Diffracted Beams


In Chapter 11 we discussed why diffraction occurs; in this chapter we give a more detailed mathematical treatment. It may be more detailed than you need at this stage. Diffraction is one of those phenomena that lends itself directly to a detailed mathematical modeling, but there is a danger: don’t become so engrossed in the math that you miss the principles involved; conversely, don’t ignore the subject because it is mathematically daunting! The topic of this chapter is one which causes major problems for many microscopists. The treatment we will follow is known as the ‘dynamical theory.’ Later we will make some gross simplifications, partly because this is instructive and partly because these simplifications do apply to some important special cases; the kinematical approximation is one such simplification. Many other texts begin with the so-called ‘kinematical’ treatment and then advance to the more realistic, more general dynamical case. We will not do this but we will introduce the words and assumptions in Chapter 27.


Direct Beam Diffract Beam Bloch Wave Total Wave Function Exit Surface 
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  1. This treatment of diffracted beams follows that given by Hirsch and Whelan and the textbook by Hirsch et al. that built on Darwin’s treatment of X-ray diffraction.Google Scholar


  1. Darwin, CG 1914 Röntgen-Ray Reflection I;II Phil, Mag. 27 315–333 and 675-690. Charles Galton Darwin was a grandson of Charles Robert Darwin and, like (the) Darwin (and Hirsch), became a Fellow of Christ's College.Google Scholar
  2. Hecht, E 1987 Optics, 4th ed., Addison-Wesley, Reading MA.Google Scholar
  3. Howie, A and Whelan, MJ 1961 Diffraction Contrast of Electron Microscope Images of Crystal Lattice Defects. II The Development of a Dynamical Theory Proc. Roy. Soc. A263 217–237.Google Scholar

The Column Approximation

  1. Howie, A and Basinski, ZS 1968 Approximations of the Dynamical Theory of Diffraction Contrast Phil. Mag. 17 1039–1063.Google Scholar
  2. Takagi, S. 1962 Dynamical Theory of Diffraction Applicable to Crystals with Any Kind of Small Distortion Acta Cryst. 15 1311–1312.Google Scholar


  1. Spence, JCH 2003 Experimental High-Resolution Electron Microscopy 3rd Ed. Oxford University Press New York. Uses the quantum-mechanical convention rather than the crystallographic one.Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.The University of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.University of ConnecticutStorrsUSA

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