Dynamical Theory

Abstract

This chapter solves the Schrödinger equation for a high-energy electron in a solid with translational periodicity — i.e., a crystal. Section 11.2.1 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: 11.3,’ the first two short subsections of 11.2.1, 11.2.3, the first subsection of 11.4.1,’ and finally 11.5. These sections offer an intuitive understanding of the issues in dynamical theory. These sections 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-scattered¹and diffracted wavefunctions (11.20). At the precise Laue condition 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, U _g , where gequals the difference in wavevector of the two coupled beams.