Bragg’s Law of 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. 

Figure 7-30. Bragg s law of diffraction. d is the separation and 0 the angle of observation.

X-ray diffraction is based on Bragg s law of diffraction which states that a set of lattice planes in a crystalline material diffracts a beam of X-rays at a certain incident angle that satisfies the Bragg equation (Cullity 1978) ... 

X-ray diffraction is described by Bragg s Law of Diffraction. The equation is pretty simple ... 

X-ray diffraction (XRD) helps scientists determine the structure of crystalline materials and the stoichiometry of the materials within the crystals. It s based on Bragg s Law of Diffraction (nA = 2d Sin0) that describes how a crystalline material diffracts light waves when they pass through it. Crystalline materials cause diffraction because of the regular and periodic placement of atoms in the crystal. The crystalline and ordered structure breaks up the solid stream of x-ray waves into fragmented streams of x-ray waves according to the crystal order in the material. 

Figure 5.7 Derivation of Bragg s law of X-ray diffraction. Parallel X-rays strike the surface at an angle 0, and are reflected from successive planes of crystals of interplanar spacing d. The path difference between reflections from successive planes is given by AB + BC, which, by geometry, is equal to 2dsin0. For constructive interference, this must be equal to a whole number of wavelengths of the incoming radiation.

Bragg s Law for Diffraction of X-rays from Crystal Planes... 

From the analysis of the Bragg s law of X-ray diffraction, the relation (5.40), there results how, excepting the interplanarJ distance as a structure characteristic (internal to the crystal), the wavelength of incident radiation X and the angle of reflection 0 rest as variable (or external) parameters. 

In simple terms we can regard the crystal as a set of lattice planes, reflecting radiation according to Bragg s law. The diffraction angle is then 26 where 2d sin 6 = 1. For electrons A very small. This means that lattice planes will diffract only if they are almost parallel to the... 

In order to determine which lattice planes give rise to Bragg diffraction, a geometrical construct known as an Ewald sphere is used. This is simply an application of the law of conservation of momentum, in which an incident wave, k, impinges on the crystal. The Ewald sphere (or circle in two-dimensional) shows which reciprocal lattice points, (each denoting a set of planes) which satisfy Bragg s Law for diffraction of the incident beam. A specific diffraction pattern is recorded for any k vector and lattice orientation - usually projected onto a two-dimensional film or CCD camera. One may construct an Ewald sphere as follows (Figure 2.44) ... 

That is to say, peaks are observed in the diffraction data when the momentum transfer vector is equal to a vector of the reciprocal lattice. Such peaks are known as Bragg peaks and Equation [34] is Bragg s law, for diffraction from a single crystal. Figure 6 shows a typical single crystal diffraction pattern for every peak in the observed diffraction pattern the momentum transfer vector, Q, satisfies Bragg s law. Each and every Bragg peak may be identified by a unique combination of indices h, k, /). 

Bragg s Law (Equation 1-11) is obeyed so well that it is possible to use x-ray diffraction from crystals for highly precise determinations either of d or of A. The former type of determination is basic in establishing crystal structure. 

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