International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.3, pp. 263-264
Section 3.3.2.1. Introduction – symmetric peak profiles^{a}Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA |
The realization that the neutron powder diffractometer at the Reactor Centrum Nederland, Petten, produced powder peak profiles that were Gaussian in shape led Rietveld (1967) to develop a full-pattern method for crystal structure refinement (Rietveld, 1967, 1969), now known as the Rietveld refinement method. The Gaussian is formulated aswhere the width of the peak is expressed as either the full width at half-maximum (FWHM = Γ_{G}) or as the variance (σ^{2}). Rietveld also recognized the earlier analysis of the resolution of a neutron powder diffractometer by Caglioti et al. (1958), who showed that the contributions from the source size, collimators and monochromator crystal mosaic spread and scattering angle could be combined analytically to givewith U, V and W adjustable during the Rietveld refinement. A modified form of this may have more stability in refinement (attributed to E. Prince by Young & Wiles, 1982):where K_{0} is arbitrarily chosen as 0.6.
Improvements in the resolution of neutron powder diffractometers and (more importantly) attempts to apply the Rietveld method to X-ray powder diffraction data required the development of new powder profile functions (Malmros & Thomas, 1977; Young et al., 1977; Young & Wiles, 1982); this is because the Gaussian function [equation (3.3.3)] gave poor fits to observed peak profiles, partially because of the Lorentzian emission line profile (G_{λ}) from laboratory X-ray tubes. Many functions were considered, including Lorentzian (`Cauchy'), various modified Lorentzians, Pearson VII and pseudo-Voigt. Of these the last two performed (on individual peak fits) about equally well; functional forms are:
pseudo-Voigt functionwhere Γ_{L} is the FWHM of the Lorentzian peak and Γ(μ) in the Pearson VII function is the Gamma function; μ may vary between 0 and ∞, and μ is the half width at (1 + 1/μ)^{−μ} of the peak height (David, 1986); P_{P7}(Δ, Γ, 1) ≃ P_{L}(Δ, Γ) and P_{P7}(Δ, Γ, ∞) ≃ P_{G}(Δ, Γ). Although the Pearson VII function performs well in individual peak fits, it is of little use for Rietveld refinements because of the difficulty in relating its coefficients to physically meaningful characteristics of the sample and will not be considered further in this discussion.
The pseudo-Voigt function is an approximation to the Voigt function, which is the convolution of a Gaussian and a Lorentzian:
A number of formulations have been proposed for the pseudo-Voigt coefficients to make the best fit to the corresponding Voigt function (Hastings et al., 1984; David, 1986; Thompson et al., 1987). The latter is most commonly used and gives overall the FWHM, Γ and the mixing coefficient, η, to be used in equation (3.3.8) as functions of the individual FWHMs Γ_{G} and Γ_{L}:
The alternative given by David (1986) uses a more generalized version of the pseudo-Voigt function,where Γ = Γ_{G} + Γ_{L}, ρ_{1} = Γ_{G}/Γ and ρ_{2} = Γ_{L}/Γ; this is claimed to match the Voigt function to better than 0.3%.
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