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

International Tables for Crystallography (2018). Vol. H, ch. 3.6, p. 293

Section Strain broadening (lattice distortions)

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Strain broadening (lattice distortions)

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A local variation of the lattice spacing (due, for example, to the presence of a defect) leads to an average phase term that, in general, is a complex quantity:[\eqalignno {\langle \exp[2\pi i \psi_{hkl}(L)]\rangle & = \langle \cos [(2\pi Ld_{\{ hkl \}}^*\varepsilon_{ \{ hkl \}} (L)] \rangle \cr&\quad+ i\langle \sin [2\pi Ld_{ \{ hkl \} }^*\varepsilon_{ \{ hkl\} } (L)] \rangle \cr & = A_{ \{ hkl \}}^D(L) + iB_{ \{ hkl \} }^D(L). & (3.6.31)}]The strain [epsilon]{hkl}(L) represents the relative displacement of atoms at a (coherence) distance L along the scattering vector hkl. Knowledge of the actual source of distortion allows the explicit calculation of the various terms (van Berkum, 1994[link]). It is quite customary to assume that the strain is the same for symmetry-equivalent reflections [[epsilon]hkl(L) = [epsilon]{hkl}(L)]: this is a reasonable hypothesis for a powder, where we assume that any configuration is equally probable.

Traditional LPA methods such as the Warren–Averbach method (Warren & Averbach, 1950[link], 1952[link]; Warren, 1990[link]) take a first-order MacLaurin expansion of equation (3.6.31)[link] to extract the microstrain contribution from the measured data:[\eqalignno{A_{ \{ hkl \} }^D(L) &\cong 1 - 2\pi^2 L^2d_{ \{ hkl \} }^{*2}\langle \varepsilon_{ \{ hkl \} }^2(L)\rangle,&(3.6.32)\cr \cr B_{ \{ hkl \} }^D(L) &\cong - {4 \over 3} \pi^3 L^3 d_{ \{ hkl \} }^{*3}\langle \varepsilon_{ \{ hkl \} }^3(L)\rangle. & (3.6.33)}]The term in equation (3.6.33)[link] would cause peak asymmetry. However, we usually consider only the second-order moment of the strain distribution (root-mean strain or microstrain) and thus symmetric peaks. Owing to the anisotropy of the elastic properties, the broadening described by equation (3.6.32)[link] is in general anisotropic: an extensive discussion of strain anisotropy and of the order dependence of strain broadening can be found, for example, in Leineweber & Mittemeijer (2010[link]) and Leineweber (2011[link]). It should be stressed that in their original form, traditional line-profile methods are unable to deal with this anisotropy (corrections have been proposed for particular cases, for example, in the so-called modified Williamson–Hall (MWH) and modified Warren–Averbach (MWA) analyses; Ungár & Borbély, 1996[link]).


Berkum, J. G. M. van (1994). Strain Fields in Crystalline Materials. PhD thesis, Technische Universiteit Delft, Delft, The Netherlands.Google Scholar
Leineweber, A. (2011). Understanding anisotropic microstrain broadening in Rietveld refinement. Z. Kristallogr. 226, 905–923.Google Scholar
Leineweber, A. & Mittemeijer, E. J. (2010). Notes on the order-of-reflection dependence of microstrain broadening. J. Appl. Cryst. 43, 981–989.Google Scholar
Ungár, T. & Borbély, A. (1996). The effect of dislocation contrast on X-ray line broadening: a new approach to line profile analysis. Appl. Phys. Lett. 69, 3173.Google Scholar
Warren, B. E. (1990). X-ray Diffraction. New York: Dover Publications. (Unabridged reprint of the original 1969 book.)Google Scholar
Warren, B. E. & Averbach, B. L. (1950). The effect of cold-work distortion on X-ray patterns. J. Appl. Phys. 21, 595–599.Google Scholar
Warren, B. E. & Averbach, B. L. (1952). The separation of cold-work distortion and particle size broadening in X-ray patterns. J. Appl. Phys. 23, 492.Google Scholar

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