Abstract
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 analytical solutions of the wave equations. Bloch waves are developed as eigenfunctions of a solid with a weak periodic potential, and are related to the diffracted beams in the solid. The fundamental differences between kinematical and dynamical diffraction are explained, as is the role of the diffraction error in both. Dispersion surface constructions are presented for dynamical theory, and are used for semi-quantitative estimates of beam amplitudes through the depth of the sample, and at defects. With absorption, the two-beam dynamical theory is developed for diffraction contrast from stacking faults, and dynamical theory is used qualitatively for dislocations and other defects. The formalism of multibeam dynamical theory is developed.
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Notes
- 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.
- 2.
The average potential is the Fourier component of the potential for g=0. Since it is handled a bit differently than the other Fourier components, however, we give it the special designation, U 00. There is no term U 0 in the following summations over g.
- 3.
Perhaps the reader is more familiar with temporal beats of acoustic waves, which occur when adding the amplitudes of two tones with close but slightly different frequencies. When the tones are in phase, the sound is loud. The sound diminishes as the tones drift out of phase. As the tones drift in and out of phase, the listener senses an average tone, but with a slow modulation of intensity. This slow modulation is the beat pattern.
- 4.
The phase factors \(\mathrm{e}^{\mathrm{i}(\boldsymbol{k}_{0}-\boldsymbol{k})\cdot \boldsymbol{r}}\) on the left-hand sides of (13.73) and (13.74) provide amplitude modulations perpendicular to \(\hat{\boldsymbol{z}}\), as shown at the tops of Figs. 13.6 and 13.7. The crests of these horizontal modulations are important for setting the precise energies of the Bloch wavefunctions (Sect. 13.4.3), but we can ignore them for now as we seek the z-dependence of the beams.
- 5.
The beat pattern would vanish if the Bloch waves were to have wavevectors such that γ (1)=γ (2), but we will see later that this condition does not occur in a periodic crystal. It occurs only in free space or in a uniform potential.
- 6.
- 7.
We constructed the Bloch waves (13.88), (13.89) or (13.91), (13.94)) with knowledge only of symmetry. The Bloch waves had the periodicity of the crystal lattice, which is a rigorous result and a robust consequence of making linear combinations of the beams Φ 0(r) and Φ g (r). There was a hidden assumption, however, that the phases of Φ 0(r) and Φ g (r) differ by ±1, and not, for example, by exp(iδ) with some arbitrary δ. The formal approach to this problem is to find eigenfunctions of the Schrödinger equation in a periodic potential. This is done in Sect. 13.4.4, using first-order degenerate perturbation theory and is, in fact, the same calculation used to derive band gaps in the nearly-free electron model of solid state physics.
- 8.
- 9.
They are, in fact, 2-dimensional rotation matrices. At the top of the sample where z=0, the orthonormal ϕ 0 and ϕ g , when rotated by the angle β/2, become ψ (2) and ψ (1), which remain orthonormal.
- 10.
In scattering physics, the word “kinematical” refers to a single event such as a single collision, whereas “dynamical” means multiple interactions before the wave leaves the interaction region.
- 11.
This same trick was used when solving for the C ij ’s in (13.146).
- 12.
Spacings of the prominent intensity oscillations are difficult to quantify because they depend on the details of how the specimen is bent.
- 13.
Except for the special case where the fault displacement is such that \(\boldsymbol{\delta r}=m (2g )^{-1}\hat{\boldsymbol{g}}\), where g is the active diffraction and m is an integer.
- 14.
Most of the irregularities in the experimental traces are caused by noise or dirt on the TEM negatives, although the doublet in the intensity of the first DF fringe is a real effect.
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Fultz, B., Howe, J. (2013). Dynamical Theory. In: Transmission Electron Microscopy and Diffractometry of Materials. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29761-8_13
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