The Principle of Calibration for Electron Diffraction Patterns in TEM Analysis:The Principle of Electron Diffraction

1.1 Types of Electron Diffraction Patterns

In transmission electron microscopy (TEM), various diffraction results can be observed depending on the sample type and the diffraction method used. These include single-crystal diffraction patterns, polycrystalline diffraction patterns, amorphous diffraction patterns, convergent beam electron diffraction (CBED) patterns, and Kikuchi patterns.

The structural characteristics of the crystal itself are reflected in the electron diffraction patterns. For instance, diffraction patterns from ordered phases have unique features. Additionally, secondary diffraction can make the patterns more complex.

Figure 1

In the figure:

(a) and (d) are simple single-crystal diffraction patterns.
(b) is a single-crystal diffraction pattern of an ordered perovskite along the [111]p direction, showing a six-fold periodicity.
(c) is an amorphous diffraction result.
(e) and (g) are polycrystalline diffraction patterns.
(f) is a secondary diffraction pattern where satellite spots appear around each main spot due to secondary diffraction.
(i) and (j) are typical Kikuchi patterns.
(h) and (k) are CBED patterns.

To understand why these different diffraction results appear, one must first grasp the principles of electron diffraction. The principles of electron diffraction are fundamentally similar to those of X-ray diffraction, but the extremely short wavelength of electrons gives electron diffraction its unique characteristics.

1.2 Imaging Principles of Electron Diffraction Patterns

When using the Ewald sphere to discuss the imaging geometry of X-ray or electron diffraction, the sample is treated as a geometric point. In reality, however, samples have a finite size, so the outgoing beams cannot be treated as single rays. The applicability of the Ewald sphere is due to the sufficiently large reflective sphere, which allows for approximations.

To fully understand the formation of electron diffraction, two concepts are crucial:

Fresnel (near-field) diffraction
Fraunhofer (far-field) diffraction

Figure 2

The direct diffraction imaging of a small aperture (without using a lens) is a typical Fresnel diffraction (near-field diffraction) phenomenon. In the imaging mode of an electron microscope, Fresnel fringes around circular apertures are often observed.

Fraunhofer diffraction (far-field diffraction) occurs when plane waves interact with an obstacle. Strictly speaking, diffraction between light beams requires superposition, which cannot occur with perfectly parallel light. Therefore, in the absence of a lens, Fraunhofer diffraction is merely a theoretical concept. However, in many cases, diffraction can be approximated as Fraunhofer diffraction. X-ray diffraction is one such example. Although X-rays are diffracted by different crystal planes within a crystal, the spacing between the planes is much smaller than the Ewald sphere (the reciprocal of the X-ray wavelength). Even the ratio of the diffraction instrument's radius to the interplanar spacing is extremely large. Hence, X-ray diffraction can be treated as Fraunhofer diffraction, as the superimposed X-rays from different crystal planes produce diffraction angles very close to the Bragg angle.

Conclusion: X-ray diffraction is not strictly Fraunhofer diffraction, but it can be approximated as such.

Electron diffraction, on the other hand, involves lenses and thus represents strict Fraunhofer diffraction in contrast to X-ray diffraction.

Figure 3

The figure above illustrates that when parallel light beams are enhanced in a specific direction within a crystal, they converge at the back focal plane of the lens to form an intensified diffraction spot. However, the directions in which the crystal generates enhanced diffraction between parallel light beams ultimately depend on whether the Bragg equation is satisfied, i.e., the Ewald geometry condition. The figure below is a schematic representation of the Ewald construction for single-crystal electron diffraction, with the scale of the reflection sphere significantly reduced in the proportional relationships depicted.

Figure 4

As shown in the figure above, if the reciprocal lattice points were idealized as discrete points, it would be impossible for any point on the zero-layer reciprocal plane to simultaneously satisfy the Bragg equation, meaning that none of the points would simultaneously lie on the Ewald sphere. The reason single-crystal electron diffraction patterns can be obtained is due to the unique characteristics of electron diffraction.

