Diffraction Lineshapes
Broadening of diffraction peaks from small crystal sizes, strain distributions, and instrument effects are discussed as convolutions. The convolution process is explained in detail. Fourier methods for deconvolution are derived, including practical effect
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Diffraction Lineshapes
9.1 Diffraction Line Broadening and Convolution This chapter begins by explaining the physical origins of three types of broadening of diffraction peaks from crystalline materials: 1) small sizes of crystallites, 2) distributions of strains in crystallites, and 3) the diffractometer itself. These sources of peak broadenings pertain to electron diffraction, but since x-ray and neutron diffractometry data are more amenable to lineshape analysis with kinematical diffraction theory, the concepts in this chapter are presented in the context of x-ray powder diffractometry. After the basics of strain and size broadening are described, this chapter explains the concept of convolution in the context of how an instrument lineshape broadens the measured diffraction peaks. The relationship between convolutions and products of Fourier transforms, the “convolution theorem,” is presented in Sect. 9.2.1. This important relationship is used frequently in the remainder of this book. Methods are described for separating the effects of simultaneous size and strain broadenings
B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials, Graduate Texts in Physics, DOI 10.1007/978-3-642-29761-8_9, © Springer-Verlag Berlin Heidelberg 2013
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9 Diffraction Lineshapes
of diffraction lineshapes. These methods make use of the different dependencies of strain and size broadening on the Δk of the diffraction.1 An analysis of diffraction lineshapes offers statistical information about unit cells from many regions in a bulk material. What proves important are the correlations in positions between unit cells in crystallites, and this topic is the central focus of Chap. 10. With statistical averages of positional information, the microstructural origins of “strain broadening” or “particle size broadening” are often unclear from diffraction lineshapes alone. Chapter 9 concludes with a reminder that x-ray lineshape analysis can be usefully complemented by TEM imaging of microstructural features.
9.1.1 Crystallite Size Broadening Recall the result of kinematical theory for the diffraction lineshape of a small crystal shaped as a rectangular prism, (6.143). In terms of the deviation vector, s = sx xˆ + sy yˆ + sz zˆ (which is the difference between the reciprocal lattice vector and the diffraction vector, s ≡ g − Δk), and neglecting structure factors, the lineshape is: I (s) = Ix (sx )Iy (sy )Iz (sz ),
(9.1)
where the three factors have the same mathematical form: Ix (sx ) =
sin2 (πNx ax sx ) sin2 (πax sx )
.
(9.2)
ˆ and Nx is the number of these Here ax is the relevant interplanar spacing along x, planes in the crystal. This function is graphed in one dimension in Fig. 9.1. Note that identical peaks appear about each reciprocal lattice point (i.e., where s = 0 at the reciprocal lattice points g = 1/a, 2/a, 3/a. . . ). The breadth of a diffraction peak in k-space is independent of the particular diffraction. We seek a relationship between the size of the crystallites and the breadth in s
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