1. Diffraction in TEM
Janez Košir

2. Contents Why use diffraction in TEM? Electron and x-ray diffraction
Theory of diffraction
Scattering from a plane of atoms
Scattering from a crystal
Dynamical diffraction
Diffraction patterns in practice
Summary
In this presentation we will look at how electrons, from the electron beam, are scattered in the transmission electron microscope and why this is important.
First we will look at why diffraction patterns are important and what they can tell us.
Second we will look at the difference between the diffraction of electrons in the TEM and X-rays in XRD .
Next we will look at how diffraction works in theory. We will start by explaining the basic concept of scattering from a single plane and work our way up to 3 dimensional crystals.
In the last segment we will see what kind of diffraction patterns we can obtain in practice and what can they tell us.

3. Why use diffraction in TEM?
Electrons scattered from crystalline samples produce diffraction patterns in the form of spots
These patterns differ from one another in the spot geometry, size and intensity
They tell us the properties of our sample such as crystallinity, morphology, grain size, etc. on the atomic level
Due to the small area they cover, these patterns are known as selective area diffraction patterns (SADP)
When an electron beam is focused on our sample, the electrons will scatter off the samples atoms based on the samples properties.
Crystalline samples will produce a diffraction pattern consisting of many spots with varying locations, intensity and size.
From these patterns we can gather information about or sample like: - Is the sample crystalline?
- Its crystallographic characteristics (lattice parameters, symmetry…)
- Grain morphology and grain size and their distribution
- How many phases are present in the sample
The great advantages of the TEM compared to other microscopes is that we are able to determine crystallographic properties on atomic levels.
Because of this, these diffraction patterns are known as selective area diffraction patterns (SADP).

4. Electrons vs. x-rays TEM XRD
We can see all diffracted beams from the start
Takes less than a second to obtain results
Electrons have a shorter wavelength
We can observe smaller areas as electrons can be easily directed
We have to rotate the sample to see all diffracted beams
Takes minutes or hours to obtain results
X-rays have a longer wavelength
Hard to focus on a small area since x-rays are hard to direct
Another method of sample characterization is X-ray diffraction (XRD). As opposed to the electron microscope, XRD uses X-ray diffraction to obtain sample properties.
In XRD we need to rotate our sample or use a range of wavelengths to obtain the sample properties. This can take several minutes or even hours to complete.
On the other hand, TEM diffraction patterns do not require any specific procedures which means that we can observe our results in less than a second.
The other differences that are important to consider are: - Electrons have a much shorter wavelength than x-rays
- Electron beams are easily directed because electrons are charged particles
- Electrons are scattered more strongly as they interact with both the nucleus and the electrons of the scattering atom

5. Scattering from a plane of atoms
Diffraction is best described with Bragg’s law (2𝑑∙𝑠𝑖𝑛𝜃=𝑛𝜆)
An initial wavefront (WI) is scattered by a plane of atoms to produce a diffracted wavefront (WD)
Scattered waves can interact constructively or destructively with each other
A diffracted beam occurs when the scattered beams are in phase (constructive)
This is determined by the initial beam, scattered beam and plane angles
Diffraction is best described using the Laue conditions and Bragg’s law. At this point we will mostly be focusing on Bragg’s law and how it works when an electron beam interacts with an atom.
From the picture we can see an initial wavefront (WI) being scattered by a plane of atoms to produce a diffracted wavefront (WD).
Weather or not WD corresponds to a diffracted beam depends on weather the scattering is in phase. This is determined by the angles between the initial beam, diffracted beam and diffracting planes
The conditions required for two beams to be in phase are known as Laue conditions

6. Scattering from a crystal
Beams that interact with each other come from the same plane as well as different planes
A diffraction beam is formed when the path difference of the beams in a certain direction is 2𝑑𝑠𝑖𝑛𝜃
The angle at which two waves scatter constructively is called the Bragg angle (θB).
In reality, this angle is smaller than 1°
When talking about scattering we have to remember that beams which interact with each other are scattered from both the same plane as well as a different plane.
The scattering will be constructive (thus forming a diffraction beam) when the path difference of the beams in a certain direction is 2𝑑𝑠𝑖𝑛𝜃.
In the picture we can see that ray R1 travels a distance of EJ and ray R2 travels a distance HF. Because these two path distances are equal the scattering will be constructive.
When two electron waves scatter in a way that they interfere constructively they scatter at an angle called the Bragg angle (θB).
Although the angles shown in the images are several °, in reality they are below 1°.

7. Scattering from a crystal
In practice there is a series of reflections periodically spaced along a line known as a systematic row of reflections
They are called high-order reflections and arise from scattered electrons from several planes that are d distances apart
They can have destructive or constructive interfaces
In practice there is not just one reflection but a series of reflections which are periodically spaced along a line, also known as a systematic row of reflections.
These reflections called high-order reflections are particularly important in TEM. They arise from the interface of electrons scattered from planes which are d distances apart.
These electrons have both constructive and destructive interfaces.
If we put a plane (P3) halfway in-between two existing planes (P1 and P2) that are distance d apart than the planes will scatter in phase when 2 𝑑 2 𝑠𝑖𝑛𝜃=𝜆
We can generalize this equation to be 2 𝑑 𝑛 𝑠𝑖𝑛𝜃=𝜆 thus 𝑑 𝑛 represents the space between two planes instead of d.

8. Dynamical diffraction
Dynamical beams are beams that are scattered more than once
Once scattered, a beam with a certain orientation can be rescattered thus turning it into a rediffracted beam
This beam can than be diffracted again thus creating the phenomenon of dynamical scattering
This process is more likely with thicker samples
A diffracted beam can be scattered more than once.
Any beam which is oriented so as to be Bragg scattered once is automatically in the ideal orientation to be rescattered, thus turning into a rediffracted beam.
The rediffracted beam is then perfectly orientated to be diffracted again, and so on.
This gives rise to the phenomenon of dynamical scattering because the beam can be scattered again and again.
The likelihood of this process occurring increases with the sample thickness.

9. Diffraction patterns in practice
Diffraction paterens in the TEM can be formed in two ways
Selected Area Diffraction (SADP)
Convergent Beam Electron Diffraction (CBED)
SADP are sharply focused spot patterns
CBDP are arrays of discs with specific distances between them
We can obtain different information from each patteren
Diffraction patterns in the TEM can be formed in two ways, Selected Area Diffraction (SADP) and Convergent Beam Electron Diffraction (CBED)
SAPD are sharply focused spot patterns that we use to select reflections for all imaging modes
CBDP are arrays of discs with certain distances between them

10. Summary Electrons are scattered from atoms within a sample
When the scatterings fit Bragg’s law a constructive interaction occurs between the beams and we get a diffraction beam
Different diffraction beams form diffraction patterns
Diffraction patterns can tell us a lot about the crystallography and morphology of our sample