Matrix Corrections

Ema data can be quantitated to provide elemental concentrations, but several corrections are necessary to account for matrix effects adequately. One weU-known method for matrix correction is the 2af method (7,31). This approach is based on calculated corrections for major matrix-dependent effects which alter the intensity of x-rays observed at a particular energy after being emitted from the corresponding atoms. The 2af method corrects for differences between elements in electron stopping power and backscattering (the correction), self-absorption of x-rays by the matrix (the a correction), and the excitation of x-rays from one element by x-rays emitted from a different element, or in other words, secondary fluorescence (the f correction). 

Since the composition of the unknown appears in each of the correction factors, it is necessary to make an initial estimate of the composition (taken as the measured lvalue normalized by the sum of all lvalues), predict new lvalues from the composition and the ZAF correction factors, and iterate, testing the measured lvalues and the calculated lvalues for convergence. A closely related procedure to the ZAF method is the so-called ())(pz) method, which uses an analytic description of the X-ray depth distribution function determined from experimental measurements to provide a basis for calculating matrix correction factors. 

Here the matrix V contains the effect of the nuclear displacements therefore the inhomogeneous first term to the right is a driving term the second term to the right is of second order in the driving effect, and could be dropped in calculations. Formally, the solution for the configuration density matrix correction is... 

As XRF is not an absolute but a comparative method, sensitivity factors are needed, which differ for each spectrometer geometry. For quantification, matrix-matched standards or matrix-correction calculations are necessary. Quantitative XRF makes ample use of calibration standards (now available with the calibrating power of some 200 international reference materials). Table 8.41 shows the quantitative procedures commonly employed in XRF analysis. Quantitation is more difficult for the determination of a single element in an unknown than in a known matrix, and is most complex for all elements in an unknown matrix. In the latter case, full qualitative analysis is required before any attempt is made to quantitate the matrix elements. 

The use of fundamental parameters is attractive for various reasons. They impose fewer restrictions on the number of standards required for analysis. This simplifies the standardisation protocol for maintaining a XRF system, and permits greater flexibility in dealing with different types of materials. Inten-sity/concentration algorithms of the fundamental type, i.e. without recourse to the use of standards, have gradually developed [238-240] and are now widely available [241]. Functionality and quality of XRF software have reached a very high level, with a large variety of evaluation procedures and correction models for quantitative analysis, and calculation of fundamental parameter coefficients for effective matrix corrections. Nevertheless, there is still a need for accuracy improvement of fundamental parameters, such as the attenuation functions. 

It is possible to determine components in complex EPs where matrix effects can be severe. For example, zinc (as zinc borate), chlorine (as dechlorane flame retardant), antimony (as oxide) and fibre-glass have been determined in nylon using just one standard. Many users have refined the universal precalibrated programmes for standardless XRF and made them more efficient for matrix correction by using variable correction coefficients. OilQuant offers possibilities for analysing polymers [243]. Software packages usually provide ... 

The composition of the analysed point is calculated from the corrected intensities by applying matrix corrections which take account of a number of factors... 

Iteration. Matrix correction factors are dependent on the composition of the specimen, which is not known initially. Estimated concentrations are initially used in the correction factor calculations and, having applied the corrections thus obtained the calculations are repeated until convergence is obtained, i.e. when the concentrations do not change significantly between successive calculations. 

Atomic absorption provides very high sensitivity but requires careful subsampling, extensive sample preparation, and detailed sample-matrix corrections. X-ray fluorescence requires little in terms of sample preparation but suffers from low sensitivity and the application of major matrix corrections. Inductively coupled argon plasma spectrometry provides high sensitivity and few matrix corrections but requires a considerable amount of sample preparation, depending on the process stream to be analyzed. 

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

Because of the lack of standards, variations in analyses made by other methods, and errors caused by coal sampling problems, it was difficult to evaluate the need for x-ray matrix corrections and to select the best method for applying them. However, corrections were necessary because some elements in whole coal such as iron, silicon, and sulfur may vary considerably. For these elements, corrections were applied indiscriminately to all samples, because it was impossible to determine the point at which matrix variations required a correction greater than the accuracy limits of the method. We elected to use the minimum number of corrections compatible with reasonably accurate results. Therefore,... 