First, the wavelength of electron waves is extremely short, resulting in a very large Ewald sphere radius (much larger than the Earth). This makes the intersection between the Ewald sphere and the reciprocal lattice appear almost planar (though this perspective is debatable, as the reciprocal lattice vectors are also very large, and the Bragg condition must still be satisfied. Moreover, it would be practically impossible to construct such a large apparatus to record the data). However, the proportional relationship between the Ewald sphere radius and the reciprocal vectors changes significantly, so Bragg angles for planes with moderate indices fall within a range of a few degrees.

The second reason is that electron microscopy typically observes thin-film samples. In the direction perpendicular to the sample thickness, the reciprocal lattice points stretch into reciprocal rods. As mentioned earlier, a standard electron diffraction pattern corresponds to a scaled representation of the zero-layer reciprocal plane. It essentially magnifies the image at the back focal plane of the objective lens in a transmission electron microscope.

The figure on the right illustrates the relationship among reciprocal lattice vectors, the wavenumber of the electron wave, the camera length, and the vectors corresponding to diffraction spots in the electron diffraction pattern. From this figure, the following proportional relationship can be immediately deduced:

Figure 5

The term K =λ L = R d is commonly referred to as the camera constant, and L is called the camera length.

Figure 6

In the above diagram, the proportional relationship is correct, but we should note that the reciprocal sphere is very large, and the camera length cannot be too large. Therefore, if the camera length is placed inside the reciprocal sphere in the diagram, it would be more accurate. In practice, during electron diffraction operations, before magnification, the diffraction pattern forms on the back focal plane of the objective lens. The camera length is the focal length f_0 of the objective lens. The focal length we obtain on the photographic film is the result of magnification by the intermediate and projection lenses. Therefore, the actual camera length used in processing is L = f0MIMP.

1.3 Advantages and Limitations of Electron Diffraction Patterns

Advantages:

Electron diffraction allows for the simultaneous analysis of morphology and structure on the same sample.
Electrons have short wavelengths, enabling direct observation of a crystal’s reciprocal lattice as a 2D projection. This makes structural and symmetry analysis simpler than with X-rays.
Materials scatter electrons approximately 10,000 times more strongly than X-rays, leading to shorter exposure times.

Limitations:

The diffraction intensity may interact with the transmitted beam, complicating intensity analysis, unlike X-ray diffraction.
High scattering intensity limits electron penetration, requiring thin samples and making sample preparation more challenging than for X-rays.
The precision of electron diffraction is lower compared to X-ray diffraction.

1.4 Selected Area Electron Diffraction (SAED)

By introducing a selected area aperture at the image plane of the objective lens, only imaging electrons from the region A'B' pass through the aperture, forming a diffraction pattern on the fluorescent screen. This diffraction pattern corresponds to the AB area of the sample. SAED is particularly useful for analyzing microstructural details of specific regions.

Figure 7

Figure 8

An example of SAED includes:

(a) a bright-field image.
(b), (c), and (d) SAED patterns from different regions in (a).

To obtain the diffraction pattern of a specific microregion in the crystal, SAED is typically employed. The aperture is placed at the image plane of the objective lens (not directly at the sample) for the following reasons:

Microregions to be analyzed are often sub-micrometer in size, making small apertures difficult to fabricate and position accurately.
Prolonged electron exposure can quickly contaminate the aperture.
Modern TEM instruments have small pole-piece gaps, leaving little space for additional apertures.

1.5 Diffraction and Selection Correspondence

Magnetic Rotation Angle:

When capturing electron micrographs and diffraction patterns, the intermediate lens currents differ, resulting in different rotation angles between the image and the diffraction pattern. This difference is called the magnetic rotation angle.
The angle ψ = ψ_i - ψ_d can be measured under different magnifications.
Some TEM instruments have automatic magnetic angle correction, eliminating the need for manual adjustments.

1.6 Steps for Accurate SAED Pattern Acquisition

Adjust the intermediate lens current to make the edge of the selected area aperture sharp on the fluorescent screen, aligning the intermediate lens object plane with the aperture plane.
Adjust the objective lens current to focus the sample image on the fluorescent screen, aligning the objective lens image plane with the intermediate lens object plane.
Remove the objective aperture and reduce the intermediate lens current for diffraction, aligning its object plane with the back focal plane of the objective lens. Adjust the center spot for roundness and minimal size.
Lower the condenser lens current to reduce the incident beam divergence angle, ensuring diffraction spots are sharp.

This process ensures precise selection and clarity in SAED patterns.