Two-dimensional diffraction

This approach may also be applied to racemic bilayers built up from homo-chiral Langmuir-Blodgett monolayers. By measuring the two-dimensional diffraction pattern from such a bilayer it is possible to deduce the molecular chirality of each of the two monolayers in the order they were inserted to construct the bilayer. This approach can be extended to multilayers. Thus, in principle, we close the circle started in Section IV-G-1. It is possible to assign the absolute configuration of chiral molecules in centrosymmetric crystals provided that one can construct the crystal (in this case the multilayer) by adding homochiral layers one by one. 

The indexing proeedure of any diffraction pattern requires the knowledge of positions of some reflections on the pattern. The relations between the position of each peak on a two-dimensional diffraction pattern and the corresponding point in reciprocal space can be established only after a successful indexing. The combined use of the reciprocal coordinates and the determined peak-shape are essential for extracting integrated intensities from diffraction data. 

For a two-dimensional array of equally spaced holes the diffraction pattern is a two-dimensional array of spots. The intensity between the spots is very small. The crystal is a three-dimensional lattice of unit cells. The third dimension of the x-ray diffraction pattern is obtained by rotating the crystal about some direction different from the incident beam. For each small angle of rotation, a two-dimensional diffraction pattern is obtained. 

Figure 2.30. The two-dimensional diffraction patterns from stationary (left) and rotating (right) single crystals recorded using a CCD detector. The incident wavevector is perpendicular to the plane of the figure. The dash-dotted line on the right shows the rotation axis, which is collinear with c. ...

Figure 2.35. The powder diffraction pattern of the polycrystalline LaB as intensity versus 20 obtained by the integration of the rectangular area from the two-dimensional diffraction pattern shown in Figure 2.34.

As mentioned previously, a crystal will diffract x-rays with an intensity proportional to the square of the structure factor and is described by Eq. (33). The abrupt termination of the lattice at a sharp boundary (i.e., a surface) causes two-dimensional diffraction features termed crystal truncation rods (CTRs). Measurements of CTRs can provide a wealth of information on surface roughness and may be useful in the determination of crystallographic phase information. ... 

For example, the diffraction integral (3.25) containing the potential function of a cusp catastrophe (zf3), F(x c) = x4 + ax2 + bx, describes ligth scattering on a two-dimensional diffraction grating, see Fig. 48. A function defined by such an integral is called the Pearcey function. 

Forty two-dimensional diffraction patterns can be recorded at a maximum rate of 3.3 exposures per second. From Amemiya et al (1989) with permission. 

Figure 2. Two-dimensional diffraction data from 50 50 w/w PVC/PS blend particles produced from an 8 pm diameter droplet (left), and a 35 pm diameter droplet, (right).

We have recently shown that the presence of phase-separated structures in heterogeneous polymer-blend microparticles can be indicated qualitatively by a distortion in the two-dimensional diffraction pattern. The origin of fringe distortion from a multi-... 

Fig. 4.12 Two-dimensional diffraction patterns of an aligned sample of the thermotropic liquid crystalline material 5-(octyloxy)-2-(4-(octyloxy)phenyl)pyrlmidine a at 100 °C in the SmA phase and b at 50 °C in the SmC phase [23]...

WAXS measurements are based on the ratio of the intensity of the crystalline peaks and the amorphous background. Fitting the amorphous background is best done with the help of a computer. Various corrections can be applied to obtain more accurate absolute crystallinities. However, most frequently we are interested in approximate absolute but accurate relative crystallinities and the corrections are not always necessary. The peak width can give information on crystal perfection - sharper peaks indicate more perfect crystals. Two-dimensional diffraction patterns of isotropic materials show rings corresponding to the diffraction peaks. Intensity variations within the rings can be used to assess crystalline orientation. WAXS only looks at material which can be penetrated by X-rays and so the depth of the analysis is limited. The patterns for filled systems can be hard to interpret but various methods have been developed - for example for use in continuous carbon-fibre composites. 

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