Parameters for specication of sharpness in powder diffraction peak shape

Ceramics Research Laboratory, Nagoya Institute of Technology
â—‹Takashi Ida

Powder diffraction peak profile is characterized by the intensity, position, line width, asymmetry and sharpness in peak shape. Those characteristics can be uniquely determined for finite-range peak profile functions by the moments of the function. The zeroth, first, second, third and fourth-order moments are respectively connected with the integrated intensity, mean position, standard deviation, asymmetry and kurtosis (sharpness in peak shape). However, it is well known that the moments higher than zeroth order cannot be defined for the Lorentzian function, while it is often observed that the experimental diffraction peak profiles have nearly Lorentzian-like characters. The higher order moments can neither be defined for the Laue function, which is the most elementary theoretical diffraction peak profile function.
In this study, the author has defined a kind of moments for the Fourier transform of the profile function (Fourier mean width and curvature width), and compared them with the integral breadth, full width at half maximum and Fourier initial slope for typical diffraction peak profile functions, Gaussian, Lorentzian, pseudo-Voigt, Voigt, Pearson VII, Laue and theoretical size broadening profile for log-normally distributed spherical crystallites (SLN profile). It is shown that the ratio of the Fourier mean width to integral breadth of a function can be treated as the indicator of the sharpness, and well represents the apparent characters of peak shape.
It is expected that application of this indicator of the sharpness will provide appropriate models or effective approximations for such a complicated theoretical peak profile functions as the SLN profile.