Diffraction patterns combined

Because the observed diffraction pattern is a product of the diffraction patterns from the two distributions, what is observed at each nonzero point in the combined transform, or diffraction pattern determined by the periodic point lattice in Figure 5.8c, is the value of the Fourier transform at that point from the continuous distribution in Figure 5.8a. That is, the combined diffraction pattern samples the continuous Fourier transform of the object making up the array, that of Figure 5.8b, but only at those discrete points permitted by the array s periodic, discrete transform seen in Figure 5.8d. 

Figure 8.3 (a) The layered structure of aromatic acid salts. The cleavage plane exposes the apolar side of the molecules. The polar parts, with the acid moieties and M cations, are bnried. The cleavage plane properties (e.g., hydrophobicity) and unit-cell parameters can be modulated to some extent by the nature (f-butyl, chlorine, etc.) and also by the position (ortho, meta, or para) of the substituent X. Reproduced from Reference [11] with permission of Pergamon Press, (b) Combined diffraction pattern of poly(ethylene terephthalate) and the sodium salt of p-chlorobenzoic acid. The salt reflections are sharp spots the PET reflections are more arced. The contact plane of PET is (100) the chain axis is vertical. 

Gelatin stmctures have been studied with the aid of an electron microscope (23). The stmcture of the gel is a combination of fine and coarse interchain networks the ratio depends on the temperature during the polymer-polymer and polymer-solvent interaction lea ding to bond formation. The rigidity of the gel is approximately proportional to the square of the gelatin concentration. Crystallites, indicated by x-ray diffraction pattern, are beUeved to be at the junctions of the polypeptide chains (24). 

Figure 1 shows the decomposition sequence for several hydrous precursors and indicates approximate temperatures at which the activated forms occur (1). As activation temperature is increased, the crystal stmctures become more ordered as can be seen by the x-ray diffraction patterns of Figure 2 (2). The similarity of these patterns combined with subtie effects of precursor crystal size, trace impurities, and details of sample preparation have led to some confusion in the Hterature (3). The crystal stmctures of the activated aluminas have, however, been well-documented by x-ray diffraction (4) and by nmr techniques (5).

The physical stmcture of mixed-layer minerals is open to question. In the traditional view, the MacEwan crystallite is a combination of 1.0 nm (10 E) non-expandable units (iUite) that forms as an epitaxial growth on 1.7 nm expandable units (smectite) that yield a coherent diffraction pattern (37). This view is challenged by the fundamental particle hypothesis which is based on the existence of fundamental particles of different thickness (160—162). 

Fraunhofer rules do not include the influence of refraction, reflection, polarization and other optical effects. Early Iziser particle analyzers used Fraunhofer approximations because the computers of that time could not handle the storage cuid memory requirements of the Mie method. For example, it has been found that the Fraunhofer-based instrumentation cannot be used to measure the particle size of a suspension of lactose (R.I. = 1.533) in iso-octane (R.I. = 1.391) because the relative refractive index is 1.10, i.e.- 1.533/1.391. This is due to the fact that diffraction of light passing through the particles is nearly the same as that passing around the particles, creating a combined interference pattern which is not indicative of the true... 

Combined analyses by XRD and TEM showed that the aurichalcite mineral was sufficiently similar to the synthetic aurichalcite to be used as a model compound, to study the microstructural changes occurring during the catalyst preparation procedures. Calcination of the mineral and synthetic samples led to highly preferred orientations of ZnO. ZnO electron diffraction patterns with [lOlO] and [3031] zone... 

Transmission electron microscopy (TEM) resembles optical microscopy, except that electromagnetic instead of optical lenses are used to focus an electron beam on the sample. Two modes are available in TEM, a bright-freld mode where the intensity of the transmitted beam provides a two-dimensional image of the density or thickness of the sample, and a dark-field mode where the electron diffraction pattern is recorded. A combination of topographic and crystallographic information, including particle size distributions, can be obtained in this way [32],... 

Structure refinement based on kinematical scattering was already applied by the Russian scientist 60 years ago. Weirich et al. (1996) first solved the structure of an unknown TinSe4by HREM combined with crystallographic image processing. Then they used intensities extracted from selected area electron diffraction patterns of a very thin crystal and refined the structure to a precision of 0.02 A for all the atoms. Wagner and Terasaki et al. (1999) determined the 3D structure of a new zeolite from selected area electron diffraction, based on kinematical approach. 

We have reconstructed the 3D structure of a complex quasicrystal approximant v-AlCrFe (P6 m, a = 40.687 and c = 12.546 A) (Zou et al, 2004). Due to the huge unit cell, it was necessary to combine crystallographic data from 13 projections to resolve the atoms. Electron microscopy images containing both amplitude and phase information were combined with amplitudes from electron diffraction patterns. 124 of the 129 unique atoms (1176 in the unit cell) were found in the remarkably clean calculated potential maps. This investigation demonstrates that inorganic crystals of any complexity can be solved by electron crystallography. 

For symmetry determinations, the choice of the pertinent technique among the available techniques greatly depends on the inferred crystallographic feature. A diffraction pattern is a 2D finite figure. Therefore, the symmetry elements displayed on such a pattern are the mirrors m, the 2, 3, 4 and 6 fold rotation axes and the combinations of these symmetry elements. The notations given here are those of the International Tables for Crystallography [1]. 

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. 

The data base of some 27,000 powder diffraction patterns that is used in the CIS (5) is in fact a direct descendant of that with which Hanawalt carried out his pioneering work. A problem that arises in connection with this particular component stems from the fact that powders, as opposed to crystals, are frequently impure and so the patterns that are obtained experimentally are often combinations of one or more file entries. A reverse searching program, that examines the experimental data to see if each entry from the file is contained in it, has been written after the general approach of Abramson (23), and seems to cope with this particular difficulty. It is currently running in test on the NIH PDP-10 and will be made available to the scientific community during the latter part of 1978. 

The phase problem of X-ray crystallography may be defined as the problem of determining the phases ( ) of the normalized structure factors E when only the magnitudes E are given. Since there are many more reflections in a diffraction pattern than there are independent atoms in the corresponding crystal, the phase problem is overdetermined, and the existence of relationships among the measured magnitudes is implied. Direct methods (Hauptman and Karle, 1953) are ab initio probabilistic methods that seek to exploit these relationships, and the techniques of probability theory have identified the linear combinations of three phases whose Miller indices sum to... 

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