Linear Analysis of Spectra

In working with blends of polymers it is sometimes useful to analyse the spectrum from a blend of two or more materials in terms of the spectra from the basic polymers. Linear Analysis is a technique for estimating the proportions of these basis spectra found in the spectrum taken from the composite material. By way of example, consider the spectra shown in Figure 1, where three possible basis spectra are used to analyse the blend.

Figure 1: Polymer blend (left) together with spectra from the basic polymers (right).

The procedure to determine the composition of the polymer blend is as follows:

1) Transform the basis spectra into energy-corrected, background-subtracted normalised spectra.

2) Background-subtract the spectrum to be analysed (Figure 2).

Figure 2

3) Load the transformed basis spectra into the active display tile as overlaid traces.

4) Select the spectra to be analysed in the right-hand-side of the Experiment Frame (Figure 3).

Figure 3

5) Press the Generate Spectra pushbutton in the Linear Analysis section of the PCA property page on the Spectrum Processing dialog window.

A new Experiment Frame in created containing the original spectra analysed, plus each of the weighted basis spectra together with the sum of all the weighted basis spectra. If overlaid in the same tile, this result is not dissimilar to the display of a peak-fitted spectrum using synthetic components. The relative contributions to the polymer blend can be determined using integration regions on each of the basis spectra, while the coefficients determined by the linear analysis are entered into the VAMAS block comment fields for each of the spectra in the new Experiment Frame.

Figure 4

What is Linear Analysis?

Linear analysis is simply a linear least square solution where the loadings for each of the basis functions are constrained to be non-negative. The non-linear least square method used to fit synthetic components allows line-shapes to move relative to one another via the position parameter and alter in shape via the FWHM parameter. While such parameters are necessary in general peak modelling scenarios, in the case of well-defined basis spectra these additional degrees of freedom are undesirable since there is an opportunity to adjust the line-shapes beyond the realms of reality. By energy-calibrating the basis spectra the positional information is fixed and so only the relative intensities of the basis spectra found in the unknown spectrum are determined using the linear analysis option. The equivalent calculation could be performed using line-shapes determined from the basis spectra, however once the line-shapes have been extracted, the curves become abstract shapes open to optimisation without direct reference to the information in the original basis spectra. The linear analysis retains all the information about the basis spectra and the resulting spectra can be managed as if they were ordinary spectra.