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. 3.2, pp. 255256
Section 3.2.2.3.1. Domain size^{a}Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA 
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), in an angledispersive measurement, the full width at halfmaximum (FWHM) in 2θ, measured in radians, is given bywhere 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). 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 angledispersive measurement, the integral breadth of a given peak centred at 2θ_{0} is defined asFrom 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 volumeaverage thickness of the crystallite in the direction of the diffraction vector, viz.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 integralbreadth version of the Scherrer equation,the shape factor K_{β} is unity for (00l) reflections from cubeshaped crystals of size L. K_{β} = 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,where and are the third and fourth moments of the size distribution (Langford & Wilson, 1978).
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; Rietveld, 1969). By way of illustration, Fig. 3.2.1 shows one Bragg peak of the computed powderdiffraction 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 × 10^{5} `atoms'. (This line shape was calculated using the Debye equation, described in Section 3.2.4.)

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 x_{0} with FWHM Γ, the normalized Gaussian function iswith σ = Γ/2(ln 2)^{1/2}. The normalized Lorentzian isThe symmetric PearsonVII function is a generalization of the Lorentzian, written asHere m is a parameter that governs the line shape (essentially the strength of the wings versus the peak), and the numerical parameters b = 4(2^{1/m} − 1) and a = (b/π)^{1/2}Γ(m)/Γ(m − 1/2). (Note the gamma function in the definition of a; the FWHM does not appear in that expression.) The PearsonVII 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 powderdiffraction 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 (x − x_{0})^{−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,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 pseudoVoigt,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 pseudoVoigt lineshape function, independent of its width. There is a computationally convenient approximate relation between Voigt lineshape parameters Γ_{L} and Γ_{G} and the pseudoVoigt Γ and η (Thompson et al., 1987).
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 ScholarLoopstra, B. O. & Rietveld, H. M. (1969). The structure of some alkalineearth uranates. Acta Cryst. B25, 787–791.Google Scholar
Patterson, A. L. (1939). The Scherrer formula for Xray 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 Xray data from Al_{2}O_{3}. J. Appl. Cryst. 20, 79–83.Google Scholar