In a transmission electron microscope (converging), and ring diffraction (parallel with many grains).|alt=Electron diffraction patterns from different types of crystals and different incident beam convergence. Electron diffraction in a
TEM exploits controlled electron beams using electron optics. Different types of diffraction experiments, for instance
Figure 9, provide information such as
lattice constants, symmetries, and sometimes to solve an unknown
crystal structure. It is common to combine it with other methods, for instance images using selected diffraction beams,
high-resolution images showing the atomic structure, chemical analysis through
energy-dispersive x-ray spectroscopy, investigations of electronic structure and bonding through
electron energy loss spectroscopy, and studies of the electrostatic potential through
electron holography; this list is not exhaustive. Compared to
x-ray crystallography, TEM analysis is significantly more localized and can be used to obtain information from tens of thousands of atoms to just a few or even single atoms.
Formation of a diffraction pattern In TEM, the electron beam passes through a thin film of the material as illustrated in
Figure 10. Before and after the sample the beam is manipulated by the
electron optics these act on the electrons similar to how glass lenses focus and control light. Optical elements above the sample are used to control the incident beam which can range from a wide and parallel beam to one which is a converging cone and can be smaller than an atom, 0.1 nm. As it interacts with the sample, part of the beam is diffracted and part is transmitted without changing its direction. This occurs simultaneously as electrons are everywhere until they are detected (
wavefunction collapse) according to the
Copenhagen interpretation. Below the sample, the beam is controlled by another set of magnetic lenses and apertures. However, projector lens aberrations such as
barrel distortion as well as dynamical diffraction effects (e.g.) cannot be ignored. For instance, certain diffraction spots which are not present in x-ray diffraction can appear, or automated diffraction tomography to solve crystal structures.
Polycrystalline pattern using simulation engine of
CrysTBox. Corresponding experimental patterns can be seen in
Figure 13. |alt=A pattern showing how diffraction patterns from different grain build up to yield a ring pattern. Diffraction patterns depend on whether the beam is diffracted by one
single crystal or by a number of differently oriented crystallites, for instance in a polycrystalline material. If there are many contributing crystallites, the diffraction image is a superposition of individual crystal patterns, see
Figure 12. With a large number of grains this superposition yields diffraction spots of all possible reciprocal lattice vectors. This results in a pattern of
concentric rings as shown in
Figure 12 and
13. Here
high-resolution transmission electron microscopy and
fluctuation electron microscopy can be more powerful, although this is still a topic of continuing development.
Multiple materials and double diffraction In simple cases there is only one grain or one type of material in the area used for collecting a diffraction pattern. However, often there is more than one. If they are in different areas then the diffraction pattern will be a combination. • Chemical ordering, that is different atom types at different locations of the subcell. • Magnetic order of the spins. These may be in opposite directions on some atoms, leading to what is called
antiferromagnetism. In addition to those which occur in the bulk, superstructures can also occur at surfaces. When half the material is (nominally) removed to create a surface, some of the atoms will be under coordinated. To reduce their energy they can rearrange. Sometimes these rearrangements are relatively small; sometimes they are quite large. Similar to a bulk superstructure there will be additional, weaker diffraction spots. One example is for the silicon (111) surface, where there is a supercell which is seven times larger than the simple bulk cell in two directions. This leads to diffraction patterns with additional spots some of which are marked in
Figure 14. Here the (220) are stronger bulk diffraction spots, and the weaker ones due to the surface reconstruction are marked 7 × 7—see note for convention comments.
Aperiodic materials In an
aperiodic crystal the structure can no longer be simply described by three different vectors in real or reciprocal space. In general there is a substructure describable by three (e.g. \mathbf a, \mathbf b, \mathbf c), similar to supercells above, but in addition there is some additional periodicity (one to three) which cannot be described as a multiple of the three; it is a genuine additional periodicity which is an
irrational number relative to the subcell lattice. The diffraction pattern can then only be described by more than three indices. An extreme example of this is for
quasicrystals, which can be described similarly by a higher number of Miller indices in reciprocal space—but not by any translational symmetry in real space. An example of this is shown in
Figure 15 for an Al–Cu–Fe–Cr decagonal quasicrystal grown by magnetron sputtering on a sodium chloride substrate and then lifted off by dissolving the substrate with water. In the pattern there are pentagons which are a characteristic of the aperiodic nature of these materials.
Diffuse scattering A further step beyond superstructures and aperiodic materials is what is called
diffuse scattering in electron diffraction patterns due to disorder, or neutron scattering. This can occur from inelastic processes, for instance, in bulk silicon the atomic vibrations (
phonons) are more prevalent along specific directions, which leads to streaks in diffraction patterns.
Convergent beam electron diffraction In convergent beam electron diffraction (CBED), As illustrated in
Figure 18, the details within the disk change with sample thickness, as does the inelastic background. With appropriate analysis CBED patterns can be used for indexation of the crystal point group, space group identification, measurement of lattice parameters, thickness or strain.
