Non-Bragg diffraction orders

Figure 6. Photograph of the transmitted beams and the non-Bragg diffraction orders. Beam (-0) is collimated, beam (+0) is diverging, beam (-1) is converging demonstrating the phase-conjugate relation to beam (+0), and beam (+1) is diverging two times faster than beam (+0), demonstrating the phase-doubling property.

Water on Smectites. Compared to vermiculites, smectites present a more difficult experimental system because of the lack of stacking order of the layers. For these materials, the traditional technique of X-ray diffraction, either using the Bragg or non-Bragg intensities, is of little use. Spectroscopic techniques, especially nuclear magnetic resonance and infrared, as well as neutron and X-ray scattering have provided detailed information about the position of the water molecules, the dynamics of the water molecule motions, and the coordination about the interlayer cations. 

Figure 5. Electric field dependence of the diffraction efficiency of the non-Bragg orders. The field polarity convention is illustrated in Figure 2.

A similar though considerably less well-ordered lamellar product is obtained when ethanol or methanol are used instead of water under solvothermal conditions Figure lb shows the powder XRD diagram of a sample prepared in ethanol at 90 °C dm = 3.50 nm). Considerably different products, however, are obtained when alcohols are used at lower temperatures, i.e. under non-solvothermal conditions. Figures lc and Id show the diffraction patterns of two example products from syntheses in methanol at 25 °C and in ethanol at 10 °C, respectively. In both cases the XRD reflections can be attributed to two distinct phases. One of these has a hexagonal symmetry with a dm value of 1.88 nm this mesophase will be discussed in detail below. An additional broad reflection is found at a Bragg angle comparable to that of the 001... 

The scattered intensity is usually represented as the total number of the accumulated counts, counting rate (counts per second - cps) or in arbitrary units. Regardless of which units are chosen to plot the intensity, the patterns are visually identical because the intensity scale remains linear and because the intensity measurements are normally relative, not absolute. In rare instances, the intensity is plotted as a common or a natural logarithm, or a square root of the total number of the accumulated counts in order to better visualize both strong and weak Bragg peaks on the same plot. The use of these two non-liner intensity scales, however, always increases the visibility of the noise (i.e. highlights the presence of statistical counting errors). A few examples of the non-conventional representation of powder diffraction patterns are found in the next section. 

A brief description of the anharmonic approximation is included here for completeness since rarely, if ever, it is possible to obtain reasonable atomic displacement parameters of this complexity from powder diffraction data the total number of atomic displacement parameters of an atom in the fourth order anharmonic approximation may reach 31 (6 anisotropic + 10 third order +15 fourth order). The major culprits preventing their determination in powder diffraction are uncertainty of the description of Bragg peak shapes, non-ideal models to account for the presence of preferred orientation, and the inadequacy of accounting for porosity. 

An ideal situation for the observation of twin structures is an array of parallel, non-periodically spaced walls. The diffraction signal has then a dog-bone structure as shown in Figure 2. The equi-intensity surface shows diffraction signals at an intensity of lO " compared with the intensity of the two Bragg peaks which are located inside the two thicker ends of the dog bone (Locherer et al. 1998). The intensity in the handle of the dog bone is then directly related to the thickness and internal structure of the twin wall. In order to calibrate the wall thickness the following theoretical framework is chosen. The specific energy of a wall follows in lowest order theory from a Landau potential... 

In any situation where there is substitutional disorder of any kind in a crystal the diffraction pattern consists of two parts the diffraction pattern of the averaged structure, which is confined to the Bragg reflections and whose analysis in terms of occupancy I have discussed and superimposed on this the diffraction pattern of the difference structure i.e. a distribution of electron density everywhere equal to the difference between the average and the actual structure. Even in the case of complete disorder this will be non-zero, but its diffraction pattern will be very weak everywhere because it will be diffused over all diffraction directions. Only if there is some kind of ordering on subsets of the crystallographic sites, which subsets define a superlattice of the crystal lattice, will the intensity of the difference pattern concentrate into definite directions and be easily detectable and interpretable. However if such a superlattice regularity is confined to 1 or 2 dimensions or to very small 3-dimensional domains then the difference diffraction will be smeared out. It may then be difficult to detect, and even when detected it can be difficult to interpret unambiguously. 

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