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Self-absorption II

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 Ii is the measured intensity, ki is a calibration constant, ci is the concentration in the sample, qM is the sputtering rate, bi is the background signal, sE is the self-absorption coefficient for the inner emission/absorption region, and sS the coefficient for the outer absorbing region.(1) Typically sS ~ 0.1xsE, 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 bi for the moment, using the expansions

[Expansion equation]

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

[Expansion equation]

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


which is valid provided ciqM is not too big, and assuming sS ~ 0.1sE. Now we can change the polynomial to be powers of Ii using an algorithm in ref (2)


where Ki = 1/ki 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 sE or ciqM increases.


Self-absorption for Zn 213 nm


  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