Precession electron diffraction Precession electron diffraction (PED), invented by Roger Vincent and
Paul Midgley in 1994, is a method to collect electron diffraction patterns in a
transmission electron microscope (TEM). The technique involves rotating (precessing) a tilted incident electron beam around the central axis of the microscope, compensating for the tilt after the sample so a spot diffraction pattern is formed, similar to a SAED pattern. However, a PED pattern is an integration over a collection of diffraction conditions, see
Figure 19. This integration produces a quasi-kinematical
diffraction pattern that is more suitable as input into
direct methods algorithms using electrons to determine the
crystal structure of the sample. Because it avoids many dynamical effects it can also be used to better identify crystallographic phases.
4D STEM 4D scanning transmission electron microscopy (4D STEM) is a subset of
scanning transmission electron microscopy (STEM) methods which uses a pixelated electron detector to capture a
convergent beam electron diffraction (CBED) pattern at each scan location; see the main page for further information. This technique captures a 2 dimensional reciprocal space image associated with each scan point as the beam rasters across a 2 dimensional region in real space, hence the name 4D STEM. Its development was enabled by better STEM detectors and improvements in computational power. The technique has applications in diffraction contrast imaging, phase orientation and identification, strain mapping, and atomic resolution imaging among others; it has become very popular and rapidly evolving from about 2020 onwards. Most of these are the same, although there are instances such as momentum-resolved STEM where the emphasis can be very different.
Low-energy electron diffraction (LEED) Low-energy electron diffraction (LEED) is a technique for the determination of the surface structure of
single-crystalline materials by bombardment with a
collimated beam of low-energy electrons (30–200 eV). In this case the Ewald sphere leads to approximately back-reflection, as illustrated in
Figure 20, and diffracted electrons as spots on a fluorescent screen as shown in
Figure 21; see the main page for more information and references. It has been used to solve a very large number of relatively simple surface structures of metals and semiconductors, plus cases with simple chemisorbants. For more complex cases transmission electron diffraction or surface x-ray diffraction have been used, often combined with
scanning tunneling microscopy and
density functional theory calculations. LEED may be used in one of two ways: is a
technique used to characterize the surface of
crystalline materials by reflecting electrons off a surface. As illustrated for the Ewald sphere construction in
Figure 22, it uses mainly the higher-order Laue zones which have a reflection component. An experimental diffraction pattern is shown in
Figure 23 and shows both rings from the higher-order Laue zones and streaky spots.
Low-energy electron diffraction (LEED) is also surface sensitive, and achieves surface sensitivity through the use of low energy electrons. The main uses of RHEED to date have been during thin film growth, as the geometry is amenable to simultaneous collection of the diffraction data and deposition. It can, for instance, be used to monitor surface roughness during growth by looking at both the shapes of the streaks in the diffraction pattern as well as variations in the intensities. A gas carrying the molecules is exposed to the electron beam, which is diffracted by the molecules. Since the molecules are randomly oriented, the resulting diffraction pattern consists of broad concentric rings, see
Figure 24. The diffraction intensity is a sum of several components such as background, atomic intensity or molecular intensity. I_\text{tot}(s) = I_a(s) + I_m(s) + I_t(s) + I_b(s) ,where I_a(s) results from scattering by individual atoms, I_m(s) by pairs of atoms and I_t(s) by atom triplets. Intensity I_b(s) corresponds to the background which, unlike the previous contributions, must be determined experimentally. The intensity of atomic scattering I_a(s) is defined as I_m(s) = \frac{K^2}{R^2} I_0 \sum_{i=1}^N \sum_{\stackrel{j=1}{i\neq j}}^N \left| f_i(s) \right| \left| f_j(s)\right| \frac{\sin [s(r_{ij}-\kappa s^2)]}{sr_{ij}} e^{-(1/2 l_{ij} s^2)} \cos [\eta _i (s) - \eta _i (s)] ,where r_{ij} is the distance between two atoms, l_{ij} is the mean square amplitude of vibration between the two atoms, similar to a
Debye–Waller factor, \kappa is the anharmonicity constant and \eta a phase factor which is important for atomic pairs with very different nuclear charges. The summation is performed over all atom pairs. Atomic triplet intensity I_t(s) is negligible in most cases. If the molecular intensity is extracted from an experimental pattern by subtracting other contributions, it can be used to match and refine a structural model against the experimental data.
In a scanning electron microscope .|alt=Kikuchi pattern, a set of line-like features from a scanning electron microscope. In a
scanning electron microscope the region near the surface can be mapped using an electron beam that is scanned in a grid across the sample. A diffraction pattern can be recorded using
electron backscatter diffraction (EBSD), as illustrated in
Figure 25, captured with a camera inside the microscope. A depth from a few nanometers to a few microns, depending upon the electron energy used, is penetrated by the electrons, some of which are diffracted backwards and out of the sample. As result of combined inelastic and elastic scattering, typical features in an EBSD image are
Kikuchi lines. Since the position of Kikuchi bands is highly sensitive to the crystal orientation, EBSD data can be used to determine the crystal orientation at particular locations of the sample. The data are processed by software yielding two-dimensional orientation maps. As the Kikuchi lines carry information about the interplanar angles and distances and, therefore, about the crystal structure, they can also be used for
phase identification == Notes ==