Line-broadening analysis, metal particle size

For supported metal catalysts, no simple calculation is possible. A direct measurement of the metal crystallite size or a titration of surface metal atoms is required (see Example 1.3.1). TWo common methods to estimate the size of supported crystallites are transmission electron microscopy and X-ray diffraction line broadening analysis. Transmission electron microscopy is excellent for imaging the crystallites, as illustrated in Figure 5.1.5. However, depending on the contrast difference with the support, very small crystallites may not be detected. X-ray diffraction is usually ineffective for estimating the size of very small particles, smaller than about 2 nm. Perhaps the most common method for measuring the number density of exposed metal atoms is selective chemisorption of a probe molecule like H2, CO, or O2. 

There are two very general X-ray techniques for study of the metal particle size distribution the line-broadening analysis (LBA) and the small angle X-ray scattering (SAXS) (lb, 170). Other methods include the radial electron distribution (RED) and the extended X-ray absorption fine structure (EXAFS), which are aimed primarily at studying the structure of catalysts (Section IV,G). 

Further problems can arise because of uncertainties concerning the stoichiometry of the adsorption reaction. For most metals it is assumed that the surface stoichiometry with H2 is H/M = 1. However, there is evidence especially for very small metal particles (of the order of 1 -5 nm) that the stoichiometry can exceed H/M = 1. For quantitative measurements of surface area it is necessary to establish the chemisorption stoichiometry and structure. In practice it is usually possible to achieve approximate estimate of the surface area by some other independent method (for example, from particle size analysis by X-ray line broadening or by TEM). In the case of CO, the CO/M ratio is generally taken as 1.0, but the true value may depend on the particle size and on the particle morphology. With N2O the N2O/M ratio at monolayer coverage is usually assumed to be 0.5, but once again there is no certainty about the validity of this particular assumption. 

In 1949, however, Warren pointed out that there was important information about the state of a cold-worked metal in the shape of its diffraction lines, and that to base conclusions only on line width was to use only part of the experimental evidence. If the observed line profiles, corrected for instrumental broadening, are expressed as Fourier series, then an analysis of the Fourier coefficients discloses both particle size and strain, without the necessity for any prior assumption as to the existence of either [9,3, G.30, G.39]. Warren and Averbach [9.4] made the first measurements of this kind, on brass filings, and many similar studies followed [9.5]. Somewhat later, Paterson [9.6] showed that the Fourier coefficients of the line profile could also disclose the presence of stacking faults caused by cold work. (In FCC metals and alloys, for example, slip on 111 planes can here and there alter the normal stacking sequence ABCABC... of these planes to the faulted... 

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