Bulk Analysis

In bulk analysis, no matter what the mode of operation, it is common practice to restrict the range of compositions so that q_M and R_i do not vary significantly. They are therefore omitted from the calibration function, which then simplifies to

where the restricted composition range means we rarely need third order.

Calibration Curves for Cast Iron, Ratioed to Total Light (Fi) using RF-GD-OES
Courtesy: Claude Blain, France

To improve the precision it is common to use ratios and there are four common ways to do this:

Normal Method

The elemental intensity is divided by a reference intensity I_j, usually the intensity of the major element or the total light (as shown above) or argon intensity

There is no satisfactory theory to describe total light or argon intensity, so their use can only be justified empirically, i.e. they are seen to improve precision in certain cases.

The use of the major element intensity is more problematic. The intensity depends on the stability of the plasma (which is the main reason we are using it) but it also depends on the product c_j.q_M. Therefore it will change if either its concentration or the sputtering rate changes from sample to sample. It is therefore most useful when there is no significant change in its concentration. But it does have the advantage that it will correct for changes in q_M.

Hence the Normal method, ratioed to the major element, is recommended when there is no significant change in the major element composition. It should then improve precision by adjusting for small changes in the plasma during measurement and for changes in sputtering rate. An example is shown for C 156 in low alloy steel:

Ratio Method

The Ratio method, sometimes called Virtual method, is used principally when there are significant changes in concentration of the major element. Both the Intensities and Concentrations are ratioed

An example is shown for C 156 in stainless steel:

The equation above can be derived readily from the accepted Calibration Function provided:

In other words, the reference intensity must be first order and with negligible background signal. This assumption then leads to

where we note the background term is no longer constant. This has major consequences at low concentrations.

If the reference signal is not first order or has significant background then the use of the ratio or virtual method will lead to systematic errors in calibration and analysis.

100% Method

This method is similar to the Normal method except that in analysis concentrations are normalised to 100%. This has the advantage of further correcting for changes in sputtering rate from sample to sample (to understand why see the discussion on quantitative depth profiling). The 100% method can therefore extend the range of samples that can be analysed by the Normal method. But it has the disadvantage that all major and minor elements must be included or the normalisation will give strange results.

References:

1. Th.Nelis, M.Aeberhard, R.Payling, J.Michler P.Chapon; Relative calibration mode for compositional depth profling in GD-OES; J. Anal. At . Spectrom., 2004 , 19 , 1354 – 1360

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Sputtering Rate Method

The sputtering rate method described below can be used for bulk analysis and quantitative depth profiling or better Content Depth profiling(CDP).

For bulk analysis it has the advantage that it will work for a wide range of materials but the disadvantage that the sputtering rates of the calibration materials have to be measured and this is time consuming and introduces significant measurement uncertainties.

For composition depth profiling (CDP) of layered materials, we have no choice, we must use the sputtering rate method.

We do this by measuring the sputtering rates in the calibration samples and plotting c_i.q_M versus I_i.

We must also deal with the fact that R_i can vary. There are three ways to do this

• choose a mode of operation where the change in R_i is negligible (no such mode exits, but some come close)
• choose calibration samples similar to the unknown samples so that they have the same R_i. This is called matrix matching
• choose a mode of operation where the change in R_i is known and correct for R_i. For this approach to work long term, the correction must be done as part of the calibration.

Two examples of the sputtering rate method are shown below, for zinc coatings on steel: materials used for calibration include stainless steel (SS...), brass (CTIF...), steel (NBS...), Zn-Al (43Z...), etc.

Nickel 341.477 nm, in first order. Vertical axis: c_Ni.q_M; horizontal axis: I_Ni (V)

Aluminium 396.152 nm, in second order, because of self-absorption. Vertical axis: c_Al.q_M; horizontal axis: I_Al (V)

Some problems inherent in calibration methods can be found by going to the examples section.

First published on the web: 1 March 2000.

