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. 3.2, pp. 255-256

Section 3.2.2.3.1. Domain size

P. W. Stephensa*

aDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA
Correspondence e-mail: peter.stephens@stonybrook.edu

3.2.2.3.1. Domain size

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In very general terms, diffraction peaks from an object of linear size L will have a width in Q of the order of 1/L. As formulated by Scherrer (1918[link]), in an angle-dispersive measurement, the full width at half-maximum (FWHM) in 2θ, measured in radians, is given by[\Gamma = {{K \lambda}\over{L \cos \theta}},\eqno(3.2.12)]where K is called the shape factor and is a number of the order of unity whose precise value depends on the shape of the particles, which are assumed to be of uniform size and shape. The FWHM shape factor for a spherical particle is K = 0.829 (Patterson, 1939[link]). Note that if a powder sample is polydisperse (i.e., it contains a distribution of grain sizes), the average grain size is not necessarily given by the Scherrer equation.

Perhaps a more useful measure of the width of a peak is the integral breadth. In an angle-dispersive measurement, the integral breadth of a given peak centred at 2θ0 is defined as[ \beta ={{1}\over{I(2 \theta_0)}} \int I(2\theta) \, {\rm d}2\theta. ]From a technical point of view, measurement of the integral breadth requires accurate measurement of the intensity in the wings of the diffraction peak, which in turn depends on accurate knowledge of the background intensity.

For any crystallite shape, it can be shown that the integral breadth is related to the volume-average thickness of the crystallite in the direction of the diffraction vector, viz.[ L_{V} = {{\lambda}\over{\beta \cos \theta}} = {{1}\over{V}} \int {\rm d}^3{\bf r} \, T({\bf r}, {\bf G}), ]where V is the volume of the crystallite and T(r, G) is the length of the line inside the crystallite parallel to G and passing through the point r. For example, if one writes an integral-breadth version of the Scherrer equation,[ \beta = {{K_\beta \lambda}\over{L \cos \theta}}, ]the shape factor Kβ is unity for (00l) reflections from cube-shaped crystals of size L. Kβ = 1.075 for a sphere of diameter L.

An important feature of the integral breadth is that it has a well defined meaning for a polydisperse sample of crystallites. Assuming that the crystallites all have the same shape,[ \beta = {{K_\beta \lambda \langle L^3 \rangle}\over{\cos \theta \langle L^4 \rangle}}, ]where [\langle L^3\rangle] and [\langle L^4\rangle] are the third and fourth moments of the size distribution (Langford & Wilson, 1978[link]).

In many applications such as Rietveld or profile refinement, it is important to treat the full shape of the diffraction peak instead of merely its width (Loopstra & Rietveld, 1969[link]; Rietveld, 1969[link]). By way of illustration, Fig. 3.2.1[link] shows one Bragg peak of the computed powder-diffraction pattern from an ensemble of spherical particles of point scatterers in a simple cubic lattice. The lattice parameter is a, and the diameter of the particles is chosen to be 100a, so that each crystallite consists of approximately 5.2 × 105 `atoms'. (This line shape was calculated using the Debye equation, described in Section 3.2.4[link].)

[Figure 3.2.1]

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Computed powder line shape from an ensemble of spherical particles of diameter 100a, including comparison to Gaussian and Lorentzian line shapes of equal FWHM.

Several different analytical functions are frequently used in powder diffraction. In terms of the independent variable x, centred at x0 with FWHM Γ, the normalized Gaussian function is[G(x-x_0) = \pi^{-1/2} \sigma^{-1} \exp - \left({{x-x_0}\over{\sigma}} \right)^2, ]with σ = Γ/2(ln 2)1/2. The normalized Lorentzian is[L(x-x_0) = {{\Gamma / 2 \pi}\over{(x-x_0)^2 + (\Gamma/2)^2}}. ]The symmetric Pearson-VII function is a generalization of the Lorentzian, written as[ P(x-x_0) = {{a}\over{\Gamma}} \left [1 + b \left({{x-x_0}\over{\Gamma}} \right)^2 \right] ^{-m}. ]Here m is a parameter that governs the line shape (essentially the strength of the wings versus the peak), and the numerical parameters b = 4(21/m − 1) and a = (b/π)1/2Γ(m)/Γ(m − 1/2). (Note the gamma function in the definition of a; the FWHM does not appear in that expression.) The Pearson-VII function with m = 1 is a Lorentzian, and it is a Gaussian in the limit m → ∞.

It must be emphasized that none of these functions has any theoretical justification whatsoever. However, in loose powders size broadening is almost always Lorentzian, while Gaussian size broadening is more commonly observed in dense polycrystalline specimens (such as bulk metals).

The Lorentzian and Gaussian functions are also plotted in the same figure for comparison to the actual powder-diffraction line shape of a spherical particle. It can be seen that the correct function has stronger tails than the Gaussian, but the Lorentzian line shape seriously overestimates the intensity in the wings. Both the Gaussian and the Pearson VII fail to capture the general feature that any compact object with a sharp boundary will give a diffraction line shape with tails that asymptotically decay as (xx0)−2.

One approach to obtaining a more accurate phenomenological description to diffraction line shapes is the Voigt function, which is a convolution of a Gaussian and Lorentzian,[ V(x-x_0) = \textstyle\int {\rm d}x'\, G(x'-x_0) L(x-x'). ]The presence of two shape parameters, the independent widths of the Gaussian and Lorentzian functions, provide independent parameters to control the width and the strength of the wings in the Voigt line shape. The Voigt is more computationally expensive than any elementary function, and so a commonly used approximation is the pseudo-Voigt,[{\rm PV}(x-x_0\semi \Gamma) = \eta L(x-x_0\semi \Gamma) + (1-\eta) G(x-x_0\semi \Gamma), ]which is a mixture of Gaussian and Lorentzian functions of the same width. The parameter η controls the shape (strength of the wings) of the pseudo-Voigt line-shape function, independent of its width. There is a computationally convenient approximate relation between Voigt line-shape parameters ΓL and ΓG and the pseudo-Voigt Γ and η (Thompson et al., 1987[link]).

References

Langford, J. I. & Wilson, A. J. C. (1978). Scherrer after sixty years: a survey and some new results in the determination of crystallite size. J. Appl. Cryst. 11, 102–113.Google Scholar
Loopstra, B. O. & Rietveld, H. M. (1969). The structure of some alkaline-earth uranates. Acta Cryst. B25, 787–791.Google Scholar
Patterson, A. L. (1939). The Scherrer formula for X-ray particle size determination. Phys. Rev. 56, 978–982.Google Scholar
Rietveld, H. M. (1969). A profile refinement method for nuclear and magnetic structures. J. Appl. Cryst. 2, 65–71.Google Scholar
Scherrer, P. (1918). Bestimmung der Grösse und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen. Nachr. Ges. Wiss. Göttingen, pp. 98–100.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








































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