Scattering Bragg equation

When X-rays illuminate a crystalline material, the atoms in the crystal act as scattering centers. Because of the periodic nature of crystals, the scatterers can be considered to be associated with periodically spaced parallel planes a distance d apart. For certain angles of incidence to these planes the X-rays are scattered coherently and in phase. The coherent scattering is known as X-ray diffraction and the geometric condition required for diffraction, the Bragg equation, is given by... 

In X-ray diffraction one is interested in exploring the intensity of X-rays diffracted from the crystal planes. Note that the Bragg equation does not contain information about the scattered intensity from a given plane. It only provides the... 

While it is very easy, when one knows the structure of the crystal and the wavelength of the rays, to predict the diffraction pattern, it is quite another matter to deduce the crystal structure in all Its details from the observed pattern and the known wavelength. The first step is lo determine the spacing of the atomic planes from the Bragg equation, and hence the dimensions of the unit cell. Any special symmetry of the space group of the structure will be apparent from space group extinction. A Irial analysis may (hen solve the structure, or it may be necessary to measure the structure factors and try to find the phases or a Fourier synthesis. Various techniques can be used, such as the F2 series, the heavy atom, the isomorphous series, anomalous atomic scattering, expansion of the crystal and other methods. 

For a uniform charge distribution within a spherical atom the Fourier transform of the density has been shown (equation 5.6) to be of the form sin a/a, for a wave of phase a in momentum space. From the Bragg equation (Figure 2.9), A = 2dsin0, it follows that electrons at a distance d = A/2sin0 apart, scatter in phase, i.e. with phase difference 27T. At a separation r the relative phase shift a, is given by ... 

Lane equations Equations that, like the Bragg equation, express the conditions for diffraction in terms of the path difference of scattered waves. Laue considered the path length differences of waves that are diffracted by two atoms one lattice translation apart. These path differences must be an integral number of wavelengths for diffraction (that is, reinforcement) to occur. This condition must be true simultaneously in all three dimensions. 

The Bragg equation for the scattering from a series of parallel planes of spacing d, is given by... 

To derive the Bragg equation, we used an assumption of specular reflection, which is borne out by experiment. For crystalline materials, destructive interference completely destroys intensity in all directions except where Equation (5) holds. This is no longer true for disordered materials where diffracted intensity can be observed in all directions away from reciprocal lattice points, known as diffuse scattering, as discussed in Chapter 16. 

Setting the magnitude of s to 1/2, we get the Bragg equation in terms of the magnitude of the scattering vector h ... 

The Bragg equation shows that diffraction occurs when the scattering vector equals a reciprocal lattice vector. The scattering vector depends on the geometry of the experiment whereas the reciprocal lattice is determined by the orientation and the lattice parameters of the crystalline sample. Ewald s construction combines these two concepts in an intuitive way. A sphere of radius 1//1 is constructed and positioned in such a way that the Bragg equation is satisfied, and diffraction occurs, whenever a reciprocal lattice point coincides with the surface of the sphere (Figure 1.8). 

Scattering from Crystalline Solids and the Bragg Equation... 

The phase difference between the waves scattered by two atoms will depend upon their relative positions in the unit cell and the directions along which the waves are superimposed. The directions of importance are those specified by the Bragg equation, equation (6.1), which, for simplicity, are denoted by the indices of the (hkl) planes involved in the scattering, rather than the angle itself. The phase of the wave (in radians) scattered from an atom A at a position xA, yA, zA, into the (hkl) reflected beam is ... 

Thus we see that a particle would be scattered by a crystal only when a diffraction equation similar to the Bragg equation for x-rays is satisfied. The wave length of light is replaced by the expression... 

The basic condition for the observation of the maxima in the scattered intensity is given by the Bragg equation... 

Here the structure factor signifies the vectorial sum of the waves scattered by the single atoms which show amplitude f and phase y. Every atom contributes a scattered wave to the whole diffraction effect, the amplitude of which is proportional to the so-called form factor. The phase is thus defined by the position of the atom in the elementary cell, whilst the form factor is a characteristic constant for every sort of atom which represents a measure of its scattering power. Hence no special differences exist in the positions of the diffracted beams, which in both X-ray and electron diffraction cases satisfy the geometric relations between lattice constant and X-ray or material wavelengths, according to the Bragg equation. However, there are definitely differences in their intensities. 

The Bragg equation provides a simpler physical relationship between the directions of incident and scattered rays by a crystal plane, i.e. a monochromic incident beam of wavelength A will be reflected by a family of parallel crystal planes (h, k, 1) if the incident angle is 0 ... 

Equation 4.70 is referred as the Vector Bragg Equation which expresses the relationship between H, a vector characterizing a crystal plane, and s, a vector characterizing the scattering geometry, for constructive interference to occur. Such a vector equation implies two conditions ... 

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