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I was reading through Eric Mittemeijer's book 'Modern Diffraction Methods' when I found the following statement in the Rietveld chapter by Robert Dinnebier.

It is common practice to use orthogonal Chebyshev polynomials of higher order (typically 5–10) in combination with an 1/(2θi) term to describe a steep increase in the background at low scattering angle [...]. If higher-order polynomials are used, different types of correlations can occur, which require careful checking of the correlation matrix.

When I am usually describing my background in FullProf I use the '6-Coefficients polynomial function'. But this seems to cause more problems the more parameters I refine. So I switched to the 'Chebyshev polynomial' and refinded all 24 parameters (just to make sure). This caused a large jump in the X² from about 10 to 6.5 while the background was much better described.

My question is, although it is mentioned in the book, that Chebyshev is more common and I should be careful with using higher order polynomials, is this something I can use in general for my refinements? Can I just automatically assume for most x-ray determined backgrounds that using a Chebyshev polynomial with as many parameters refined as possible will give me useful results? By useful I mean good and especially 'allowed' results. Because sometimes FullProf refines parameters into regions where they are not defined like a negative η for the Pseudo-Voigt function.

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If I recall correctly, Chebyshev polynomials are preferred over simple polynomials because of their termination behavior (i.e. they tend not to diverge wildly), although I don't have a reference for this handy.

Dinnebier's admonition to use higher-order polynomials with care is effectively a warning not to over parameterize your fit. If you inspect the two fits with 6- and 24-term backgrounds, my guess is that you will find the over parameterized version nudged your Bragg peaks up or down as necessary to better match your observed data. Hence, your residual has diminished but your fit is not more meaningful.

An overfitted background precludes accurate extraction of the observed Bragg intensities, leading to correlated errors in related factors (e.g. occupancy, position, etc.).

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  • $\begingroup$ In my understanding, Chebyshev polynomials diverge as wildly as other polynomials, but they are orthogonal (in some sense), and as such, produce more independent refinement variables. $\endgroup$ – Ivan Neretin Jun 1 at 19:56

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