International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.3, pp. 263-264

Section 3.3.2.1. Introduction – symmetric peak profiles

R. B. Von Dreelea*

aAdvanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA
Correspondence e-mail: vondreele@anl.gov

3.3.2.1. Introduction – symmetric peak profiles

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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[link]) to develop a full-pattern method for crystal structure refinement (Rietveld, 1967[link], 1969[link]), now known as the Rietveld refinement method. The Gaussian is formulated as[\eqalignno{{P}_{G}(\Delta, {\Gamma }_{G}\,{\rm or}\, {\sigma }^{2})&= {{(8\ln2)^{1/2}}\over{{\Gamma }_{G}(2\pi)^{1/2 }}}\exp\left({{-4\ln2{\Delta }^{2}}\over{{\Gamma }_{G}^{2}}}\right)&\cr &= {{1}\over{(2\pi {\sigma }^{2})^{1/2}}}\exp\left({{-{\Delta }^{2}}\over{2{\sigma }^{2}}}\right), &(3.3.3)\cr}]where the width of the peak is expressed as either the full width at half-maximum (FWHM = ΓG) or as the variance (σ2). Rietveld also recognized the earlier analysis of the resolution of a neutron powder diffractometer by Caglioti et al. (1958[link]), who showed that the contributions from the source size, collimators and monochromator crystal mosaic spread and scattering angle could be combined analytically to give[{\Gamma }_{G}^{2} =U\tan^2 \theta +V\tan\theta+W\eqno(3.3.4)]with 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[link]):[{\Gamma }_{G}^{2}=U' (\tan\theta-K_0)^2 +V' (\tan\theta -K_0) +W', \eqno(3.3.5)]where K0 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[link]; Young et al., 1977[link]; Young & Wiles, 1982[link]); this is because the Gaussian function [equation (3.3.3)[link]] gave poor fits to observed peak profiles, partially because of the Lorentzian emission line profile (Gλ) 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:

Lorentzian `Cauchy' function[{P}_{L}(\Delta, {\Gamma }_{L})= \left({{{\Gamma }_{L}}\over{2\pi }}\right)\left\{{{4}\over{\left[{\Gamma }_{L}^{2}+{\left(2\Delta \right)}^{2}\right]}}\right\},\eqno(3.3.6)]

Pearson VII function[{P}_{{\rm P}7}(\Delta, \xi, \mu )= {{\Gamma(\mu )}\over{\xi \Gamma \left(\mu -{{1}\over{2}}\right)(\mu \pi)^{1/2 }}}/\left(1+{{\Delta ^{2}}\over{\mu \xi ^{2}}}\right)^{\mu },\eqno(3.3.7)]

pseudo-Voigt function[{P}_{\rm PV}(\Delta, \Gamma, \eta )= \eta {P}_{L}(\Delta, \Gamma )+(1-\eta ){P}_{G}(\Delta, \Gamma ),\eqno(3.3.8)]where ΓL 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[link]); PP7(Δ, Γ, 1) ≃ PL(Δ, Γ) and PP7(Δ, Γ, ∞) ≃ PG(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:

Voigt function[\eqalignno{{P}_{V}(\Delta, {\Gamma }_{L},{\Gamma }_{G})&=\int\limits_{-\infty }^{\infty }{P}_{L}(\Delta, {\Gamma }_{l}){P}_{G}(\Delta -\delta, {\Gamma }_{G})\,{\rm d}\delta&\cr &=\left({{4\ln2}\over{\pi \Gamma _{G}^{2}}}\right)^{1/2}{\rm Re}[\exp(-{z}^{2}){\rm erfc}(-iz)],&(3.3.9)\cr}]where [z=\alpha +i\beta], [\alpha =(4\ln2)^{1/2} \Delta /{\Gamma }_{G}] and [\beta =(\ln2)^{1/2} {\Gamma }_{L}/{\Gamma }_{G}].

