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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.

ciqM or Ii?

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

[Ii=f(ci.qM)]

or
 [ci.qM=f(Ii)]

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

[Ai][Bi][Di]

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 cAl.qM as independent variable.

[Graph of IAl vs cAl.qM][Graph of cAl.qM vs IAl]

Then with IAl as independent variable.

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

[cAl.qM]

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 2003

First published on the web: 1 June 2000.

Authors: Richard Payling & Thomas Nelis