Laue zone

Figure 6 CBED patterns of aluminum oxynitride spinel along the [001] direction. Symmetries in the patterns contributed to the determination of the point group and space group (a) whole pattern showing 1st Laue zone ring and (b) 0th order Laue zone. Both patterns show a fourfold rotation axis and two mirror planes parallel to the axis. (Courtesy of V. P. Dravid)...

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

Lewis, A.L., Villagrana, R.F. and Metherall, A.J.F. (1978) A description of electron diffraction from higher order Laue zones, Acta Cryst. A, 34, 138-140. 

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). 

The Whole-Pattern symmetry is the symmetry which takes into account all the features present on a high S5mimetry zone axis diffraction pattern (i.e. the disks, the lines inside the disk and the Kikuchi lines). As mentioned above, in order to identify a 3D S5mimetry, the pattern should, at least, display the First-Order Laue Zone. In the example given on figure 2a, this FOLZ is weak, but clearly visible and the Whole Pattern displays a 3D-4mm S5munetry. 

For the other examples (Figures 3b, c, d), in addition to broad black and white fringes, the patterns also display sharp lines which are produced by interactions of the reflections located in the High-Order Laue Zones (HOLZ). The corresponding 3D symmetry is 6mm. 

By definition, a zone axis is normal to both g and h and other reciprocal lattice vectors in the plane defined by these two vectors. The reciprocal lattice plane passing through the reciprocal lattice origin is called the zero-order zone axis. A G-vector with z - G=n with n O is said to belong to a high order Laue zones, which separate to upper Laue zones (n>0) and lower Laue zones (n<0). 

A diffraction pattern with diffraction spots belonging to both zero-order Laue zone and higher order Laue zones can be used to determine the three-dimensional unit-cell of the crystal. 

These lines, named as high order Laue zone (HOLZ) lines, are very useful for measuring lattice parameters and local strain. The sensitivity of the lines... 

The dynamic calculations include all beams with interplanar distances dhki larger than 0.75 A at 120 kV acceleration voltage and thickness between 100 A and 300 A for the different zones. The structure factors have been calculated on the basis of the relativistic Hartree - Fock electron scattering factors [14]. The thermal difiuse scattering is calculated with the Debye temperature of a-PbO 481 K [15] at 293 K with mean-square vibrational amplitude

= 0.0013 A following the techniques of Radi [16]. The inelastic scattering due to single-electron excitation (SEE) is introduced on the base of real space SEE atomic absorption potentials [17]. All calculations are carried out in zero order Laue zone approximation (ZOLZ). 

In CBED, zone-axis patterns (ZAP) can be recorded near the relevant zone axis and the pattern may also include a higher-order Laue zone (referred to as a HOLZ). The radius of the first HOLZ ring G is related to the periodicity along the zone axis [c] and the electron wavelength, by = 2/kc. CBED can thus provide reciprocal space data in all three (x,y,z) dimensions, typically with a lateral resolution of a few nanometres. As in any application, corroborative evidence from other methods such as HRTEM and single-crystal x-ray diffraction, where possible, can be productive in an unambiguous structural determination of complex and defective materials such as catalysts. We illustrate some examples in later sections. 

Figure 4 (a) Schematic illustration of large-angle CBED technique and (b) an example of LACBED pattern taken from Si near [113] zone axis showing high order Laue zone lines in the disk... 

Powder XRD shows one-dimensional patterns created from three-dimensional structures, while SAED patterns are two-dimensional pictures reflecting two-dimensional structural information. The disadvantage of SAED is that the structural information along the projected direction is lost, unless a special technique is used, for example high-order Laue zone (HOLZ) diffraction [14, 15]. The advantage of SAED is that the relation between any two diffraction spots in a pattern, i.e. an interplane angle, can be easily revealed. This helps greatly in determination of a unit cell. 



Figure 3.12. Facing page, (a) Diagram showing the Ewald sphere cutting the reciprocal lattice rods of zero and higher order Laue zones and (b) a schematic diagram of the corresponding diffraction pattern.

Fig. 3 Ewald construction. The white half-circle indicates the Ewald sphere in two dimensions. The points of intersection between the reciprocal lattice rods and the Ewald sphere form the set of reciprocal lattice points (bright) which obey Bragg s law and appear as diffraction spots in the diffraction pattern. Zero-, first- and second-order Laue zone are indicated. Eor electron diffraction in TEM, the ratio between the radius of the Ewald sphere and the reciprocal lattice unit is larger than visualized in the figure. (View this art in color at www.dekker. com.)...

Fig. 40. (b) High magnification image of the [lOT] zone axis pattern related to the diffraction in (a). The lack of symmetry about an axis normal to the image is due to the influence of the image of the nodes of the upper Laue zones. Arrows indicate the position of the bend contours due to these nodes. 



Fig. 41. Intersection of the Ewald sphere with the reciprocal lattice of B, when the Ewald sphere is tangent to the (lOl) plane. This situation occurs for the electron beam parallel to the [101] axis. The [201] direction is slightly different from the direction of the radius of the Ewald sphere. Nodes of the upper Laue zones give extra spots in the [202] direction, but not near the origin of the reciprocal lattice. The asymmetry of the upper Laue zones is generally visible in electron diffraction.

Fig. 53. (a) General feature of the fine structure near a <111) zone axis pattern. This fine structure is due to the influence of the upper Laue zones, the image has only a three-fold symmetry, (b) Relation between two fine structures occurring on each side of a twin of C. 



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