The rigorous relationship existing between effective fundamental coefficients, as expressed by Tertian's identities, and Sherman's basic equations for x‐ray fluorescence intensities is recalled. The properties of these coefficients in the case of binary systems are described, for usual as well as for some anomalous situations, and a general, accurate hyperbolic equation is suggested to account for coefficient variation with composition. The properties of effective coefficients in the general case of multicomponent systems (alloys, etc.) are then examined, and illustrated on the typical FeNiCr ternary system, confirming the essential exactness of this approach. As a consequence, the authors propose to make use of calculated effective coefficients to apply mathematical matrix corrections, and give a first outline of the analytical procedure.
Accurate fluorescent intensities of a number of binary and ternary alloys have been calculated by relying on the classical expressions of primary and secondary fluorescence and assuming standard spectral intensity distributions for the primary radiation. It could then be shown that under polychromatic excitation the enhancement effect on the one hand, and on the other hand, the absorption effect with the complications (drift of equivalent wavelength) related to polychromatic radiation, became very much alike on practical terms, apart from generally opposed signs. Both effects could then be dealt with in a homogeneous way and a simple method was derived using (K − 1) coefficients ‐ see definition in the text ‐ which are fixed from a careful study of binary compositions. Moreover, making provision for fluorescence ‘crossed effects’, the principle of a general and accurate procedure was then established to the end of computerized non‐destructive analysis. The name of ‘generalized iteration method’ is proposed for this procedure.
The complexity of the intensity–concentration relationship in x‐ray fluorescence analysis as well as new requirements for speed and accuracy will induce analysts to resort more and more to mathematical methods for effecting matrix correction. A thorough study of Lachance–Traill coefficients, based on theoretically calculated fluorescence intensities, shows that all ‘binary’ coefficients vary systematically with composition under the usual working conditions with polychromatic excitation, while ‘multicomponent’ coefficients can no longer be represented (i.e. in the form of curves or diagrams) owing to the intricacy of third element effects. For these reasons, we are compelled to turn to a fundamental approach, where two methods are now competing: on the one hand, the classical fundamental parameter method, which permitted the abovementioned investigation, but does not in itself offer a sufficiently coherent solution to the problem of matrix correction; on the other, the new fundamental coefficient method, which accounts for all theoretical implications, but preserves the flexibility and adaptability of the older influence coefficient procedures. Utilizing such effective coefficients in a comparison standard correction algorithm allows one to solve any analytical problem, from the most complicated to the simpler ones, through a kind of ‘à la carte’ procedure. The so‐called empirical algorithms are in principle ineffective for treating the complex corrections which belong to the analysis of compact specimens (e.g. metal alloys). They must be reserved for simpler problems, notably the study of diluted specimens where an experimental calibration remains feasible.
Apart from their general importance, an accurate definition of spectral intensity distributions from x‐ray tubes is required to support the development of mathematical matrix correction procedures for x‐ray fluorescence analysis. Since a direct measurement is costly, time consuming and occasionally difficult, and a purely theoretical approach is not suitable for current utilization, a simple, readily adaptable calculation procedure is very desirable. Concerning the continuum or white radiation from the x‐ray tube, such an evaluation was worked out by effecting (1) a modification to Kramers' law and (2) a refinement of the absorption correction regarding the absorption by the target itself. The new equation accounts very closely for the best known, carefully measured and corrected experimental distributions by Brown et al. and Loomis and Keith. Extended experiments are required to ensure systematization, and special steps are needed for the treatment of the characteristic radiation.
A fast, efficient calibration procedure is proposed based on the accurate analysis of multicomponent systems by means of the influence coefficient method. It relies on a series of well designed additions made to a suitable standard composition, the latter also being used as a reference sample. We show that, for an n element composition, using the hypothesis of constant coefficients, (n + 1) preparations – allowing n(n + 1) intensity measurements – are in principle sufficient to calculate the n2 calibration parameters [i.e. n(n − 1) influence coefficients and n referenc relative intensities] which are necessary for a complete analysis. Actual analyses are carried out by measuring the fluorescent intensities using the ratio (i.e. unknown/standard) method, and then calculating the concentrations from the usual relationships through an iterative calculation, preferably executed by a computer program. A detailed description of a typical application is given, which involved the study of fused specimens (borax fusion) of bauxite and resulted in the very accurate analysis of calcined samples, as well as the direct analysis of raw bauxites following an adequate treatment of the ignition loss problem. While the method is immediately applicable to every system studied in either solid or liquid solution, it clearly extends to compact systems over a suitable compositional range, provided that carefully controlled additions can be achieved and that all prepared samples are homogeneous.
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