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. 263-265

Section 3.3.2. Peak profiles for constant-wavelength radiation (X-rays and neutrons)

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. Peak profiles for constant-wavelength radiation (X-rays and neutrons)

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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%.

3.3.2.2. Constant-wavelength powder profile asymmetry

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Rietveld (1969[link]) noted that at very low scattering angles the peaks displayed some asymmetry, which shifted the peak maximum to lower angles. He ascribed the effect to `vertical divergence' and proposed a purely empirical correction for it. Subsequent authors (Cooper & Sayer, 1975[link]; Howard, 1982[link]; Hastings et al., 1984[link]) offered semi-empirical treatments of the profile shape that results from the intersection of a Debye–Scherrer cone with a finite receiving slit, which is described as `axial divergence'. A more complete analysis of the problem in neutron powder diffraction was offered by van Laar & Yelon (1984[link]), who considered the effect of a finite vertical slit (2H) intercepting a set of Bragg diffraction cones generated from a finite sample length (2S) within the incident beam for a goniometer radius (L). As seen in Fig. 3.3.1[link], this gives peak intensity beginning at 2ϕmin < 2θ via scattering from only the ends of the sample; at 2ϕinfl the entire sample scatters into the detector. The resulting intensity profile is then convoluted with a Gaussian function to give the resulting asymmetric powder line profile (Fig. 3.3.2[link]). This approach was then considered by Finger et al. (1994[link]) for synchrotron powder diffraction and they created a Fortran code that was subsequently adopted via convolution with a pseudo-Voigt function [equation (3.3.12)[link]] for use by many Rietveld refinement codes. Although originally formulated for parallel-beam neutron optics, it was shown by Finger et al. (1994[link]) that it could be equally well applied to diverging X-ray and neutron optics by allowing the sample length to vary during the Rietveld refinement. They also showed that it could be applied to the asymmetry observed at low angles with Bragg–Brentano instrumentation. In that case the detector height is defined by the diffracted-beam Soller slits.

[Figure 3.3.1]

Figure 3.3.1 | top | pdf |

The band of intensity diffracted by a sample with height 2S, as seen by a detector with opening 2H and a detector angle 2ϕ moving in the detector cylinder. For angles below 2ϕmin no intensity is seen. For angles between 2ϕinfl and 2θ, scattering from the entire sample can be seen by the detector. Figure and caption adapted from Finger et al. (1994[link]).

[Figure 3.3.2]

Figure 3.3.2 | top | pdf |

Low-angle synchrotron powder diffraction line (2θ ≃ 4.1°) fitted by the Finger et al. (1994[link]) axial divergence powder line-shape function. The observed points (+), calculated curve, background and difference curves are shown. Note the offset of the peak top from the Bragg 2θ position (vertical line).

Clearly, this asymmetric peak-shape function properly represents the offset of the peak top from the peak position, in contrast to functions such as the split Pearson VII function. Consequently, single peak fits using this function will give peak positions that are more readily indexed using methods such as those described in Chapter 3.4[link] .

3.3.2.3. Peak-displacement effects

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The position of the peak is also affected by various instrumental and geometric effects. For example, the sample position in a Bragg–Brentano experiment is ideally tangent to the focusing circle (Parrish, 1992[link]). A radial displacement, s, of the sample will shift the Bragg peaks according to[\Delta 2\theta =360s\cos\theta /\pi R,\eqno(3.3.13)]where R is the goniometer radius. This is the major peak-displacement effect and can be detected for sample displacements as small as 10 µm.

A similar effect can be observed for Debye–Scherrer instrumentation when the goniometer axis is not coincident with the sample axis; this is a more common problem for neutron powder diffraction instruments where accurate placement of very massive goniometers can be difficult. In this case the peak displacement is[\Delta 2\theta ={{180}\over{\pi R}}({s}_{x}\cos2\theta +{s}_{y}\sin2\theta ),\eqno(3.3.14)]where sx and sy are displacements perpendicular and parallel to the incident beam, respectively, all in the diffraction plane.

In high-resolution instrumentation (even at a synchrotron) goniometer axis displacements less than 10 µm can be detected.

Specimen transparency in Bragg–Brentano diffraction can also cause peak displacements arising from the shift in effective sample position to below the surface at high scattering angles. This shift for a thick specimen is[\Delta 2\theta =90\sin2\theta /{\mu }_{\rm eff}\pi R,\eqno(3.3.15)]where μeff is the effective sample absorption coefficient taking into account the packing density.

3.3.2.4. Fundamental parameters profile modelling

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An alternative method for describing the source and instrumental part of the powder peak profile is to develop a set of individual functions that form the part of the profile arising from each of the instrumental components that shape the beam profile (Cheary & Coelho, 1992[link], 1998a[link],b[link]). Ideally, each function is parameterized in terms of the physical parameters of the corresponding instrument component (e.g. slit width and height, sample dimensions and absorption, source size and emission characteristics, etc.), which are known from direct measurement. The set of functions are then convoluted via fast mathematical procedures to produce a line profile that matches the observed one. Any remaining profile-broadening parameters (e.g. for sample crystallite size and microstrain, see Section 3.3.5[link] for details) are then allowed to adjust during a Rietveld refinement. By employing this fundamental parameters (FP) approach, these parameters are unaffected by any instrumental parameterization.

The FP method offers two clear advantages over the more empirical approach outlined in Sections 3.3.2.1[link]–3.3.2.3[link] above: (i) it can more closely describe the actual instrumental effects that contribute to the profile shape, thus improving the precision of the fit to the observed data and (ii) it can be used to describe a source characteristic or an instrumental arrangement that is outside the normally used configuration, yielding a result that would be difficult to obtain otherwise (Cheary et al., 2004[link]).

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