There are several analytical approaches and commercial software for analyzing spectral peaks. The simplest one assumes a linear background under the peak shown in Fig. 9 as a straight line "c""d".
Fig. 9. Expanded INS spectrum in Fig. 4 shows oxygen – and carbon-peaks. The trapezoidal background under the carbon peak is bound between ‘a" and "b".
Thus, the net number of counts in the peak is defined as the total counts minus the background delimited by the area enclosed by the trapezoid "abcd". This approach is valid provided that the peak is clearly defined and there are no overlapping peaks distorting the area with extraneous counts. The least-squares method offers a more advances analysis wherein the peak is fitted with one or more Gaussian functions, thus partly resolving the problem of overlapping peaks. However, with complete overlap, as might occur when an identical gamma-ray is produced by an interfering element, it is more difficult to resolve. However, since an excited element generally produces more than a single gamma-ray with fixed ratios among them, it is possible to resolve interfering peaks by fitting an entire spectrum instead of the peak alone; this is the library-least-squares method (LLS) [Arinc at al., 1976]. The measured spectra from pure elements, referred to as elemental standard library, are least-squares-fitted to an unknown measured spectrum revealing any discrepancies between the synthetically modeled spectrum and the measured one. The fundamental assertion in the LLS method is that a measured unknown spectrum is a linear superposition of standard reference libraries plus an error term. The multipliers of the standard libraries are found by minimizing the error term, or of the reduced x2 given by equation Eq. 3 [Gardner et al., 1975; Wielopolski, 1981; Wielopolski and Cohn].
bi – counting rate in channel i for the composite spectrum;
aij – counting rate of pure element j in channel i per unit amount of component j; xj – amount of component j in the unknown; m – number of components; n – number of channels;
oi2 – variance of the random error in channel i.
The LLS method requires an extra effort in deriving good elemental libraries with good statistics; in turn, this approach reduces the reported error for the analyzed peak intensities.