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[link]; David, 1986[link]; Thompson et al., 1987[link]). 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)[link] as functions of the individual FWHMs ΓG and ΓL:[\eqalignno{\Gamma &= [({\Gamma }_{G}^{5}+2.69269\Gamma _{G}^{4}\Gamma _{L}+2.42843\Gamma _{G}^{3}\Gamma _{L}^{2}+4.47163{\Gamma }_{G}^{2}{\Gamma }_{L}^{3}&\cr &\quad +0.07842{\Gamma }_{G}\Gamma _{L}^{4}+{\Gamma }_{L}^{5})]^{1/5},&(3.3.10)\cr}][\eta =1.36603\left({{\Gamma _{L}}/{\Gamma }}\right)-0.47719\left({{\Gamma _{L}}/{\Gamma }}\right)^{2}+0.11116\left({{\Gamma _{L}}/{\Gamma }}\right)^{3}.\eqno(3.3.11)]

The alternative given by David (1986[link]) uses a more generalized version of the pseudo-Voigt function,[\eqalignno{&{P}_{\rm PV}(\Delta, {W}_{G},{W}_{L},{\eta }_{G},{\eta }_{L})= {\eta }_{L}{P}_{L}(\Delta, {W}_{L})+{\eta }_{G}{P}_{G}(\Delta, {W}_{G}),&\cr &{\eta }_{G}=0.00268\rho_{1}+0.75458\rho_{1}^{2}+2.88898\rho_{1}^{3}-3.85144\rho_{1}^{4}&\cr &\quad\quad -0.55765\rho_{1}^{5}+3.03824\rho_{1}^{6}-1.27539\rho_{1}^{7},&\cr &{\eta }_{L}=1.35248\rho_{2}+0.41168\rho_{2}^{2}-2.18731\rho_{2}^{3}+6.42452\rho_{2}^{4}&\cr &\quad\quad -10.29036\rho_{2}^{5}+6.88093\rho_{2}^{6}-1.59194\rho_{2}^{7},&\cr &{W}_{G}=\Gamma (1-0.50734\rho_{2}-0.22744\rho_{2}^{2}+1.63804\rho_{2}^{3}&\cr &\quad\quad -2.28532\rho_{2}^{4}+1.31943\rho_{2}^{5}),&\cr &{W}_{L}=\Gamma (1-0.99725\rho_{1}+1.14594\rho_{1}^{2}+2.56150\rho_{1}^{3}&\cr &\quad\quad -6.52088\rho_{1}^{4}+5.82647\rho_{1}^{5}-1.91086\rho_{1}^{6}),&(3.3.12)}]where Γ = ΓG + ΓL, ρ1 = ΓG/Γ and ρ2 = ΓL/Γ; this is claimed to match the Voigt function to better than 0.3%.

References

Caglioti, G., Paoletti, A. & Ricci, F. P. (1958). Choice of collimators for a crystal spectrometer for neutron diffraction. Nucl. Instrum. 3, 223–228.Google Scholar
David, W. I. F. (1986). Powder diffraction peak shapes. Parameterization of the pseudo-Voigt as a Voigt function. J. Appl. Cryst. 19, 63–64.Google Scholar
Hastings, J. B., Thomlinson, W. & Cox, D. E. (1984). Synchrotron X-ray powder diffraction. J. Appl. Cryst. 17, 85–95.Google Scholar
Malmros, G. & Thomas, J. O. (1977). Least-squares structure refinement based on profile analysis of powder film intensity data measured on an automatic microdensitometer. J. Appl. Cryst. 10, 7–11.Google Scholar
Rietveld, H. M. (1967). Line profiles of neutron powder-diffraction peaks for structure refinement. Acta Cryst. 22, 151–152.Google Scholar
Rietveld, H. M. (1969). A profile refinement method for nuclear and magnetic structures. J. Appl. Cryst. 2, 65–71.Google Scholar
Thompson, P., Cox, D. E. & Hastings, J. B. (1987). Rietveld refinement of Debye–Scherrer synchrotron X-ray data from Al2O3. J. Appl. Cryst. 20, 79–83.Google Scholar
Young, R. A., Mackie, P. E. & von Dreele, R. B. (1977). Application of the pattern-fitting structure-refinement method of X-ray powder diffractometer patterns. J. Appl. Cryst. 10, 262–269.Google Scholar
Young, R. A. & Wiles, D. B. (1982). Profile shape functions in Rietveld refinements. J. Appl. Cryst. 15, 430–438.Google Scholar








































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