Bragg reflections multiple

Polymers invariably form helical structures, and the helix symmetry is denoted by u, indicating that there are u repeat units in V turns of the helix. The helix pitch is denoted by P and the molecular repeat distance is c = vP. X-ray diffraction patterns from non-crystalline specimens contain diffracted intensities restricted to layer lines that are spaced by 1/c. On a diffraction pattern from a polycrystalline specimen, diffraction signals, or Bragg reflections, occur only at discrete positions on the layer lines, the positions being related to the lateral dimensions of the unit cell of the crystal. The meridian (vertical axis) of the diffraction pattern is devoid of diffracted intensity unless the layer line number J, is a multiple of u, so that u can be determined straightforwardly. The diffracted intensities can be calculated using standard expressions (2), for model structures (i.e. given the atomic coordinates). 

Extinction An effect of dynamical diffraction whereby the incident beam is weakened as it passes through the crystal. If the crystal is perfect there may be multiple reflection of the incident beam, which then is out of phase with the main beam and therefore reduces its intensity (primary extinction). If the crystal is mosaic, one block may diffract the beam, and is not then available to a second block similarly aligned (secondary extinction). Both effects result in a diminution of the intensities so that the most intense Bragg reflections are systematically smaller than those calculated from the crystal structure. The effect can be reduced by dipping the crystal in liquid nitrogen, thereby increaising its mosaicity. 

A general practice is to classify those Bragg reflections that have measured intensities less than an arbitrary multiple (usually two or three) of their estimated standard deviations as unobserved reflections. The term unobserved " is an unfortunate one since it also includes some weak reflections whose intensities have been measured. Reflections that are classified as weak or unobserved cannot simply be discarded from the data set they are needed for statistical analysis purposes and may contain relevant information about the structure. ... 

Multiple reflections, described in Chapter 3, can cause enhancement in the intensities of some reflections and diminution in intensities of others. They are most clearly seen when reflections expected to be systematically absent have weak intensity. Generally only a few Bragg reflections are significantly affected, but there are various methods that can be used... 

Measurement of multiple Bragg diffraction. Some sets of phases may be derived from experimental measurements of i/ -scan profiles of reflections exhibiting the effects of multiple Bragg diffraction. Groups of relative phases have absolute values that can be detected by this method which is currently in the developmental stage, and is only applicable to those sets of Bragg reflections that display the effect. 

FIGURE 8.28. Multiple Bragg reflections. Shown are four idealized V scans (in degrees), and the values of Osum derived from the peak profile, (a) -90°, (b) 0°, (c) -(-90°, and (d) 180°. Values intermediate between these relative phase angles these can be measured for noncentrosymmetric crystals. 

Multiple Bragg reflections Needs very special and precise equipment. In developmental stages. 

Multiple Bragg diffraction Further diffraction of a Bragg reflection by a second set of lattice planes. This occurs when two reciprocal lattice planes lie simultaneously on the surface of the Ewald sphere. It affects the intensity of the Brage reflection and a detailed analysis of the effect can lead to some phase information. 

As we established earlier, a powder diffraction pattern is one-dimensional but the associated reciprocal lattice is three-dimensional. This translates into scattering from multiple reciprocal lattice vectors at identical Bragg angles. Consider two points in a reciprocal lattice, 00/ and 00/. By examining Eqs. 2.29 to 2.34 it is easy to see that in any crystal system l/c/ (00/) = l/c/ (00/). Thus, Bragg reflections from these two reciprocal lattice points will be observed at exactly the same Bragg angle. 

In the second approach, the total intensity of the diffraction peak is equally divided among the individual reflections, so that /total = Yet another approach in a blind division is to account for the multiplicity factors of different Bragg reflections, so that /total = where w,- is the multiplicity factor of the z reflection, which depends on symmetry and combination of indices (see sections 2.10.3 and 2.12.2). No obvious preference can be given to any method of intensity division, as each of them is quite arbitrary. This way of handling the overlapped intensities, instead of simply discarding them is most beneficial in the Patterson method. 

The division of intensities of the overlapped Bragg reflections is only critical when they are needed to calculate Patterson-, Fourier- or E-map(s). There is no need in their separation during a least squares refinement of structural parameters because each point of the diffraction profile is simply taken as a sum of contributions from multiple Bragg reflections. ... 

An automatic search-and-match can be done much faster, and most importantly, using multiple Bragg reflections by seeking through enormous arrays of data, which a typical database contains. The algorithms employed to conduct automatic searches vary extensively, however, parameters that are critical in any search include the following ... 

Figure 4.23. The results of a qualitative analysis of a multiple phase sample. Three crystalline phases are clearly identifiable lithium silicate - Li2Si03, silicon oxide - SiOj (quartz), and a different pol)imorph of silicon oxide - tridymite. A low quality diffraction pattern collected during a fast experiment was employed in this example. The data shown on top were smoothed, the background was subtracted, and the Ktt2 components were stripped before the digitized pattern (shown below the smoothed profile) was obtained using an automatic peak search. Note, that many weak Bragg reflections were missed in the peak search,...

R3c, and many others, which contain multiple glide planes and/or screw axes, distinguishable from the list of unobserved (extinct or forbidden) Bragg reflections. 

Due to the complexity of the pattern, multiple overlaps (e.g. about 90 Bragg reflections are possible in the range of the first 40 observed peaks below 20 = 40°) and the relatively broad peaks, the pattern decomposition was carried out using a semi-manual profile fitting. For each group of Bragg reflections, located within the manually selected ranges of the powder... 

Considering Eq. 2.48 and taking into account the one-dimensionality of powder diffraction data, which introduces multiple Bragg reflection overlaps, the expanded form of Eq. 7.2 in the simplest case, i.e. when the powder is a single phase crystalline material, becomes... 

Ann is the nearest-neighbor distance in the crystal. Because of the ABC stacking along the surface normal direction, the unit cell contains three monolayers and the Bragg reflections are spaced apart by multiples of three in 1. The basic scattering equation for an unrelaxed, unmodified (111) surface along the CTRs shown in Fig. 2(a) is then given by... 

For certain applications the polarization of the x radiation is important. As follows from theory, synchrotron radiation is perfectly linearly polarized in the plane of the electron orbit and elliptically polarized outside the plane (Figure 4). However, in practice one has to take into account the finite size and position stability of the radiation source as well as the polarizationchanging properties of monochromators. Therefore it is desirable to determine the actual polarization experimentally. The linear polarization can be measured by diffraction methods, such as, e.g., by Bragg reflection at 2d = 90° or by observing the high Laue transmission for the polarization component with electric vector parallel to the crystal lattice (Borrmann effect). By employing multiple reflection arrangements polarization ratios can be determined even at the level of A simple and fast method... 

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