Scherrer line broadening

The crystallite dimension is calculated from the degree of broadening of the (002) reflection using the Scherrer line broadening equation namely ... 

X-Ray diffraction has an important limitation Clear diffraction peaks are only observed when the sample possesses sufficient long-range order. The advantage of this limitation is that the width (or rather the shape) of diffraction peaks carries information on the dimensions of the reflecting planes. Diffraction lines from perfect crystals are very narrow, see for example the (111) and (200) reflections of large palladium particles in Fig. 4.5. For crystallite sizes below 100 nm, however, line broadening occurs due to incomplete destructive interference in scattering directions where the X-rays are out of phase. The two XRD patterns of supported Pd catalysts in Fig. 4.5 show that the reflections of palladium are much broader than those of the reference. The Scherrer formula relates crystal size to line width ... 

The Crystallite Size and the Scherrer Formula. The Scherrer formula is based on a restricted assumption assuming that the peak shape is dominated by size effects. The problem of Bragg line broadening, originating from particle size, was first investigated by Scherrer [30] who derived the well-known and widely used law ... 

There are a number of reasons why the XRD peaks of LDH samples are often rather broad. The relatively small domain size, particularly in the (00/) direction, leads to line broadening. The Scherrer equation may be used to estimate the domain size in the a and c directions from the width of the (110) and (00/) reflections, respectively [119,120], although the inherent approximations in this method should always be borne in mind [121,122]. 

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... 

An estimate of the surface area of Fe oxides in a mixture of other fine grained compounds, e. g. in soils or sediments, may be obtained from the crystal size calculated from XRD line broadening using the Scherrer formula (see Chap. 7) or from the difference in area before and after selective removal of the iron oxides divided by the amount of oxides that were extracted (see Chap. 16). The latter method assumes that the areas of the various components are additive (Schwertmann, 1988). 

The crystal size of soil Fe oxides usually ranges from a few to several hundred nm. A survey of 256 goethites, 101 hematites and 72 lepidocrocites from soils around the world showed maxima in the mean coherent length (MCL) perpendicular to (101), of 15-20 nm for goethite and ca. 40 nm perpedicular to (110), for hematite (Fig. 16.10). These values have been deduced from XRD line broadening using the Scherrer for-... 

In the Scherrer formula j3 is proportional to sec 0. For other causes of line-broadening, the relation is different therefore, in studying a particular substance, if /J is found to be proportional to sec 6, it is probably justifiable to assume that the broadening is due to the small size of the crystals. 

The Scherrer equation [Eq. (6-2)] indicates that measuring at smaller wavelengths gives sharper peaks, not only because X becomes smaller but also because the diffraction lines shift to lower angles, which decreases the 1/cos 9 term in Eq. (6-2). Both effects help to reduce line broadening. Thus, by using Mo Ka X-rays (17.44 keV X 0.07 nm), one can obtain diffraction patterns from smaller particles than with Cu Ka radiation (8.04 keV 0.15 nm). 

In order to study the growth of particles with temperature, anatase powder (preheated to 150°C) was heated for a period of 3 h at 400, 600, 800 and 1000°C. Marked increase in particle size was noticed in the 600-1000°C region, as indicated by the photo-micrographs. The specific surface area (B.E.T.) of anatase heated at 400°C was 55 m2/g and decreased markedly for samples heated to higher temperatures. The crystallite size normal to the (101) and (110) reflecting planes of anatase and rutile samples was calculated by measuring the X-ray diffraction line-widths of the samples heated at 200, 400, 600, 800 and 1000°C for 3 h. The Scherrer equation corrected for instrumental line-broadening by Warren s equation was employed for the calculation.16 The line-width of the sample heated at 1000°C was taken as the reference. The crystallite size increases rapidly after 600°C (fig. lb). The transformation of pure anatase also starts only above 600°C. 

