Bragg’s reflection

The WAXS patterns of two nanocomposites at different LDH concentrations are shown in Fig. 55. The similarity of these patterns between the two systems is that the first three Bragg s reflections of LDH-C10 can be detected in both. This means that the LDH particles are not fully exfoliated in any of the matrices. 

The curve in Fig. 56b shows entirely different conditions. The first maximum is displaced much nearer to the point of penetration. This indicates the very frequent occurrence of some distance in the liquid which exceeds the intramolecular distances of the individual atoms for according to Bragg s reflection law... 

Therefore, when the Ewald s sphere contains, passes by, or touches points of the reciprocal lattice the possibility of Bragg s reflection is predicted the point from the reciprocal space, on the Ewald sphere, touched... 

FIGURE 5.13 The Bragg s reflection rationalization through the Ewald s sphere of diffraction, after (Matter Diffraction, 2003 X-Rays, 2003 Pufz Lacrama, 2005 HyperPhysics, 2010). 

At this point the second possibility of reproduction of the diffraction picture comes into attention, by fixing the wavelength of the incident X-radiation, but rotating the crystal, i.e., varying the angle of Bragg s reflection, Figure 5.17. 

The unit-cell parameter (a, mn) for SBA-15 (equation 3) was evaluated using the interplanar spacing (d, mn) corresponding to (100) Bragg s reflection assessed from the X-ray diffraction profile. 

Fig. 6.6. Bragg s reflection from a set of hkl planes having interplanar distance d...

It can be rightly said that The reciprocal lattice is as important in crystal structure analysis as the walking stick of a blind man moving in a narrow lane having frequent turns. It is extremely difficult if not impossible to picture the different intersecting crystal planes satisfying the Bragg s reflection in three-dimensional lattice from the two-dimensional array of spots or lines. 

Therefore, to get (6.4) and (6.5) simultaneously satisfied, we get BC = l/dhki, i.e., B is a point on the normal drawn on the plane hkl and its distance from the plane is reciprocal of the interplanar distance dhki- SOj the point B is the reciprocal lattice point of the plane hkl, and when there is a Bragg s reflection from the plane for a definite wavelength then it should lie on the surface of the corresponding Ewald s sphere. 

Conclusion. The Bragg s reflection will occur from a plane for a definite wavelength of X-ray, incident in a particular direction only when the reciprocal lattice point of the plane concerned lies on the surface of the corresponding Ewald s sphere. 

A Laue pattern which is shown in Fig. 7.2 is symmetrical arrangement of diffraction spots on the flat film either in the transmission mode or in back reflection mode. The spots when connected generate either an ellipse or a hyperbola. These spots originate due to Bragg s reflection from sets of intersecting planes which are parallel to an axis known as zone axis (Fig. 7.3). Planes having common zone axis generate symmetrical spots, and in most of the cases they show a center of symmetry. 

The scattered X-ray wave from such modulated structures is in the form of Bragg s reflection and the scattering vectors of Bragg s reflections are... 

An external electric field interacts with the local dielectric anisotropy of a blue phase and contributes e E /An to the energy of the liquid crystal [45]. The field distorts the cubic lattice and results in a change in the angular (or spectral) positions of Bragg s reflections. Moreover, field-induced phase transitions to novel phases have been observed [42, 46, 47]. The field can also induce birefringence parallel to the field direction, due to the optical biaxiality of the distorted cubic lattice [48]. 

This simplified derivation is based on Bragg s reflection analogy. Its main insufficiencies are (1) While the number of planes N corresponds to the dimensions of the crystal perpendicular to these planes, no account is taken of the lateral extent of the planes, that is, the dimensions of the crystal parallel to the planes. (2) Absolute values for the reflected intensity can not be calculated a is unknown). For the present calculation this is, however, of minor importance since a cancels out. The same final formula could be derived by a more exact but less convenient derivation (3). 

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