Scattering domains

An average crystal size can also be obtained from XRD line broadening using the Scherrer formula vhich describes the corrected width of an XRD line at an angle 9 as a function of the mean size of the coherently scattering domain perpendicular to the hkl plane MCLhki... 

Plots of In I as a function of sin2 (0/2) were made by a computer for all of the films studied. One such plot for an SA-modified PMMA is shown in Figure 3. The initial and tail portions have different slopes. The mean sizes ax and a2 of the scattering domains were calculated from these slopes and are given in Table I. 

Pinnavaia etal. used neutral alkylamines as templates to form disordered mesoporous silica, named hexagonal molecular sieves (HMS). The S°I° formation mechanism was proposed between neutral amine micelles (S°) and neutral inorganic precursors (1°). The interactions between S° and 1° were assumed to be hydrogen bonding. The resulting HMS has a worm-like pore structure, with thicker framework walls and smaller X-ray scattering domain sizes compared to M41S. 

Mesoporous silicate molecular sieves were characterized by an arsenal of techniques including XRD, SEM, TEM, adsorption measurements, Si NMR, IR, Raman and XANES. XRD patterns of all nanoporous phases are dominated by low angle peaks. Figure 11 shows typical patterns for MCM-41, MCM-48, MCM-50 and SBA-2 phases. HMS, MSU-n and some "MCM-41" exhibit only the 100 peak either because of too small scattering domain sizes [69] or because of poorly ordered pore system [55,72,80]. 

The XRD patterns of Ti containing samples consist of a main peak at 20 < 3° corresponding to the 100 diffraction, sometimes accompanied by much weaker 110, 200 and 210 reflections in the 20 range of 4 to T. HMS based catalysts exhibit only the 100 diffraction peak because of excessive broadening of the hkO reflections due to too small scattering domain sizes [69,71,139,142] or more likely to the presence of a poorly ordered pore system [55]. Indeed, TEM studies indicated that the pore structure of Ti-HMS is much less ordered than that of Ti-MCM-41 [147]. In addition, SEM [143] and TEM [147] show that Ti-HMS is comprised of spherical particles with 0.2-0.3 pm in diameter. N2 adsorption isotherms obtained by Pinnavaia et al. [71,139,142] showed that in addition to the framework-confined mesoporosity due to the presence of parallel channels, HMS materials display a well developed textural mesoporosity. However, other workers found that the N2 adsorption-desorption isotherm of Ti-HMS is reversible [143,147]. The pore distribution was broader for HMS as compared to MCM-41 materials. The Ti-MSU-1 material also exhibited only the 100 diffraction peak due to the occurrence of disordered, hexagonal-like packing channels [72]. These samples displayed reversible N2 adsorption-desorption isotherms with no hysteresis. 

The corrected FWHHs can be used to ftirther characterize the crystal size (more precisely the size of the mean coherently scattering domain, MCD) in the various crystallographic directions if this is smaller than ca.lOO nm. For example, small hematite crystals are often less developed in the [001]... 

Strictly speaking, the term crystallite size here refers to the dimension of a coherently scattering domain. Only in a perfect crystal, is this the grain size. 

If one considers each crystalline domain as made of columns of unit cells (actually the unit cells are projected along the specific [hkl direction considered), then the scattering domain can be described by a column length distribution p L), which gives the fraction of columns of length L. Continuing the analysis introduced by Bertaut it can be demonstrated that p L) is proportional to the second derivative of... 

Information about both the size and the geometry of the scattering particles can be obtained from different scattering domains. In the Guinier range with QR < < 1 the scattering function can be written... 

Lanson B, (Champion D (1991) The I/S-to-illite reaction in the late stage diagenesis. Am J Sci 291 473-596 Lanson B, Kibler B (1994) Experimental determinations of the coherent scattering domain size distribution of natural mica-like phases with the Warren-Averbach technique. Clays Clay Minerals 42 489-494 Li G, Peacor DR, Merriman RJ, Roberts B (1994) The diagenetic to low grade metamorphic evolution of matrix white micas in the system muscovite-paragonite in a mudrock from Central Wales, U.K. Clays Clay Minerals 42 369-381... 

Method Discrimination based on Scattering domain size... 

X-ray diffraction analysis is used routinely by every catalyst manufacturer to determine the phase composition of the catalysts produced and the size of coherently scattering domains, and hardly needs a detailed description. An aspect that we would like to emphasize concerns the influence of the enviromnent on the oxidation state of carbon-supported metal nanoparticles. Quite often, authors try to correlate electrochemical performance with the phase composition of as-prepared samples. It has, however, been demonstrated convincingly in a number of publications by both x-ray diffraction [155] and x-ray absorption spectroscopy [156] that as-prepared fuel cell catalysts and samples stored under ambient conditions are often in the form of metal oxides but are reduced under the conditions of PEMFC or DMFC operation. The most dramatic changes are observed for samples with high metal dispersions, while larger particles are affected only marginally [17]. One should keep in mind, however, that the extent of the particle oxidation depends critically on the preparation procedure. 

FWHMcorr is corrected for the instrumental resolution and K varies with Ad/d and the FWHM. Setting the Scherrer constant K to unity one then plots FWHMcorr cost against sin0 as abscissa. One can then extract the scattering domain radius A from the intercept with the ordinate and Ad/d from the slope (see Figure 4). 

Method Origin of contrast Scattering domain size (pm)... 

TABLE 11.1 Characteristics of the phases present, corresponding lattice parameters and the coherently scattered domain size of the PVA stabilized Ag-Cu alloy NPs... 

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