Thomas Noel William
Fachhochschule Koblenz, Fachbereich Ingenieurwesen, Fachrichtung Werkstofftechnik Glas und Keramik, Rheinstrasse 56, 56203 Höhr-Grenzhausen, Germany.
Acta Crystallogr A. 2011 Nov;67(Pt 6):491-506. doi: 10.1107/S0108767311029825. Epub 2011 Oct 13.
A fast Fourier transform algorithm is introduced into the method recently defined for calculating powder diffraction patterns by means of the Debye scattering equation (DSE) [Thomas (2010). Acta Cryst. A66, 64-77]. For this purpose, conventionally used histograms of interatomic distances are replaced by compound transmittance functions. These may be Fourier transformed to partial diffraction patterns, which sum to give the complete diffraction pattern. They also lead to an alternative analytical expression for the DSE sum, which reveals its convergence behaviour. A means of embedding the DSE approach within the reciprocal-lattice-structure-factor method is indicated, with interpolation methods for deriving the peak profiles of nanocrystalline materials outlined. Efficient calculation of transmittance functions for larger crystallites requires the Patterson group symmetry of the crystals to be taken into account, as shown for α- and β-quartz. The capability of the transmittance functions to accommodate stacking disorder is demonstrated by reference to kaolinite, with a fully analytical treatment of disorder described. Areas of future work brought about by these developments are discussed, specifically the handling of anisotropic atomic displacement parameters, inverse Fourier transformation and the incorporation of instrumental (diffractometer) parameters.
一种快速傅里叶变换算法被引入到最近通过德拜散射方程（DSE）计算粉末衍射图谱所定义的方法中[托马斯（2010年）。《晶体学报》A66卷，64 - 77页]。为此，传统使用的原子间距离直方图被复合透射函数所取代。这些函数可以进行傅里叶变换得到部分衍射图谱，将这些部分衍射图谱相加就得到完整的衍射图谱。它们还引出了DSE求和的另一种解析表达式，揭示了其收敛行为。指出了一种将DSE方法嵌入倒易晶格结构因子方法的途径，并概述了用于推导纳米晶体材料峰形的插值方法。对于较大的微晶，高效计算透射函数需要考虑晶体的帕特森群对称性，如α - 和β - 石英的情况所示。通过参考高岭石证明了透射函数适应堆积无序的能力，并描述了对无序的完全解析处理。讨论了这些进展带来的未来工作领域，特别是各向异性原子位移参数的处理、傅里叶逆变换以及仪器（衍射仪）参数的纳入。
