Bragg-relation

X-ray diffraction occurs in the elastic scattering of X-ray photons by atoms in a periodic lattice. The scattered monochromatic X-rays that are in phase give constructive interference. Figure 4.4 illustrates how diffraction of X-rays by crystal planes allows one to derive lattice spacings by using the Bragg relation ... 

If one measures the angles, 26, under which constructively interfering X-rays leave the crystal, the Bragg relation (1) gives the corresponding lattice spacings, which are characteristic for a particular compound. 

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... 

Figure 23 Bragg relation for crystalline coherent scatter.The atomic distances are comparable to the X-ray wavelength. When the difference in path lengths from reflections off adjacent crystal planes is a multiple of the wavelength, reinforcement occurs. Scatter is significantly favored when the X-ray energy and scatter angle obey the Bragg Law.

Crystalline coherent scatter obeys the Bragg relation that is shown schematically in Figure 23. The photon energy, E, at which coherent scatter will occur off a target crystal is... 

Consider a heteroepitaxial film on a single crystal substrate. For a given family of planes (hkl), specific to the crystal phase that comprises the film, when the sample is irradiated at an incidence angle enabling these (hkl) planes to diffract, each crystal within the film diffracts and contributes to the intensity of a peak located at the angle 20 given by the Bragg relation. 

When a beam of white radiation impinges, as in Figure 2.9a, on a crystal with its surface cut parallel to a crystallographic plane (hkl), reflection can take place only for the component of the beam having a single wavelength X that satisfies the Bragg relation... 

The harmonics of the fundamental of interest (for the Mn K-edge, this is at 6.54 keV) may also satisfy the Bragg relation. An examination of Fig. 10 shows that for a Wiggler magnet the content of the first harmonic (e.g., 13.08 keV for Mn absorption) can be nearly equal to the flux of the... 

The average lamellae thickness distribution may be described quantitatively by small angle X-ray diffraction (SAXS), or indirectly by differential scanning calorimetry (DSC). According to the Bragg relation, the SAXS measurements permit the determination of the structure units with dimension of some hundreds angstrom (lnm = lO m = lOA). The... 

Four- circle diffractometers (Figs. 32 and 33) are free of these disadvantages. The (p and x circles set the crystal in such a way that the reciprocal lattice vector of the reflection being measured lies in the equatorial plane of the in.strument, while the counter tube moves along this circle (2 0 circle). The 0 and 2 0 circles make it possible to set up the Bragg relation. The x circle, called the Eulerian cradle, can be either closed or open (as in Fig, 32). A closed Eulerian cradle has the advantage that any reciprocal lattice vector can. in principle, be... 

Figure 2. Graphical test of modified Bragg relation (eq 1) , D = 285.0 nui , D = U58.6 my Oand , D = 712.8 my, first- and second-order reflections, respectively.

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