Numerical Solution for Local Emission Coefficients in Axisymmetric Self-Absorbed Sources

Abstract

This paper presents a method for determining local emission and absorption coefficients in an axisymmetric, self-absorbed source which is applicable to laboratory arc plasma jets in current use. In spectrometric diagnosis of arc plasma, physical and chemical properties are established through their theoretical relation to experimental line emission coefficients. These coefficients, however, are not measured directly. The measured line intensity is a result of emitters along the spectroscopic line of sight and is affected by optical thickness, a result of physical depth and absorption in the source. Previous techniques (References 1 through 6), involving a single Abel integral equation, neglect the effect of absorption. There is evidence (References 7 and 8), however, that certain lines are absorbed. In order to minimize the errors in the resulting plasma properties, as discussed in Reference 9, it is desirable to perform an accurate mathematical inversion and account for self-absorption. This requires the solution of two integral equations of the first kind. The method developed here requires spectroscopic observation of the source with and without the presence of a plane mirror (Figure 1). This technique, proposed by Pearce [10], supplies the additional information necessary to experimentally determine the local absorption coefficients. These are used in the solution for emission coefficients. The method is based on an exact solution of the equation of transfer in local radial zones of constant emission and absorption coefficients. The technique has been applied to analytical functions which approximate data to be expected. The accuracy of the numerical methods was determined by comparison with the analytical solutions.