Bragg orientation

What happens if the set of (hkl) lattice planes is not exactly at the Bragg orientation As shown on figure 2b, the position of the two spots is hardly affected but the intensity of the diffracted beam is strongly modified. This behavior can be explained by means of the Ewald sphere construction. 

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). 

If the set of (hkl) lattice planes is not exactly in Bragg orientation, then the two excess and deficiency lines are shifted inside the disks as shown on figure 4b. Depending on their intensity, two t5 es of line are observed. If the intensity is weak, then a quasi-kinematical behavior occurs and the lines are sharp and look like the ones on figures 4a and b. Lines with a strong intensity have a dynamical behavior and they display a set of black and white fringes (figure 4c). 

The Dark-Field symmetry is observed inside a hkl diffracted disk (often characterized by its diffraction vector g) which is exactly in Bragg orientation. This situation occurs when the hkl Bragg line goes through the center of its hkl diffracted disk. 

For particles in a matrix, e.g. Au or Fe particles embedded in MgO, " even with very thin samples the matrix contrast is still too strong compared with the contrast of the particles. Tilting the sample by a few degrees from the Bragg orientation enables attenuation of the image of the matrix before that of the particles, which are smaller and less sensitive to the orientation. HRTEM is the only technique which can be used to determine the size and the edges of particles of this type of sample. 

Special Effects [189], For foil orientations close to an exact Bragg orientation, the transmitted and scattered wave fields both propagate along the lattice planes, in a model... 

Bragg-Brentano Powder Diffractometer. A powder diffraction experiment differs in several ways from a single-crystal diffraction experiment. The sample, instead of being a single crystal, usually consists of many small single crystals that have many different orientations. It may consist of one or more crystalline phases (components). The size of the crystaUites is usually about 1—50 p.m in diameter. The sample is usually prepared to have a fiat surface. If possible, the experimenter tries to produce a sample that has a random distribution of crystaUite orientations. 

X rays and so that the angle between the difiracting plane and the incident X rays is equal to the Bragg angle For a single crystal or epitaxial thin film, there is only one specimen orientation for each (hkl) plane where these difiraction conditions are satisfied. 

Fig. 2. (continued)—(d) an aggregate of microcrystallites whose long axes are parallel, but randomly oriented (left), diffracts to produce a series of layer lines (right) and (e) a polycrystalline and preferentially oriented specimen (left) diffracts to give Bragg reflections on layer lines (right). The meridional reflection on the fourth layer line indicates 4-fold helix symmetry. 

For materials which are available not in the form of substantial individual crystals but as powders, the technique pioneered by Debye and Scherrer is employed (Moore, 1972). The powder is placed into a thin-walled glass capillary or deposited as a thin film, and the sample is placed in the X-ray beam. Within the powder there are a very large number of small crystals of the substance under examination, and therefore all possible crystal orientations occur at random. Hence for each value of d some of the crystallites are correctly oriented to fulfil the Bragg condition. The reflections are recorded as lines by means of a film or detector from their positions, the d values are obtained (Mackay Mackay, 1972). 

XRD on battery materials can be classified as powder dififaction, a technique developed by Peter Debye and Paul Scherrer. In powder dififaction the material consists of microscopic crystals oriented at random in all directions. If one passes a monochromatic beam of X-rays through a fiat thin powder electrode, a fraction of the particles will be oriented to satisfy the Bragg relation for a given set of planes. Another group will be oriented so that the Bragg relationship is satisfied for another set of planes, and so on. In this method, cones of reflected and transmitted radiation are produced (Fig. 27.2). X-ray diffraction patterns can be recorded by intercepting a... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

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