Correction of interelement effect
Quantitative spectral analysis began with introduction of concept of the homologous line-pair in 1924 by Gerlach. Methods of exciting spectra were spark and arc. Until now, these methods remain the most common and most difficult for calculating results of Optical Emission Spectrometry (OES). These methods have a generally nonlinear calibration line. Whereas the newer methods ICP and GDL have practically linear calibration functions. Therefore, calculations of interelement effects for spark are quite difficult in general form. Nevertheless the mathematical algorithms used for them can be used for GDL and ICP as a special case.
Interelement effect can be caused by overlap of spectral lines or by matrix effect, i.e. changing of the physical and chemical processes in discharge zone due to influencing element. Correction of the interelement effect can be carried out via intensity or concentrations.
Overlap of the spectral lines causes the addition of line intensity, so it is natural to calculate the additive effect via intensity:
where Inc is the uncorrected line intensity
Imc is the line intensity corrected for the additive effect
j is the number of the influencing element
Bj is the coefficient of the additive component
Ij - intensity of the influencing element
practically j is not greater than 10
Change of parameters: density of discharge cloud, plasma temperature, diffusion rate, formation of chemical compounds, etc., depend on concentrations. So it is natural to calculate such effects via concentrations.
Matrix effect correction algorithms were originally developed for X-ray fluorescence analysis (XRF) in the second half of the last century, and then transferred to OES. The physical processes that cause the matrix effect are different for these types of spectral analysis, but attempts to develop special mathematics for the OES have not been widely used, and so far, mainly, the equations for XRF are used.
A significant contribution to study of matrix effect was made by Sherman, Lachance, Traill, Lucas-Tooth, Price, and others. As a results of their works, similar equations for the multiplier effect were obtained, which can be presented in general form for OES:
where Cnc is the uncorrected concentration value
Cmc is the concentration corrected for the multiplicative effect
Cj is the concentration of the influencing element
Mj is the coefficient of the multiplicative component
practically j is not greater than 10
In the initial equations, on the right-hand side of the equation were not concentrations Cnc but there were intensity. Generally, for the linear calibration function for XRF it does not matter.
In OES with spark exitation, a linear graph is a special case. In the general case, the function is approximately described by the Lomakin-Scheibe equation, and is not linear:
I = f + aC (3)
where I is the line intensity
f is background
a is the coefficient of proportionality
C is the concentration of the element being analyzed
b is coefficient of reabsorption
Difficulty of practical application of equation (2) is that the interelement influences can form a closed cycle. To work around this problem, usually appropriate intensities are used on the right-hand side of eq. (2) instead of Cnc or instead of Cnc and Cj. This limits the validity of applying this equation to spark spectrometers. In this case, the coefficients Mj will depend on concentrations and it can not be uniquely determined for the entire calibration range.
To solve this problem, in software SESA the function C in equation (2) is recurrent, i.e. it calls itself. However, the maximum number of calls is limited to avoid stack overflow.
Approximated calibration function
is found by the method of least squares. To achieve high accuracy of spectral analysis, calculations are performed by Chebyshev orthogonal polynomials. To determine the minimum of the multidimensional function, is used the Rosenbrock method. Of course, the methods for determining the calibration function and the multidimensional regression functions Dmin = f (I | B) and Dmin = f (C | M) are made in feedback via the variance. Here D is the variance, B and M are the vectors of the parameters B and M in equations (1) and (2).
The above-described methods for correcting the matrix effect provide a good result only with a sufficiently large number of standards participating in the calibration. With a small number of standards, the solution is ambiguous. In real conditions, the same errors can be caused by different influences. With increasing number of standards, we tighten the conditions of the task.
To test the accuracy of interelement effect correction, the file MATRIX EFFECT TEST.ars is used. This file and his description file “Interelement effect test.doc” are in the archive with the demo software.Download