Peak Search by Second Derivatives

The peak search by 2nd derivatives represents a kind of sharpening (deconvolution), i.e. a division by the Fourier transform of a certain peak shape in the frequency domain. This is possible only if this Fourier transform has no zeroes, i.e. if it monotonically approaches zero. Bromba and Ziegler (1984) report such an algorithm, but its usability for X-ray patterns was not proved until now. 

The 2nd derivative can also be used for the estimation of half-widths by looking for its zeroes at both sides of a minimum. These correspond to the points of inflexion. As they are placed somewhat higher than the points of half heights (in 61% of the height of Gaussian peaks and in 75% of Lorentzian peaks) the distances between both zeroes are smaller than the half widths (for Gaussian peaks 85% of FWHM, and for Lorentzian peaks 58%). In practice. 

Error in the determination of the position of a simulated, asymmetrical reflection (without noise) at 29 = 20° and with 0.17° half width (W). Top With minimum of 2nd derivative calculated with a polynomial of 2nd/3rd order. Middle With zero of the 1st derivative calculated with a polynomial of 3rd/4th. order (best result as long as the filter width does not appreciably surpass the half width) Bottom With zero of 1st derivative with a polynomial of lst/2nd order. (After Huang, 1988, or Huang and Parrish, 1984. ) 

3 Peak Search vdth a Predefined Peak Shape 

The advantage of X-ray powder patterns over other spectra is the roughly common shape of the individual reflections (equal half width and equal shape of the flanks). Therefore, one can use peak search methods that presume a special peak shape. Sanchez (1991) reports a peak search algorithm for Gaussian peaks with an average half width 2D. This method can be easily adapted for Lorentzian peaks (y = A/[l + x — u)/b) ] with FWHM = 2b) or Pearson-VII peaks (y = A/[l + ((x- )/b) f with FWHM = 2b-7( 72-l)). X-ray peaks very often exhibit a peak shape with m between 1.5 and 2. 

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