Diffraction curve

We can also visualise the subsurface as being made up of an enormous number of point scatterers or diffractors. (Fig b). Each contributes a diffraction curve (hyperbola) to the reflection section. Migration focuses the energy in these curves to a single point. 

The former way is not useful when X-ray diffraction is used, because the difference between the scattering curves of Si and A1 is too small. If neutron diffraction is used, the neutron diffraction curves differ by as much as 25 %, so that the Si/Al ratio can be satisfactorily refined. Unfortunately, large crystals are needed at present (nearly 1mm in volume), therefore this method can only be applied to a restricted number of zeolites. 

Fig. 7. X-ray diffraction curves for (Ga,Mn)As films obtained with Cu Kor radiation, (a) Mn concentration dependence of peak positions [(004) reflection] of ISO-nm thick (Ga,Mn)As grown on GaAs with compressive strain (Ohno et al. 1996a). (b) (Ga,Mn)As grown on (In.Ga)As buffer layer with tensile strain, (c) Double-crystal x-ray diffraction curves for a 2 /rm-thick (Ga,Mn)As showing the asymmetric (224) reflection with high- and...

According to an x-ray diffraction curve, a slight amount of crystallinity was present... 

The information given by an X-ray diffraction curve for a solution is one dimensional only. By a Fourier inversion it can be transformed into... 

A typical diffraction curve is shown in Fig. 2. It gives the measured intensity from a solution as a function of the scattering vector... 

The structure-dependent part of the diffraction curve, the reduced intensity function, t(s), is obtained by subtracting the independent coherent scattering from the normalized experimental values (Fig. 2) ... 

Two atoms, p and q, bonded together at a distance rpq give a contribution, ipq(s), to the diffraction curve, which can be calculated according to the Debye expression... 

Two straightforward structure derivations can be used to illustrate the analysis of a diffraction curve and the structural information that can be extracted when conditions are favorable. 

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. 

The other example is the chloro complexes of platinum(II) and palla-dium(II), which also have square-planar structures (13). The PtCl42-and the PdCl42 complexes have identical structures, with the same bond lengths, and form isostructural solutions. By using the difference between their normalized diffraction curves contributions from interactions not involving the metal atoms can be eliminated and precise structures can be derived, not only for the complex but also for the structure beyond the first coordination sphere of the metal ion. 

The PtCl42- and PdCl42 complexes have the same square-planar structure as the gold(III) halide complexes and concentrated solutions of each of them can be prepared. In crystal structures the Pt—Cl and the Pd—Cl bond lengths in these complexes do not differ significantly and their solution diffraction curves show this to be true also in solution. They can, therefore, be expected to form isostructural solutions and the difference between the normalized diffraction curves for a platinate(II)... 

In the presence of a complex-forming anion the number of interactions is increased, and in order to get information beyond a simple determination of average bond lengths, this method cannot be used. Comparisons between diffraction curves for solutions of different compositions are then needed in order to separate and analyze specific interactions. 

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... 

The internal structure of a polyatomic ligand is obtained as an additional result in these investigations. Its high concentration and the sharp intramolecular distances often result in dominant contributions to the diffraction curves and make possible a precise determination of its bonding distances. When careful analyses have been made, no significant differences from values found in crystals have been found, however. Therefore, the derived structure for the ligand can serve as an internal check on the quality of the data. Large deviations from values found in crystal structures may be an indication of errors in the data or in the method of analysis. 

The stability constants indicate that solutions can be prepared in each of which one of the complexes is solely dominant (Fig. 29). Diffraction curves for two such solutions, and for an acid solution containing no hydrolysis complexes, give RDFs as shown in Fig. 30 (217). The Pb-Pb intramolecular interactions are very distinct and are clearly seen in the two hydrolyzed solutions. For the Pb4 (OH)44 + solution the single... 

The combination of information from different sources thus makes it possible to determine the structure of the complexes in solution even when the information contained in the solution diffraction curves is too limited for a complete and unambiguous structure determination. 

Although the metal-oxygen distances in the hydrolysis complexes are usually too weak and too irregular to give distinct features to a diffraction curve they can be observed and can sometimes be used to choose between different conceivable models for the structure. In hydrolyzed bismuth(III) solutions a dominant complex containing six Bi atoms has been shown to occur (201-204, 226,228). The octahedral arrangement of the six Bi atoms can easily be proved from the diffrac-... 

X-ray diffraction measurements can give direct structural information on complexes in solution, which cannot be obtained by other methods. Although the complete structure of a solution cannot be derived from the limited information in a single one-dimensional diffraction curve, a separation of the distinct contributions given by intramolecular interactions can often be made. This makes it possible to determine structures of complexes even in solutions with chemically complicated compositions. Comparison of diffraction curves for different solutions can give information on structural changes caused by changes in concentration of a particular atomic species. 

TABLE 2 FWHM of X-ray diffraction curve versus GaN thickness on sapphire with GaN buffer layer [19],... 

Drits VA, Sakharov BA, Manceau A (1993) Stmctnre of feroxyhite as determined by simulation of X-ray diffraction curves. Clay Minerals 209-222... 

Figure 9. WAXS diffraction curves for the a) M-B-23/25-48 series and b) M-E-23/25-48 series at different levels of catalyst (DHTDL).

Figure 2. Far-field diffraction curve recorded with the full thermal distribution of the Cm beam [Nairz 2003]. The velocity is centered around v = 200 m/s and has a FWHM of Av/v = 0.6.

The width of the diffraction curve of Fig. 3-15(a) increases as the thickness of the crystal decreases, because the angular range (20, — 202) increases as m... 

Fig. 3-15 Effect of fine particle size on diffraction curves (schematic).

Figure 8-36(b) is also a rocking curve, for a calcite crystal reflecting in transmission. The surface of this crystal had been roughened by grinding it on emery paper, and the half-width of the diffraction curve is about 2 minutes (0.03°). A similar crystal that had been etched rather than ground showed a width of only 0.2 minutes (0.003°), very near the perfect-crystal value [8.25]. 

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