On the Intrinsic Limits of the Convolution Method to Obtain the Crystallite Size Distribution from Nanopowders Diffraction

The present work briefly reviews the convolution of crystallite shape functions and discusses its experimental limitations. The diffraction from a theoretical spherical shape powder is used to exemplify the limits of the convolution procedure. Mean lattice distortions were not considered since the discussed limitations are inherent to the convolution method using Fourier transforms. The diffraction pattern and the convolution were calculated using appropriate macros for the Topas program. It is shown that very small crystallites require a large 2θ convolution span and the smallest subdivision for the distribution will depend on this convolution span. To show the importance of the convolution limits and its application, the nanocrystalline CeO2 round-robin diffraction pattern was evaluated. The chord frequency distribution by XRD showed conformity with the chord distribution calculated from a grain size histogram obtained by transmission electron microscopy for this sample.

本研究简要综述了微晶形状函数的卷积方法，并探讨了其实验局限性。以理论球形粉末的衍射结果为例，阐释了卷积流程的局限性。由于所讨论的局限性是采用傅里叶变换（Fourier transforms）的卷积方法所固有的，因此未考虑平均晶格畸变。本研究借助针对Topas程序开发的专用宏命令，计算了衍射图样与卷积结果。研究表明，尺寸极小的微晶需要较大的2θ卷积跨度，而分布的最小细分步距将取决于该卷积跨度。为阐明卷积局限性的重要性及其应用场景，研究人员对纳米晶二氧化铈（CeO₂）的循环比对衍射图样开展了评估。本样品通过X射线衍射（XRD）得到的弦频率分布，与通过透射电子显微镜（TEM）获得的晶粒尺寸直方图计算得到的弦分布结果一致。