Diffraction patterns lattice planes

The geometrical aspect concerns the position of the diffracted beams on a pattern it only depends on the direct lattice of the crystal through the Bragg law =2dhkisin9B - dhu being the interplanar distance of the diffracted (hkl) lattice planes and 0b the Bragg angle. In other words, it only depends on the lattice parameters of the crystal a, b, c, a, P and y. 

It is important to note the features of diffraction pattern of a single crystal in TEM the diffraction pattern represents a reciprocal lattice plane with reciprocal lattice points which are shown as the diffraction spots, and the reciprocal lattice plane contains the diffraction of lattice planes belonging to one crystal zone, of which the axis is parallel to the transmitted beam. These features are well illustrated in Figure 2.9 in which the zone axis is [001] of a cubic crystal and the reciprocal lattice plane consists of diffraction spots from lattice planes (hkO). 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

Because the electrons do not penetrate into the crystal bulk far enough to experience its three-dimensional periodicity, the diffraction pattern is determined by the two-dimensional surface periodicity described by the lattice vectors ai and ai, which are parallel to the surface plane. A general lattice point within the surface is an integer multiple of these lattice vectors ... 

Figure 12. Cross-sectional TEM images of a silica sample implanted with Ag and S (a) high-resolution image showing the lattice planes of the Ag2S shell (b) bright-field showing the contrast between the Ag core and the Ag2S shell (c) and (d) are the diffraction pattern of the sample sequentially implanted with S followed by Ag and with Ag followed by S, respectively. (Reprinted from Ref [1], 2005, with permission from Italian Physical Society.)...

Usually many set of lattice planes can simultaneously be exactly or close to the Bragg orientation and give a multi-beam pattern made of several diffracted beams as shown in the example on figure 2c. A special type of multi-beam pattern concerns Zone-Axis Patterns (ZAP). This type of pattern is observed when a high symmetry [uvw] direction of the crystal is parallel to the incident beam. In this case, the spots on the pattern are arranged along Laue zones (Figure 2d). 

One of the basic features of high-energy ED is the short wavelength of electrons used 0.05 A (accelerating voltage lOOkV). Therefore Ewald s sphere practically degenerates into a plane, and the electron diffraction pattern (ED) is the planar cross-section of the reciprocal lattice (Fig.2.). 

Figure 4. Indexing of an electron diffraction pattern representing a coordinate and noncoordinate planes of reciprocal lattice.

If unit cell is orthogonal there are layers lines on oblique texture electron diffraction pattern. These lines occur when certain reciprocal lattice planes lie perpendicular to the texture axis. In this case period c may be more accurately determined by measuring the minor semi-axis R of any ellipse (in the presence of layer lines it is measured directly, since there is a zero line with /=0) and H of any reflection on that ellipse (preferably with a large l) ... 

Geometrically, electron diffraction patterns of crystals can be approximated as sections of the reciprocal lattice, since the Ewald sphere can be regarded as a plane (i.e. the radius of the Ewald sphere, 1/2, is much larger than the lengths of low-index reciprocal lattice vectors). 

Fig. 2.13 Lattice image of slightly reduced WO,. The black lines show a kind of planar defect, which corresponds to the shear plane 120. The inset shows a streaked diffraction pattern for the crystal parallel to g(120), indicating the random distribution of shear planes (see also Fig. 2.16).

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