Single diffraction

Symmetrical tetrasubstituted Pcs recently characterized by X-ray single diffraction techniques are mainly the 1,8,15,22-tetrasubstituted Pcs, which arise from... 

One example is the square-planar tetrachloro- and tetrabromo-complexes formed by gold(III) in aqueous solution, which can be uniquely and precisely structure determined from a single diffraction curve (18). The heavy atoms in the complexes and the highly concentrated solutions, that can be prepared, result in dominant contributions from the complexes to the scattering curve. 

A summary of results reported in the literature is given in Table III. Most of these have been derived from analyses of single diffraction curves and only a limited number are based on diffraction data for solutions of different metal ion concentration and halide to metal ratios, which are needed in order to determine structures of individual complexes. The values obtained are, therefore, usually averages over the different complexes, that may be present, and give only limited information on coordination geometry. 

Since the metal-sulfur distance in an inner-sphere sulfate complex appears in a region where a large number of other distances in the solution will also occur, a unique identification and structure determination of sulfate complexes is difficult to make unless difference methods can be used. Definite conclusions cannot generally be made on the basis of an analysis of a single diffraction curve only. Data for several solutions of different compositions are needed and a careful analysis... 

Small-angle X-ray scattering patterns exhibit a weak single diffraction peak at ( =0.68 and 0.66nm" for SBAla and SBAlb respectively, characteristic of the wormlike stracture (Fig. 1) with a regular pore-pore distance [15], The corresponding d value was calculated to be 9.2 and 9.5 mn respectively. 

In the imaging modes used most commonly for imaging crystal defects (such as dislocations and stacking faults), the image is formed using only the transmitted beam or a single diffracted beam. The way in which defects are revealed in these images is discussed qualitatively in the first part of Chapter 5. This is followed by an explanation of how the mathematical description of the distortion around a defect in a crystal is incorporated... 

Finally, we consider the nature of the image formed with a single diffracted beam, say the first-order beam at ii = +no- Under these conditions, the first two terms of Eq. (1.31a) are eliminated and... 

In x-ray diffraction, primary extinction is rarely encountered with powder specimens because the individual crystals are very small. However, primary extinction can become pronounced for single crystals of the order of 1 mm thick. For such crystals, the diffracted intensity is critically dependent on the crystal perfection. In a distorted crystal, the singly diffracted wave Si may not be at the exact Bragg angle at A. This wave may, therefore, pass through the crystal without having its intensity significantly reduced by interference with a transmitted wave T. 

Double reflection X rays diffracted by one set of crystal lattice planes may have sufficient intensity to be diffracted again by a second set of planes that are, by chance or design, in exactly the correct orientation. The twice-diffracted beam emerges in a direction that corresponds to a third set of planes with Miller indices equal to the sum of the indices of the two planes causing the double reflection. This double reflection, travelling in the same direction as a singly diffracted beam, will enhance or weaken the intensity of the latter. 

In summary then, a crystal can be conceived of as an electron density wave in three-dimensional space, which can be resolved into a spectrum of components. The spectral components of the crystal correspond to families of planes having integral, Miller indexes, and these can, as we will see, give rise to diffracted rays. The atoms in the unit cell don t really lie on the planes, but we can adjust for that when we calculate the intensity and phase with which each family of planes scatter X rays. The diffracted ray from a single family of planes (which produces a single diffraction spot on a detector) is the Fourier transform of that family of planes. The set of all diffracted rays scattered by all of the possible families of planes having integral Miller indexes is the Fourier transform of the crystal. Thus the diffraction pattern of a crystal is its Fourier transform, and it is composed of the individual Fourier transforms of each of the families of planes that sample the unit cells. 

However, this simple explanation of Talbot images still requires plane waves corresponding to a highly collimated and therefore weak input beam. The full intensity gain of the Talbot effect is only deployed when it is applied to uncollimated and therefore much more intense molecular beams [Clauser 1994 Brezger 2002], This is realized if the single diffraction grating is re-... 

The chief problem presented by a fiber texture is the identification of the fiber axis uvw This can be done fairly easily with a single diffraction photograph, and the procedure is described in this section. If, in addition, we wish to determine the amount of scatter in the texture, a diffractometer method is preferable (Sec. 9-9). 

The application of DM to powder data requires the previous application of a full pattern decomposition procedure (see Chapter 5) in the following we will suppose that single diffraction intensities are available for each reflection in the measured 26 range. Owing to the peak overlap the estimates of the diffraction moduli will be affected by unavoidable errors this weakens the efficiency of DM (naively, wrong moduli will produce wrong phases), and still today makes crystal structure solution from powder data a challenge. 

Generally speaking. X-ray beams produced with a tube are polychromatic, in other words, they contain characteristic emission peaks in addition to a continuous spectrum over a wide range of wavelengths. In order to characterize the material to be studied, it is important to have a monochromatic beam which makes it possible to associate a single diffraction peak with each family of crystal planes. Therefore, it is necessary to select one peak among all of those emitted by the tube. Naturally, the most intense one is chosen. 

In the diffraction patterns in Figure 4.23, the platinum-iridium bimetallic clusters exhibit a single diffraction line (middle field of figure) about midway between the lines (upper field of figure) for bulk platinum and bulk iridium. The line for the clusters is broader than the lines for the bulk metals because... 

In studies of the platinum-iridium system in the bulk, Raub and Platte (47) have reported a miscibility gap at temperatures lower than about 975°C. At 500°C the gap extends over the composition range from 7 to 99% iridium. Nevertheless, the results presented in Figure 4.23 indicate that it is possible to prepare platinum-iridium catalysts for which X-ray diffraction patterns do not reveal separate lines for platinum-rich and iridium-rich phases, despite the fact that the catalysts have been heated to only 500°C in their preparation. Instead, single diffraction lines are observed. 

Room temperature measurements by Kjaer et al. of arachidic acid on water at Ti = 27mNm" also show a single diffraction peak. It lies at... 

Of course, if only one single diffraction maximum is observed, a strict assignment of a certain lattice type cannot be made. The assignment of triclinic, orthorhombic, monoclinic or hexagonal lattices, as they are frequently found in lipids, requires the indexing of several reflections, which in practice, however, are often rather obscure. The analysis can be aided considerably by the comparison to the diffraction from single crystals... 

A powdered crystaUine material contains many thousands of tiny crystals. These crystals are oriented in all possible directions relative to the beam of X-rays. Hence, instead of the sample generating only single diffraction spots, it generates cones of diffracted X-rays, with the point of all of the cones at the sample (Fig. 8.42). Each family of planes will have a different circular diameter, so the result is a series of concentric... 

---