and thermal parameters should be ignored unless extreme precautions are taken to produce <5 (μm particles and high packing densities.