Authors: Richard Payling & Thomas Nelis

References and further reading:

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Dependent variables in calibration curves

In calibration, concentration (or concentration times sputtering rate) is plotted as the dependent variable (y-axis) and Intensity is plotted as the independent variable (x-axis). Why? And why do some text books show calibration curves the other way round? Are they equivalent? In general, they are not equivalent.

The reason we plot Intensity as the independent variable: it is convenient for analysis. When an analyte intensity is measured, it can then be converted directly into concentration (or concentration times sputtering rate) using the calibration function.

c_iq_M or I_i?

In GD-OES, a second order calibration function could be expressed either as

or

And either could be transformed into the other using the algorithm in ref (1). For example, the first equation would transform into the second with

So, mathematically, if the coefficients of the polynomials are known then one equation could be transformed into the other. The problem is that the coefficients are not known, until after regression. And, in general, the values of the regression coefficients will depend on how the regression is carried out. Crucial to this is the choice of the dependent variable.

Example

Consider the following calibration for Al 396 nm, made with RF GD-OES, using a variety of matrices, after DC bias correction. For simplicity I have subtracted the background signal.

First the calibration is plotted with c_Al.q_M as independent variable.

Then with I_Al as independent variable.

If we transform the equation in the first graph to match the second we get

which is significantly different from the equation shown with the graph. This transformed equation is plotted as the dotted line in the second graph. The two calibration curves are clearly not the same.

Explanation

The inverse function of a 2nd order polynomial involves necessarily a square root. The inverted 2nd order polynomial is therefore not exactly another second order polynomial.

The exact form of GD calibration curve involving self-absorption is not even a second order polynomial but a rather complex function. If we can use 2nd order polynomial to fit the calibration data, this only means that out model reproduces the data within the limits of the uncertainty.

When calibration function rather than analytical function are used to fit the calibration data, the inversion process just needs to be sufficiently precise to reproduce the data within the limits of measurement uncertainty, which may involve a higher order polynomial for the inverted function. In general, however, a second order polynomial will be sufficient, to reproduce the data, within the limits. For more detailed discussion check our the references given below.

References

1. M Abramowitz and I A Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, John Wiley, New York (1972), p 16.
2. Th. Nelis, R. Payling; Glow Discharge Optical Emission Spectroscopy: A practical Guide; RSC Analytical Spectroscopy Monographs; RSC 2004

First published on the web: 1 June 2000.

Authors: Richard Payling & Thomas Nelis

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Problems in Calibration using a non-linear reference line in relative methods

Consider a hypothetical series of calibration samples with only three main elements: e1, e2 and e3. Let e2 and e3 vary randomly between 0% and 30%, and e1 is the remainder. The elements could be, for example, Fe, Cr and Ni in a range of high alloy steels.

Let us assume there is no error in the intensities or concentrations and that the intensities of e2 and e3 are linear with concentration but that our reference element e1 is in second order.

If we use the Normal method ratioed to e1, we get something like this:

and it would be tempting to delete one of the upper points to improve the fit. Though this would not be justified since there is no error in our data and we have simply used the wrong calibration function.

If we use the Ratio or Virtual method, we get something like this:

and it would be tempting to use second order to get a better fit near 0, though this would not completely remove the scatter. The scatter is there simply because of the limitations of the Ratio or Virtual method when the reference intensity is not proportional to its concentration.

This problem is typically encountered when using resonant lines as an internal standard. Examples are: Fe 372nm for steel, Cu 325nm for copper base material, Al 396nm for aluminium alloys etc.

References:

1. Th.Nelis, M.Aeberhard, R.Payling, J.Michler P.Chapon; Relative calibration mode for compositional depth profling in GD-OES; J. Anal. At . Spectrom., 2004 , 19 , 1354 – 1360, DOI: 10.1039/b406187j

First published on the web: 15 May 2000

Authors: Richard Payling & Thomas Nelis

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Effect of Self-absorption on analytical characteristics of an emission source

After having described the effect of self-absorption on the emission line shape The Spectroscopy Net; it is now interesting to look at the effect of self-absorption on the spectrochemical analysis, i.e. the shape of the calibration curves. We can model the effect of self-absorption on the calibration curve if we assume that nothing changes in the emission source (discharge) as we introduce more material. In particular if the spatial distribution of the emitters and absorbers does not change and their relative abundance is constant.
Summarising the effect of self-absorption on the line shape :

• In the case of the separated emitter-absorber model a clear dip in the emission line develops with increasing density of the optical active species.
• In the second case where emitter and absorber are simultaneously present in the same region a flat top profile develops with increasing atom density.

