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There are various methods for diffraction pattern calculation of powders, differing in Q-averaging procedures. Three of these were studied in detail: (1) The structure factor formula sums up all the hkl planes for a Q-vectors despite their respective orientations. Only peak intensities are calculated and the peak shape is arbitrary. (2) A Monte Carlo (MC) method of averaging used i.a. by the program SASSENA involves a number of random Q-points over which the computed intensity is averaged. (3) The Debye Formula considers only interatomic distances and the integration is taken with respect to scattering angle and the scattering is weighted as the fraction of the Ewald sphere surface area.
The structure factor intensity has to be normalized on the Ewald sphere surface area in order to yield the same result as the Debye formula – with the drawback that the peak shapes are arbitrary. Only a sample of one million Q-points was enough to reproduce the Debye formula for monoatomic powders by the MC method, but still insufficient for biatomic crystals. The time for the monoatomic compound there exceeded an hour (C++) while the Debye calculation is finished on the minutes time scale (python).
Summing up, the Debye formula appears to be the best and the most time-efficient way for crystalline powder diffraction modeling.
