diffraction determination of mean size of coherent scattering regions
(rus. дифракционное определение среднего размера областей когерентного рассеяния)
—
indirect method of determining the mean size of small particles (or, more correctly, coherent scattering regions) from the broadening of X-ray diffraction reflections with decreasing particle (grains) size in compact or powdered nanostructured substances and materials.
Description
The diffraction method enables the estimation of the size of the particles (grains) averaged over the volume of the studied substance and somewhat lowered as compared to the results of electron microscopy.
The small particle size is not the only possible reason for the broadening of diffraction reflections. The broadening of the reflections is also due to microstrain and chemical inhomogeneity, i.e. heterogeneity of the compound in the volume of the test sample. The values of the broadening caused by small grain size, strain and inhomogeneity are proportional to
, 
and 
, respectively, where 
is the diffraction angle. Due to different angular dependence, the three types of broadening can be distinguished.
The profile of diffraction reflection is characterised by the Full Width at Half-Maximum (FWHM). The best description of the reflection profile is provided by the pseudo-Voigt function, which is a superposition of Lorentz and Gauss functions. In a real experiment, due to the finite resolution of the diffractometer, the reflection width can not be smaller than the instrument width. This means that the reflection broadening b shall be determined in relation to the instrument width, i.e. the diffractometer resolution function FWHMR, as follows
.
The procedure of the diffraction experiment for determining the mean size of coherent scattering regions (particle size), microstrain and inhomogeneity from the reflection broadening values is as follows:
1) measure the X-ray diffraction pattern of the reference substance and determine the resolution function of the diffractometer;
2) measure the X-ray diffraction pattern of the substance under study and determine the reflection width;
3) determine the reflection broadening of the substance under study as a function of the diffraction angle;
4) determine the contributions to the broadening by small particle size, microstresses and inhomogeneity of the studied substance;
5) estimate the mean size of coherent scattering regions (particles, grains), the values of microstresses and inhomogeneity.
The small particle size is not the only possible reason for the broadening of diffraction reflections. The broadening of the reflections is also due to microstrain and chemical inhomogeneity, i.e. heterogeneity of the compound in the volume of the test sample. The values of the broadening caused by small grain size, strain and inhomogeneity are proportional to
, and , respectively, where is the diffraction angle. Due to different angular dependence, the three types of broadening can be distinguished. The profile of diffraction reflection is characterised by the Full Width at Half-Maximum (FWHM). The best description of the reflection profile is provided by the pseudo-Voigt function, which is a superposition of Lorentz and Gauss functions. In a real experiment, due to the finite resolution of the diffractometer, the reflection width can not be smaller than the instrument width. This means that the reflection broadening b shall be determined in relation to the instrument width, i.e. the diffractometer resolution function FWHMR, as follows
. The procedure of the diffraction experiment for determining the mean size of coherent scattering regions (particle size), microstrain and inhomogeneity from the reflection broadening values is as follows:
1) measure the X-ray diffraction pattern of the reference substance and determine the resolution function of the diffractometer;
2) measure the X-ray diffraction pattern of the substance under study and determine the reflection width;
3) determine the reflection broadening of the substance under study as a function of the diffraction angle;
4) determine the contributions to the broadening by small particle size, microstresses and inhomogeneity of the studied substance;
5) estimate the mean size of coherent scattering regions (particles, grains), the values of microstresses and inhomogeneity.
Illustrations
|
Spreading of n-WC tungsten carbide nanopowder diffracting peaks with mean size of particles <D> = 20 ± 10 nm in comparison with coarse-grained WC carbide powder diameter (D = 6 um). Microstresses (ε = 0,35 ± 0,03%) make a contribution to spreading of diffracting peaks n-WC besides the small size of the particle. |
Author
- Gusev Alexander I.
Sources
- Gusev A. I. Nanomaterials, Nanostructures, and Nanotechnologies (in Russian) // Fizmatlit, Moscow (2007) - 416 pp.
- Gusev A.I., Kurlov A. S. Certification of nanocrystalline materials by particle (grain) size (in Russian) // Metallofizika i novejjshie tekhnologii. 2008. V. 30. No5. 679–694 pp.