To estimate the effect of self-absorption on calibration curves for spectrochemical analysis using a discharge source, we will first look at the evolution of the transmitted light across the discharge source, when the concentration of the analyte species is increased.
We will consider a configuration close to a Grimm-type discharge cell. The sample material is introduced into the discharge cell on one side and the emitted light is observed from the opposite side of the tubular discharge cell.

Case 1

In case 1 the emitter and absorber are separated in space. In the emitter region the light intensity, directed to the detector increases linearly across the region, once the absorbing region is reached the intensity will decrease again, but exponentially now. The light generated in the emitting region, detectable at the end of the emitting region will increase linearly with the abundance of the analyte species. At the same time the absorption coefficient describing the absorption in the absorbing region will also increase linearly, the effect on the transmitted light is, not linear due to the exponential character of the Lambert-Beer law.

The different behaviour of the emission and absorption can be illustrated choosing reference conditions for the analyte density. We define ‘IE’ ‘T ‘ the emitted light and transmission coefficient of the absorbing layer respectively. We express any analyte density ’c’ relative to the reference density ‘cr’.

When we double the analyte density, the emitted light will be multiplied by a factor of two, the transmission coefficient, which is smaller than one, will be squared.
As a consequence of this non linear dependence of the transmission coefficient on the species density the observed integral intensity goes through a maximum before it decreases as yet more material is introduced to the light source. The calibration curve is reversed. For optical thin media, the absorption is a minor effect only and the calibration curve is nearly linear. For optical thick layers the absorption effect is significant.

 

Case 2

In the second case, when emitting and absorbing species are present in the same region, the line intensity first increase with the increasing number of emitting and absorbing species without a significant change in the line shape, the calibration curve is nearly linear. At a certain point, however, the flat top profile will be more pronounced and the increase in integral intensity will slow down. The figure below illustrates the increase of intensity across the emission source, for different species density. We consider here light directed towards the right, where the imaginary detector sits. For optically thin sources, the light intensity increases linearly from the left to the right. For increasing densities, or optical thickness, the saturation effect becomes noticeable. The limit for very large sources does not depend on the analyte species density, but only on the ratio of absorbing and emitting species. The analyte density only determines how quickly this limit is reached. The trend, however, will not be reversed, the calibration curve will not show a local maximum.

When calibrating an optical emission spectrometer we relate the detected line intensity to the atomic concentration. A “normal” spectrometer used for this purpose will not resolve the actual line shape, but detect the integral intensity. As the concentration of the analyte species increases more photons will be emitted, but also more will be absorbed.
If we assume no self absorption at all (no absorbing atoms present in the source), the detected (transmitted) light intensity will increase linearly with the density of analyte species in the discharge.

In case 2, the mixed emitter/absorber, the transmitted line intensity first increases “almost” linearly with the analyte density, the calibration curve is expected to be linear. As the analyte density increases the emission line develops a flat top shape, hardly increasing in intensity as the density increases, the integral line intensity, however increases slowly, because the line becomes broader. There is no inversion of the calibration curve in this case. In the first case, the separated emitter/absorber, again the line intensity increases linearly with the analyte density as long as the effect of self-absorption is small, but eventually the calibration curve deviates from linearity, as the dip in the line centre develops, the calibration curve is inverted, the transmitted intensity drops as the analyte density increases. Measuring the transmission line profile with a “low resolution” spectrometer typically used for analytical purposes will not necessarily show the dip, because the line profile is actually determined by the instrumental spectral resolution and not by the characteristics of the emission source. The situation is a real discharge source is of course more complex. Neither is the distribution of emitters and absorbers in the discharge volume constant, nor is their relative distribution constant. The gas temperature will vary in the different regions of the discharge. The temperature will be high relatively close to the cathode area to drop further away from it. It is therefore obviously difficult to predict the exact line shape of the emitted (transmitted) light and consequently to derive a general form of the calibration curve, how they deviate from the ideal linear case, when no self-absorption is observed to the non-linear case in the presence of self-absorption.

