Crystallographic planes diffraction

Both ultrasonic and radiographic techniques have shown appHcations which ate useful in determining residual stresses (27,28,33,34). Ultrasonic techniques use the acoustoelastic effect where the ultrasonic wave velocity changes with stress. The x-ray diffraction (xrd) method uses Bragg s law of diffraction of crystallographic planes to experimentally determine the strain in a material. The result is used to calculate the stress. As of this writing, whereas xrd equipment has been developed to where the technique may be conveniently appHed in the field, convenient ultrasonic stress measurement equipment has not. This latter technique has shown an abiHty to differentiate between stress reHeved and nonstress reHeved welds in laboratory experiments. 

Transmission electron microscopy ( ) analysis reveals that these materials crystallize as hexagonal planar particles with marked anisotropic shape,8,37 as shown in Figure 6. When appropriate preparation methods are used, plate-like crystals are obtained with small thickness of about 20-30 nm and an aspect ratio D/h=5-10. Selected area diffraction (SAD) patterns of incident beams perpendicular and parallel to the large hexagonal facet show that they correspond to the crystallographic planes perpendicular to the c axis. The anisotropic shape of the... 

Definition rocking curve (RC) is a function of the total intensity of X-rays reflected by a sample versus its angular position in rotation around the axis perpendicular to the diffraction plane. The sample is adjusted to have diffracting crystallographic planes perpendicular to the diffraction plane. 

Dark-field methods Dark field images are obtained by admitting only diffracted electrons and excluding directly transmitted electrons. Dark-field imaging selectively detects crystallites with crystallographic plane spacings within a relatively narrow range. 

Figure 1. Expression for the diffracted x-ray intensity from each crystallographic plane (h,k, ). [Note the role of the structure factor, which includes the scattering contributions from all atoms in the unit cell.]...

The crystallographic plane is a geometrical concept (mathematical abstraction) introduced to illustrate the phenomenon of diffraction from ideal crystal lattices since algebraic equations that govern diffraction process are difficult to visualize. It is important to realize and remember that no real ... 

According to the Braggs, diffraction from a crystalline sample can be explained and visualized by using a simple notion of mirror reflection of the incident x-ray beam from a series of crystallographic planes. As established earlier (see Chapter 1, section 1.14.1), all planes with identical triplets of Miller indices are parallel to one another and they are equally spaced. Thus, each plane in a set hkJ) may be considered as a separate scattering object. The set is periodic in the direction perpendicular to the planes and the repeat distance in this direction is equal to the interplanar distance dhki- Diffraction from a set of equally spaced objects is only possible at specific angle(s) as... 

X-ray beams incident on a crystalline solid will be diffracted by the crystallographic planes as illustrated in Figure 2.6. Two in-phase incident waves, beam 1 and beam 2, are deflected by two crystal planes (A and B). The deflected waves will not be in phase except when the following relationship is satisfied. 

A crystallographic plane (hkl) is represented as a light spot of constructive interference when the Bragg conditions (Equation 2.3) are satisfied. Such diffraction spots of various crystallographic planes in a crystal form a three-dimensional array that is the reciprocal lattice of crystal. The reciprocal lattice is particularly useful for understanding a diffraction pattern of crystalline solids. Figure 2.7 shows a plane of a reciprocal lattice in which an individual spot (a lattice point) represents crystallographic planes with Miller indices (hkl). 

Crystal planes belonging to one crystal zone (a group of crystallographic planes of which normal directions are perpendicular to one direction called the zone axis direction) form one reciprocal lattice plane. The zone axis direction is perpendicular to that reciprocal lattice plane. For example, Figure 2.9 shows an example of the relationship between crystal zone [001] and the reciprocal lattice plane. The reciprocal lattice plane is composed of the diffraction spots of crystallographic planes belonging to the [001] zone. 

Figure 2.18 The experimental diffraction pattern of a silicon and AI2O3 mixture. Numbers with three digits mark the Miller indices of corresponding crystallographic planes.

Any factor that changes the lattice parameters of crystalline specimens can also distort their X-ray diffraction spectra. For example, residual stress in solid specimens may shift the diffraction peak position in a spectrum. Residual stress generates strain in crystalline materials by stretching or compressing bonds between atoms. Thus, the spacing of crystallographic planes... 

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