In near-infrared reflectance, the matrix correction is made possible by the additivity of each weighted contribution for that wavelength in the spectrum being observed. The low absorptivity in the near-infrared contributes to this necessary "additivity" and eliminates, also, the necessity of sample dilution prior to sensing. 

A numerical matrix correction technique is used to linearise fluorescent X-ray intensities from plant material in order to permit quantitation of the measurable trace elements. Percentage accuracies achieved on a standard sample were 13% for sulfur and phosphorus and better than 10% for heavier elements. The calculation employs all of the elemental X-ray intensities from the sample, relative X-ray production probabilities of the elements determined from thin film standards, elemental X-ray attenuation coefficients, and the areal density of the sample cm2. The mathematical treatment accounts for the matrix absorption effects of pure cellulose and deviations in the matrix effect caused by the measured elements. Ten elements are typically calculated simultaneously phosphorus, sulfur, chlorine, potassium, calcium, manganese, iron, copper, zinc and bromine. Detection limits obtained using a rhodium X-ray tube and an energy-dispersive X-ray fluorescence spectrometer are in the low ppm range for the elements manganese to strontium. 

Measured and computed values of the matrix coefficient are shown in Table 7.13. The values agree within a few percent except for Fe and Mn in radishes, where the difference is 6%. A matrix correction factor of 2 means that the combined attenuation of the exciting and fluorescent X-rays is 50%. In radishes, about half of this figure is from the cellulose and the other half from the presence of 6% potassium. 

In order to determine the effectiveness of the matrix correction procedure for elements fighter than manganese, a sample of known concentration was measured. The mean values and standard deviations of five separate measurements of NBS orchard leaves (SRM 1571) are shown in Table 7.14. 

Table 7.13. Measured and calculated matrix correction coefficients (from [111])...

For these samples, matrix corrections were not made. The matrix effects are most severe at the lower energy levels and are considered to be of almost negligible importance for elements having an atomic weight greater than that of iron. Thus, concentrations of Al, Si, S, and Cr may be in error by up to 50% of the amount present because of matrix effects and relatively poor counting statistics. All other reported elements are estimated to be accurate to within 10% of the correct values. 

Inouye, T., Harper, T., and Rasmussen, N. C., Application of Fourier transforms to the analysis of spectral data. Nucl. Instrum. Methods 76, 125-132 (1969). Jenkins, R., and Campbell-Whitelaw, A., Determination of interelement correction factors for matrix correction procedures in X-ray fluorescence spectrometry. Can. Specbrosc. 15, 32-38 (1970). 

A useful method of analysis of cmde and wear metals is the sample slurry method. It is well known that cmde and wear oils contain soluble, suspended and insoluble metal particulates and analysis of oils containing the soluble and suspended metals can be successfully carried out using a slurry method, provided that they are less than 4.0 pm in size and at a concentration that can be detected. An internal standard can be added to the oil sample slurry and nebulised along with the sample for matrix correction. 

KEY WORDS gypsum. X-ray fluorescence, thermogravimetry. analysis of gypsum, borates. Claisse fluxer, synthetic standards, matrix corrections... 

Survey of Canada in Ottawa [4]. The matrix correction model used in this work was that described by DeJongh [5]... 

We would like to thank E>r. Bruce A. Hudgens of the U.S. Gypsum Co. Research Center, IL. for the help with the round-robin analyses of several gypsum specimens and stimulating discussions on the subject. We are also grateful to Dr. Richard Rousseau and Mr. G. R. Lachance of the Canadian Geological Survey in Ottawa, who calculated the alpha coefficients used in the matrix corrections procedure and provided many valuable suggestions. 

In this analysis, three wavelength-dispersive X-ray spectrometers were used to simultaneously measure the characteristic X-ray intensities for copper, barium, and yttrium, at each point in the scan, producing two-dimensional X-ray intensity arrays. Complete quantitative analysis corrections, using the NBS theoretical matrix correction procedure FRAME (2), were performed at each picture element (pixel) in the image scan. 

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