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:

(1)

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:

(2)

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.

Shamil
Musin

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