Bragg-type scattering

Cholesteric liquid crystals, e.g., those of cholesteroylnonaoate (see Sec. 3.2), produce a Bragg-type scattering, which depends on temperature and angles of incidence and observation. Either total reflection or total transmission of circular polarized light is observed, which effect provides the basis of the dark-bright liquid crystal display in the Schadt-Helfrich cell (Fig. 3.5.3) as well as color reflection. 

The reflection from cholesteric and blue phases is Bragg-type scattering, similar to the diffraction of X-rays by crystals. The wave vector of the incident light Ko, the wave vector of the scattered light Kg, and the wave vector of the dielectric constant component q must satisfy the Bragg condition ... 

The Bragg-Brentano type of diffractometer is composed of an x-ray tube with a metallic anode that supplies x-rays that are scattered from the sample and focused at the slit before hitting the detector. In some cases, a monochromator capable of yielding a monochromatic x-ray beam is added. The sample is rotated, relative to the x-ray at angles from 0° to 90° with the help of a goniometer, where the powdered sample is placed on the sample holder. Electronic equipment is used to amplify and filter signal pulses from the detector. 

The displacements of atoms may take several forms (Figure 13.2). These have been described by Jack Dunitz, Verner Schomaker, and Kenneth N. Trueblood as follows The perfectly ordered crystal would have every atom firmly fixed to its own perfectly defined site in each unit cell for the entire period of observation. There are, however, various types of disorder from unit cell to unit cell. If the atom jumps to a different site, that is one kind of disorder [a mixture of static and dynamic disorder ] if it moves to and fro, that is another kind of disorder [ dynamic disorder] if it is forever in one site in a certain unit cell and in a different site in another cell, that is still another kind [static disorder]. Each of these types of vibrations, displacements, and disorder has somewhat similar effects on the intensities of Bragg reflections the effect they have in common is that they reduce these intensities by an amount that increases with increasing scattering angle, 26, as shown in Figure 13.1. 

In this equation f is the atomic scattering factor or atomic form factor, h,k,l are the Miller indices of the reflecting plane, and x ,y ,z are the coordinates of the scattering atom in decimal fractions of unit cell parameters, a,b,c, respectively. For simple structures the structure factor indicates what types of Bragg planes in a given kind of structure can produce a diffraction peak, i.e., have non-cancelled, coherent scattering, and also indicates the relative intensity of the allowed peaks (Warren 1969). A few structure factors for simple crystal structures are shown below. 

In powder diffraction, the scattered intensity is customarily represented as a function of a single independent variable - Bragg angle - 20, as modeled in Figure 2.33 for a polycrystalline copper. This type of the plot is standard and it is called the powder diffraction pattern or the histogram. In some instances, the diffracted intensity may be plotted versus the interplanar distance, d, the g-value Q = /d = d ) or sin0/A,. 

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