The purpose of the above figure [1] is to illustrate the different spatial distribution of absorbers (left) and emitters (right) in a discharge cell. The sample is situated in the centre at the bottom of the discharge cell and acts as the discharge cathode, For end-on operation, the photon detector would be situated at the top of the image. The figure shows, in fact, the calculated spatial distribution of ground-state atoms (left) and ions (right) in a discharge cell. The ion-density distribution does certainly not represent the excited atom (emitter) distribution, but as the recombination reaction of the ions with free electrons in the discharge will generate excited atoms, we just pretend it does, for this qualitative discussion assumption is justified. The atom density is highest close to the cathode (sample) where the ion density maximises at 8 mm into the discharge volume. Comparing the two figures clearly shows that the emitter and absorber density both vary within the discharge volume, but they vary differently. The expected shape of the calibration curve is therefore somewhere in between the two cases described earlier.

Refs:
[1] Annemie Bogaerts Renaat Gijbels, Glen P. Jackson ; Modeling of a millisecond pulsed glow discharge: Investigation of the afterpeak  J. Anal. At. Spectrom., 18, 2003, 533–548;  DOI: 10.1039/b212606k

First published on the web: 15. 03. 2008.

Author: Thomas Nelis. The text resumes and extends topics presented by Prof. E.B Steers during the first GLADNET Training course held in Antwerp, Be, in September 2007.

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Self-absorption in a Glow Discharge

Despite the complexity of selfabsorption on the anaylitcal characteristics of a glow discharge source, Richard Payling has developped a semi-empirical formula for the effect of self absorption on calibration curves and has applied this with some success in many applications.

Modelling the glow discharge as an inner emitting/absorbing region and outer absorbing region, where the probability of self-absorption increases exponentially with the number of absorbers, led to the following equation

where I_i is the measured intensity, k_i is a calibration constant, c_i is the concentration in the sample, q_M is the sputtering rate, b_i is the background signal, s_E is the self-absorption coefficient for the inner emission/absorption region, and s_S the coefficient for the outer absorbing region.⁽¹⁾ Typically s_S ~ 0.1xs_E, ie, most of the absorption occurs in the emission region where the density of absorbers is higher.

We could use this equation directly but it would involve non-linear regression and non-linear regression can sometimes lead to funny results. One way around the problem is to expand the right hand side as a polynomial, neglecting b_i for the moment, using the expansions

(this unusual expansion is explained in ref (1)) and

where a₁ = 0.9664 and a₂ = 0.3536.⁽²⁾ The resulting polynomial equation is

which is valid provided c_iq_M is not too big, and assuming s_S ~ 0.1s_E. Now we can change the polynomial to be powers of I_i using an algorithm in ref (2)

where K_i = 1/k_i and we have brought the background term back in the form of a concentration (BEC).

One thing worth noting about this final polynomial is that other than the background term which is negative, all the other terms are positive, which means it is a very well behaved polynomial. It won't go off and do funny wobbles or big side-trips the way free polynomials can! And it doesn't matter how many orders the polynomial is expanded into, the higher order terms always remain positive. When self-absorption is small it is therefore possible to start with a first or second order and then include higher orders as s_E or c_iq_M increases.

Self-absorption for Zn 213 nm

References:

1. R Payling, M S Marychurch and A Dixon, in R Payling, D G Jones and A Bengtson (Eds), Glow Discharge Optical Emission Spectrometry, John Wiley, Chichester (1997), pp 376-91;
2. R Payling, Spectroscopy 13, 36 (1998).
3. M Abramowitz and I A Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, John Wiley, New York (1972), pp 16, 71.

First published on the web: 1 June 2000.

Author: Richard Payling