Diffraction from Crystals
This chapter develops diffraction from crystals in the Laue formulation, using both the Born approximation from Chap. 4 , and a general formulation for the scattering of any wave from a potential. The concept of the reciprocal lattice is presented, and d
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Diffraction from Crystals
6.1 Sums of Wavelets from Atoms Chapters 6–8 are concerned with the angular dependence of the diffracted wave, ψ(Δk), emitted from different arrangements of atoms. The underlying mechanism is coherent elastic scattering from individual atoms, the topic of Chap. 4. Diffraction itself, however, is a cooperative phenomenon based on phase relationships between the wavelets1 scattered coherently by the individual atoms. This chapter explains how a translationally-periodic arrangement of atoms in a crystal permits strong constructive interferences between individual wavelets, creating the familiar Bragg diffractions. 1 We call the outgoing waves from individual atoms “wavelets,” to distinguish them from their coherent sum, the total diffracted wave, that is measured at the detector. The “wavelets” are in fact full wavefunctions, but each contributes a small amplitude to the total wave.
B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials, Graduate Texts in Physics, DOI 10.1007/978-3-642-29761-8_6, © Springer-Verlag Berlin Heidelberg 2013
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Diffraction from Crystals
The diffraction theory developed here is “kinematical theory.” As discussed in Chap. 4, the validity of kinematical theory for electron diffraction is contingent on the validity of the first Born approximation (presented as (4.74), leading to (4.82)). The assumption that the incident wave is scattered weakly by the material is also used when developing kinematical theories of x-ray and neutron diffraction. For the strong Coulomb interactions between incident electrons and atoms, however, kinematical theory must be used with caution. It is usually reliable for calculating the structure factor of the unit cell. For electron diffraction contrast from larger features such as crystal shapes and crystalline defects, however, kinematical theory is often only qualitative. Kinematical theory is more quantitative for x-ray diffraction because x-ray scattering is much weaker than electron scattering. Kinematical calculations can be highly reliable for neutron diffraction. For electron diffraction, kinematical theory can be improved considerably by redefining the extinction length as is done in Sect. 8.3, but quantitative results generally require the dynamical theory developed in Chap. 13 or the physical optics approach of Chap. 11.
6.1.1 Electron Diffraction from a Material Diffraction is a wave interference phenomenon. To form diffraction patterns, we must have more than one scattering center. Consider the geometrical array of scattering centers in Fig. 6.1. We use the same coordinates as in Fig. 4.7, but now we have a set of vectors {R j }, which mark the centers of the atoms in the material. In Sect. 6.2 we impose the crystal symmetry on the vectors {R j } (specifically, the translational periodicity), but this comes later. Our scattered electron wave in the first Born approximation is (4.82): −m eik·r ψscatt (Δk, r) = 2π2 |r|
V r e−iΔk·r d3 r .
(6.1)
An important step in calcu
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