Dynamical Theory
The Born approximation and its Fourier transform relationships between waves and structure formed the basis for kinematical theory. Chap. 13 presents the alternative dynamical theory of electron diffraction, starting with the two-beam model that has analy
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Dynamical Theory
13.1 Chapter Overview This chapter solves the Schrödinger equation for a high-energy electron in a solid with translational periodicity—i.e., a crystal. Section 13.2.1 derives the dynamical equations (the “Howie–Whelan–Darwin equations”) from the Bethe treatment of the Schrödinger equation, and contains the most condensed mathematics in the book. For a first approach to this chapter, the authors recommend reading the following sections in this order: 13.3, the first two short subsections of 13.2.1, Sect. 13.2.3, the first subsection of 13.4.1, and finally 13.5. These sections present an overview of the concepts of dynamical diffraction theory. They show how the wavefunction of the high-energy electron is affected by the potential energy of the crystal—specifically, the periodicity of the potential energy that originates with the periodicity of the atom arrangements. It turns out that the periodic potential causes the amplitude of the high-energy electron to be transferred back-and-forth (“dynamically”) between the forward-scattered1 and diffracted wavefunctions. At the precise Laue condition 1 It is no longer proper to use the term “transmitted beam” as we did for kinematical theory because
the beam leaving the sample in the forward direction has undergone many interchanges of energy with the diffracted beams. B. Fultz, J. Howe, Transmission Electron Microscopy and Diffractometry of Materials, Graduate Texts in Physics, DOI 10.1007/978-3-642-29761-8_13, © Springer-Verlag Berlin Heidelberg 2013
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Dynamical Theory
for strong diffraction (s = 0), the physical distance over which the wave amplitude is transferred back-and-forth once is called the “extinction distance.” The extinction distance is shown to be inversely proportional to the Fourier component of the crystal potential, Ug , where g equals the difference in wavevector of the two coupled beams. Quantum mechanics allows an electron wavefunction to be described by different “representations,” which employ different sets of orthogonal basis functions. The “beam representation” {Φ(g)}, and the “Bloch wave representation” {Ψ (r)}, are the two representations used in this chapter. The reader is already familiar with the forward and diffracted wavefunctions Φ0 (r) and Φg (r) of the beam representation, whose amplitudes, φ0 (z) and φg (z), vary with depth z into the specimen. In its simplest form, the Bloch wave representation uses two Bloch wavefunctions, Ψ (1) (r) and Ψ (2) (r). It is a convenient representation for an electron that propagates in a crystal because the amplitudes of the Bloch wavefunctions, ψ (1) and ψ (2) , are constant throughout a perfect crystal. Bloch waves are eigenfunctions of an infinite, periodic crystal. Although the different Bloch waves have the same total energy, their electron density is distributed differently within the unit cell. The different Bloch waves therefore have slightly different balances between potential energy and kinetic energy. Our two Bloch waves therefore have wavevectors differing sl
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