The common application of the Scherrer formula in catalyst structure determination is a crude approximation to microstructural analysis. Strain and particle size give rise to the same effects, namely, line broadening, but fortunately causing different variations with diffraction angle, as shown by Equations (3) and (4). The methodologies implied by Wil-liamson-Hall plots and Warren-Averbach profile analyses provide access to the strain and size parameters in the commonly encountered case that both phenomena contribute to an experimental line broadening. 

The width of diffraction peaks carries information on the dimensions of the reflecting planes. Diffraction lines from perfect crystals are very narrow. For crystallite sizes below 100 nm, however, line broadening occurs due to incomplete destructive interference in scattering directions where the X-rays are out of phase. The Scherrer formula relates crystal size to line width ... 

Diffraction line broadening has been adapted to light diffraction to measure ordered domain size by Chang and Ring [56]. The first treatment of diffraction line broadening was by Scherrer [57], who derived the equation... 

Powder X-ray diffraction (XRD) is performed on a Siemens D5005 diffractometer with Cu Ka radiation. The particle size is calculated from the X-Ray line broadening, using the Debye-Scherrer equation. DRS is measured by Perkin-Elmer Lambda 20 UV-visible spectophotometer at room temperature in the wavelength region between 200 and 800 nm. Raman spectra are recorded on a Broker RFS 100 with 2 cm resolution. 

Selected chars were examined by x-ray diffraction (XRD) to identify Ca and Mg containing species present in the chars. Average crystallite diameters were determined using the line broadening concept and the Scherrer equation ( ). 

If the dimensions of the scattering particles fall below about 500 nm the interference pattern shows line broadening. Hence, the full width at half maximum (F WHM) value, /) is dependent on particle size D. This relation has been expressed by the Scherrer equation (Scherrer, 1918) as... 

Lundy and Eanes (1973) reported lengths in the c-axis direction of 200-400 A and microstrains of 3 to 4 x 10 determined from line broadening of the 002 and 004 apatite reflections from mineralized turkey leg tendon. They also tabulated earlier measurements of the dimensions of biological apatites in the c-axis direction. Wide-angle XRD also shows that older bone mineral is better crystallized than newly formed mineral. For example, Bonar et al. (1983) measured the improvement in crystallinity for bone from embryonic to 2-year-old chickens. They reported that the coherence length (neglecting microstrain) calculated from the Scherrer formula from the 002 peak increased from 107 to 199 A. In a similar study of bone from 3-week to 2-year-old rats (Ziv and Weiner 1994), the change in the same quantity was from approximately 200 to 280 A (means of 3 measurements). 

XRD patterns were recorded on a Philips X-ray generator with a Debye-Scherrer camera. CuKa X-rays (X, = 1.5418 A) were used as the X-ray source. XRD measiuements were performed on powdered washcoat scraped fi om the comer region of exposed channels on the outer surface of the cores. The XRD apparatus used to analyze the samples have been described in detail previously [8]. Particle sizes were estimated using the Scherrer equation with correction for instrumental line broadening. 

Phase identification of the resultant powders were performed by X-ray diffractometry (XRD, Model X pert PRO MPD, PANalytical, Almelo, the Netherlands) using CuK radiation. Crystallite size of the calcined powders was determined by X-ray line broadening and calculated using the Scherrer equation ... 

Though the crystallinity of cw-polyacetylene is remarkably high, there are strong indications of disorder. It is generally accepted that diffraction line broadening is a combined result of crystallite size and disorder in polyacetylene, A quantitative evaluation of the separate contributions is hardly feasible due to the limited number of separate reflections, but it is common practice to consider the number obtained from the Scherrer equation to be a diffraction coherence length rather than a crystallite size. No more than approximately 10 peaks are observed in diffraction, and there is considerable overlap of reflections.. The number of higher layer lines observable in oriented patterns is only... 

Average crystallite size of Ce02 calculated by the Scherrer equation from the XRD line broadening was 46 nm for powder A and 34 nm for powder B. The crystallite size increases as the calcined temperature increases. 

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