Chapter 2: Diffraction
2. Chapter 2: Diffraction#
part of
MSE672: Introduction to Transmission Electron Microscopy
Spring 2024
| Gerd Duscher | Khalid Hattar |
| Microscopy Facilities | Tennessee Ion Beam Materials Laboratory |
| Materials Science & Engineering | Nuclear Engineering |
| Institute of Advanced Materials & Manufacturing | |
Background and methods to analysis and quantification of data acquired with transmission electron microscopes.
Content
The Diffraction chapter has the following sections:
Kinematic Scattering
Dynamic Scattering
2.1. Basics#
Diffraction is the direct result of the interaction (with or without energy transfer) of electrons and matter.
Kinematic diffraction theory describes only the Bragg angles (the position of the Bragg reflections) but not the intensity in a real diffraction pattern.
Dynamic theory is responsible for the intensity variation of the Bragg reflections
2.1.1. Diffraction and Imaging#
To achieve an image from a diffraction pattern only a Fourier Transformation of parts of the diffraction pattern is needed.
Any image in a TEM can be described as Fourier Filtering, because we select the beams which form the images. The knowledge of which and how many diffracted beams contribute to the image formation is crucial.
Because the intensity of selected diffracted beams is necessary to calculate image intensities, dynamic diffraction theory is necessary.
Understanding difraction theory of electrons is at the core of the analysis of TEM data.
According to Fourier-Optical view of the TEM, we first form a diffraction pattern (in Fourier space) and after another (inverse) Fourier Transformation of parts of this diffraction pattern we are back in real space and can observe an image.
2.1.2. Dynamic and Kinematic Theory#
Kinematic Theory is based on a single scattering event per electron
Kinematic theory is used in neutron and X-ray diffraction almost exclusively.
Dynamic theory incorporates multiple scattering events.
Dynamic theory results in Rocking Curves (oscillations) of intensities of diffracted beams with sample thickness.
Dynamic theory can analytically be solved only for the two beam case, strangely the basis for conventional TEM.
2.1.3. Diffraction and Scattering#
Electrons can be viewed as particles and/or as waves.
2.1.4. Particle-wave dualism:#
| Scattering | $\leftrightarrow$ | Diffraction |
|---|---|---|
| Particle picture | $\leftrightarrow$ | Wave picture |
We will switch back and forth between these two pictures, depending on which is mathematically easier to express.
2.1.5. Kinematic Diffraction Theory Buzz Words#
The following terms will be important in Kinematic Theory:
| f | atomic scattering factor | scattering strength of an atom |
| F | form factor | combination of symmetrry and atomic scattering factor |
| forbidden reflection | a direct result of the form factor | |
| $\sigma$ | cross-section | scattering probability expressed as an effective area |
| $\frac{\partial \sigma}{\partial \Omega}$ | cross-section | scattering probability in a solid angle |
| $\lambda$ | mean free path | scattering probability expressed as a path-length between two scattering events |
We will start discussing these terms in the Atomic Form Factor and following notebooks