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. 289

Section Peak profile and the convolution theorem

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Peak profile and the convolution theorem

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Each peak profile h(s) in a powder diffraction pattern can be described as the convolution of an instrumental profile g(s) with a function f(s) accounting for sample-related effects (microstructure; see, for example, Jones, 1938[link]; Alexander, 1954[link]; Klug & Alexander, 1974[link]; and references therein):[h(s) = \textstyle\int\limits_{-\infty}^\infty f(y)g(s - y)\,{\rm d}y = f \otimes g(s). \eqno (3.6.4)]The calculation of the integral in equation (3.6.4)[link] can be simplified through the use of a Fourier transform (FT). In fact, the convolution theorem states that the FT of a convolution is the product of the Fourier transforms of the functions to be folded:[{\cal C}(L) = {\rm FT}[h(s)] = {\rm FT}[f(s)] \times{\rm FT}[g(s)]. \eqno (3.6.5)]In this equation, L is the (real) Fourier variable conjugate to s. The properties of the Fourier transform allow equation (3.6.4)[link] to be rewritten as[h(s) = {\rm FT}^{-1} [{\cal C}(L)] = {\rm FT}^{-1}\big[{\rm FT}[h(s)]\big] = {\rm FT}^{-1}\big[{\rm FT}[f(s)] {\rm FT}[g(s)]\big]. \eqno (3.6.6)]This equation is the basis of the Warren–Averbach approach and also of all modern LPA methods.


Alexander, L. (1954). The synthesis of X-ray spectrometer line profiles with application to crystallite size measurements. J. Appl. Phys. 25, 155–161.Google Scholar
Jones, F. W. (1938). The measurement of particle size by the X-ray method. Proc. R. Soc. Lond. Ser. A, 166, 16–43.Google Scholar
Klug, H. P. & Alexander, L. E. (1974). X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials, 2nd ed. New York: Wiley.Google Scholar

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