Bragg-reflection

A cholesteric forms a helical structure and its optical properties are characterised by the tensor of dielectric permittivity rotating in space. We are already familiar with the form of the cholesteric tensor (see Section 4.7). It was Oseen [1] who suggested the first quantitative model of the helical cholesteric phase as a periodic medium with local anisotropy and very specific optical properties. First we shall discuss more carefully the Bragg reflection from the so-called cholesteric planes . 

The most characteristic features of cholesteric liquid crystals are as follows  

The regions of rotation with different handedness are not separated by an absorption band as in typical gyrotropic materials. Instead, there is a band of a selective reflection of the beam with a particular circular polarization, curve R in Fig. 12.1. The beam with the opposite circular polarization is transmitted without any change, therefore the reflection is negligible and not shown in the plot. Only one band is observed in the wavelength spectmm without higher diffraction orders. 

The wavelength of selected reflection A,q (in vacuum) depends on the angle of light incidence / measured from the layer normal, namely, Xq = 2 P(J2) n cos/. It is the same Bragg condition discussed in Section 5.2.2, Xq = 2i/sin . However, in the case of the X-ray diffraction on a stack of the layers in vacuum, we used refraction index n = 1, sliding angle 0 = (n/2) — i, and interlayer 

We see that the optical properties of cholesterics are quite peculiar. How to explain them on the quantitative basis  

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Percent Crystallinity. For samples that consist of a mixture of crystalline and amorphous material, it is possible to determine the percent of crystallinity by measuring the integrated intensity of sharp Bragg reflections and the integrated intensity of the very broad regions due to the amorphous scattering. 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

The observation of the departure from cubic symmetry above Tm co-incident with the appearance of the central peak scattering serves to resolve the conflict between dynamic and lattice strain models. The departure from cubic symmetry may be attributed to a shift in the atomic equilibrium position associated with the soft-mode anharmonicity. In such a picture, the central peak then becomes the precusor to a Bragg reflection for the new structure. 

Such off-zone-centre, soft-mode systems offer the most favourable conditions for a test of the hypothesis that the central peak is a precursor to a Bragg reflection in the transformation phase. Zone-centre softening, such as occurs in NbaSn, results in the central mode scattering emerging from an existing Bragg peak, which ultimately splits in the lower symmetry transformation structure, which presents a problem with resolution. 

As one may infer from the quotation, W. L. Bragg realized that a crystal can act as an x-ray grating made up of equidistant parallel planes (Bragg planes) of atoms or ions from which unmodified scattering of x-rays can occur in such fashion that the waves from different planes are in phase and reinforce each other. When this happens, the x-rays are said to undergo Bragg reflection by the crystal and a diffraction pattern results. 

Provision is sometimes made for varying 6 by continuous rotation of the crystal. From what was said above, it follows that the detector must be rotated at twice the angular velocity of the crystal to obtain Bragg reflection of all wavelengths in the range being investigated. 

To summarize Filtering is an effective way of producing intense monochromatic beams, but it is severely limited because it cannot be used at all wavelengths and cannot achieve high spectral purity at any wavelength. The analysis of a spectrum, that is, the selection of a line and the measurement of its intensity, requires Bragg reflection. 

Satisfactory monochromatization by Bragg reflection from a flat crystal presupposes parallel beams hence the necessity for collimation. This necessity is implicit in Bragg s Law (Equation 1-11), according to which the wavelength singled out for first-order reflection by a crystal of spacing d depends only upon 0, the angle of incidence. One may turn... 

Fig. 4-6. Diffraction of a beam from a point source by a large crystal. The crystal is positioned for the Bragg reflection of wavelength X2 at angle 02 Without a slit, Bragg reflection of all wavelengths between Xi and X3 will occur because the crystal receives x-rays at all angles between 0i and 03. A slit at A or B will collimate the beam and remove the unwanted wavelengths.

Bragg reflection (1.14) can accomplish, as filtering (4.6) cannot, both monochromatization and spectral analysis. With a well-collimated beam, this can be done with a flat crystal, though not without further losses in intensity. 

We say then that a crystal is satisfactory for purposes of chemical analysis if the beam it reflects is monochromatic within the limits established by the collimating system. As theory shows,15 some broadening is to be expected on Bragg reflection even from perfect crystals, but this broadening is so small (not over 0.001°) that we need not consider it. Relatively few crystals, notably some diamonds and calcites, approach perfection. Sodium chloride, more useful in x-ray spectrog-raphy, broadens monochromatic x-rays appreciably, but the. total broadening can be smaller than 0.30°,16 the collimator a perture. See Figure 4-9. 

For the purposes of analytical chemistry, four kinds of monochromatic beams need to be considered. (The quotation marks are to remind the reader that the beams under discussion are not always truly monochromatic.) Three kinds of beams—those produced by Bragg reflection (4.9), filtered beams (4.6), beams in which characteristic lines predominate over a background that can be neglected— will be discussed later ( 6.2). The fourth kind of beam contains monochromatic x-rays that are a by-product of our atomic age and that promise to grow in importance they are given off by radioactive isotopes. These x-rays must not be confused with 7-rays (11.1), which also originate from radioactive atoms but which differ from x-rays because the transformation that leads to radiation involves the nucleus. 

Table 8-1. Higher-Order Bragg Reflections and Aluminum Ka...

Bornite, determination by x-ray emission spectrography, 207, 208 Boundary layers in gases, studied by x-ray absorptiometry, 84 Bragg reflection, 22, 115-124 from a curved crystal, 118-124 from a flat crystal, 115-118 order of, 23... 

Higher-order Bragg reflections, interferences with aluminum Ka by, 218 High-temperature alloys, analysis by x-ray emission spectrography, 179-183... 

Monochromatization by Bragg reflection, see Bragg reflection Monochromators, relation of wavelength to diffraction angle for common, table, 318-327... 

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