Trend-oriented Peak Search

The refinement of a preliminary peak position can easily be obtained by fitting a parabola a + bv + cx into 5 (or more) points around the local maximum, using the solutions for a, b, and c after Savitzky and Golay. The maximum of this parabola is the wanted refinement of the peak position and is calculated as the root of the 1st derivative y ( max) = 0 = b + 2c.Xinax- So far the 5-point appoximation one gets  

As additional criterion for the acceptance of an X-ray reflection a certain minimal size of the 2nd derivative can be used (denominator in the above-mentioned equation 0, e.g. 2c —1 or —2). This is certainly necessary to avoid a possible division by 0. The points following a maximum must be discarded as further peaks as long as the 1st derivative stays negative, i.e. as one remains in the decreasing slope. 

After peak recognition both flanks are searched for the half height (above background, possibly by interpolation) and the difference of the corresponding x-values is taken as full width at half maximum of this peak (FWHM, short half-width). This difference should surpass a certain threshold (e.g. 0.06° in 20) to accept a found elevation above background as a true reflection. Thus, one has a further possibility at this point to eliminate possible outliers because they cause much narrower peaks than real reflections. 

The right and left half-widths of a peak should not differ too greatly. If one half-width is greater than 1.5 or 2-times the other, there probably exists a shoulder, i.e. a superimposition with a weaker reflection without an individual local maximum. In such a case the half-width estimated by the half-heights on both flanks systematically becomes too large and should be corrected using the smaller of both half-widths only (which hopefully is not affected by a shoulder). 

Shoulders cannot be recognized as separate reflections by this method, but only peaks with individual local maxima in the pattern (roots of the 1st derivative). 

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