Selfabsorption IISelfabsorption in a Glow DischargeDespite the complexity of selfabsorption on the anaylitcal characteristics of a glow discharge source, Richard Payling has developped a semiempirical 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 selfabsorption 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 selfabsorption 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 nonlinear regression and nonlinear 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 a_{1} = 0.9664 and a_{2} = 0.3536.^{(2)} The resulting polynomial equation is
which is valid provided c_{i}q_{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 sidetrips 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 selfabsorption is small it is therefore possible to start with a first or second order and then include higher orders as s_{E} or c_{i}q_{M} increases.
Selfabsorption for Zn 213 nm References:  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 37691;
 R Payling, Spectroscopy 13, 36 (1998).
 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
