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. 3.6, pp. 289-290

Section 3.6.2.3. The Warren–Averbach method and its variations

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Matteo.Leoni@unitn.it

3.6.2.3. The Warren–Averbach method and its variations

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The convolution theorem can be employed to disentangle the specimen-related broadening contributions described by f(s). In fact, let us suppose, as in the Williamson–Hall method, that size and microstrain are the only two sources of specimen-related broadening. We call the Fourier transform of the profiles broadened by size and distortion effects only [A_{hkl}^S(L)] and [A_{hkl}^D(L)], respectively. As the size and distortion profiles are folded into f(s), the following holds:[A(L) = {\rm FT}[f(s)] = {\rm FT}[h(s)]/{\rm FT}[g(s)] = A_{hkl}^S(L)A_{hkl}^D(L). \eqno (3.6.7)]The separation of the size and distortion terms is straightforward for spherical domains: the size effects for a sphere are independent of the reflection order, whereas those related to distortions (causing the change in the slope of the Williamson–Hall plot) are order-dependent. To describe the distortion term it is convenient to follow the idea of Bertaut (1949a[link],b[link], 1950[link]), considering the specimen as made of columns of cells along the c direction. The profile due to distortions is calculated by taking the average phase shift along the column due to the presence of defects. The analytical formula for the distortion term is thus of the type [A_{hkl}^D(L)] = <exp(2πiLn[epsilon]L)>, where [epsilon]L = ΔL/L is the average strain along c calculated for a correlation distance (i.e. Fourier length) L.

As a first-order approximation, the distortion terms give no profile asymmetry; [A_{hkl}^D(L)] is just a cosine Fourier transform. We can thus expand it as (Warren, 1990[link])[A_{hkl}^D(L) = \langle\cos (2\pi Ln\varepsilon_L)\rangle = 1 - 2\pi^2 L^2 n^2\langle \varepsilon_L^2 \rangle. \eqno(3.6.8)]If we now rewrite equation (3.6.7)[link] on a log scale, taking equation (3.6.8)[link] into account, we obtain[\eqalignno {\ln [A_{hkl}(L)] & = \ln [A_{hkl}^S(L)] + \ln [A_{hkl}^D(L)] \cr & = \ln [A_{hkl}^S (L)]+ \ln [1 - 2\pi 2 L^2 n^2 \langle \varepsilon_L^2 \rangle] \cr & = \ln [A_{hkl}^S(L)] - 2\pi^2 L^2 n^2\langle \varepsilon_L^2\rangle. & (3.6.9)}]Equation (3.6.9)[link] represents a line in the variable n2: the intercepts at increasing L values provide the logarithm of the Fourier size term, whereas the slopes give the microstrain directly (Warren, 1990[link]). From the size coefficients, we can obtain an average size, again following the idea of Bertaut (1949a[link], 1950[link]), related to the properties of the Fourier transform:[\langle D \rangle = \left [- \left. {{\partial A_{hkl}^S(L)} \over {\partial L}} \right |_{L = 0} \right]^{-1}. \eqno (3.6.10)]This average size is thus related to the initial slope of the Fourier coefficients [assuming that they are well behaved, i.e. that the tangent is always below the [A_{hkl}^S(L)] curve].

A long chain of operations is needed to obtain the size and the strain contributions; there is thus a risk that the final result will no longer be compatible with the experimental data. Only a few years after the introduction of this method, Garrod et al. (1954[link]) wrote

Hence, in any attempt to distinguish between particle size or strain broadening from a particular material, the use of the one or the other of these functions [Gaussian and Lorentzian] (together with the appropriate relationship between B, b, and β) involves an intrinsic initial assumption about the cause of the broadening, when the object of the investigation is to discover the cause. Such an assumption must inevitably weight the experimental results, partially at least, in favour of one or the other of the two effects. In this connexion it is therefore perhaps significant that in most previous work on the cause of line broadening from cold-worked metals, those investigators who have used the Warren relationship between B, b, and β have concluded that lattice distortion was the predominant factor, whilst those who have employed the Scherrer correction found that particle size was the main cause. The best procedure in such work therefore is to make no assumptions at all about the shape of the experimental line profiles…

This is owing to the fact that a direct connection between the experimental data and the final microstructural result does not really exist in those methods and that the whole information contained in the pattern is not exploited.

References

Bertaut, E. F. (1949a). Etude aux rayons X de la répartition des dimensions des cristallites dans une poudre cristalline. C. R. Acad. Sci. 228, 492–494.Google Scholar
Bertaut, E. F. (1949b). Signification de la dimension cristalline mesurée d'apres la largeur de raie Debye-Scherrer. C. R. Acad. Sci. 228, 187–189.Google Scholar
Bertaut, E. F. (1950). Raies de Debye–Scherrer et repartition des dimensions des domaines de Bragg dans les poudres polycristallines. Acta Cryst. 3, 14–18.Google Scholar
Garrod, R. I., Brett, J. F. & MacDonald, J. A. (1954). X-ray line broadening and pure diffraction contours. Aust. J. Phys. 7, 77–95.Google Scholar
Warren, B. E. (1990). X-ray Diffraction. New York: Dover Publications. (Unabridged reprint of the original 1969 book.)Google Scholar








































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