Scherrer s equation

Table 1 shows that the physicochemical properties of the support material were modified by the pre-treatment process. The particle sizes. Dp, which are summarized in the Table 1 were calculated from the X-ray diffraction patterns of prepared catalysts and a commercial catalyst(30 wt% Pt-Ru/C E-TEK) by using Scherrer s equation. To avoid the interference from other peaks, (220) peak was used. All the prepared catalysts show the particle sizes of the range from 2.0 to 2.8nm. It can be thought that these values are in the acceptable range for the proper electrode performance[7]. For the prepared catalysts, notable differences are inter-metal distances(X[nm]) compared to commercial one. Due to their larger surface areas of support materials, active metals are apart from each other more than 2 3 times distance than commercial catalyst. Pt-Ru/SRaw has the longest inter-metal distances. 

The properties of titania particles were investigated using X-ray diffraction (XRD, Model D/MAX-RB, Rigaku Ltd.), scanning electron microscopy (SEM, Model 535M, Philips Ltd.), transmission electron microscopy (TEM, Model 2000EX, JEOL Ltd.). The crystallite sizes were estimated by Scherrer s equation and the composition of rutile phase in titania were estimated from the respective integrated XRD peak intensities. 

In order to deduce Scherrer s equation first an infinite crystal is considered that is, second, restricted (i.e multiplied) by a shape function (cf. p. 17). Thus from the Fourier convolution theorem (Sect. 2.7.8) it follows that in reciprocal space each reflection is convolved by the Fourier transform of the square of the shape function - and Scherrer s equation is readily established. 

TiOj, both anatase and rutile structures were found. On the contrary, the TiO (ST) sample showed only anatase structure. The average crystallite size was 9.3 nm for TiOj (ST) samples by using Scherrer s equation [8], which was significantly smaller compared to the P25 powder, with an average crystallite size between 15 and 25 nm [9]. 

Figure 12.11 shows the XRD patterns of a nanocrystalline Al film obtained at a constant potential of —1.7V for 2h at 100°C in the ionic liquid [Pyip] TFSA containing 1.6 M AICI3 on a glassy carbon substrate. The XRD patterns show the characteristic diffraction patterns of crystalline Al, furthermore the peaks are rather broad, indicating the small crystallite size of the electrodeposited Al. The grain size of Al was determined using Scherrer s equation to be 34 nm. For more information on the electrodeposition of nanocrystalline aluminum in the employed ionic liquid we refer to Refs. [3, 4]. 

Fe. The increase of the pore volume was mostly due to the production of mesopores. The crystallite size of a-Fe in the samples was calculated to be 15 30 nm from the 110 reflection based on the Scherrer s equation. 

Thermogravimetric analysis of the composite catalysts revealed that calcination at 400 °C doesn t provide any degradation of the support materials. XRD patterns reveal that only anatase phase can be identified for the prepared Ti02 and composite catalysts (rutile and brookite phases of Ti02 not being observed). In Table 1 are the surface areas of the catalysts as well as the anatase crystallite dimensions estimated by Scherrer s equation from the (101) reflection plane (20 25 ). 

The X-ray diffraction (XRD) patterns were obtained by Philips X pert Pro X-ray diffractometer equipped with a Cu-K source at 40 kV and 40 mA. The crystalline sizes of R particles were calculated from Scherrer s equation [15]. Transmission electron microscopy (TEM) images were obtained using the G2 FE-TEM Tecnai microscope at an accelerating voltage of 200 kV. The content of platinum and carbon in the sample was determined by inductively coupled plasma atomic emission spectroscopy (ICP-AES, RF source Jobin Yvon 2301, 40.68 MHz). 

Powder morphology was investigated using a transmission electron microscope (TEM, Model JEM-IOOCXII). Crystallite size of the powders and grain size of Nd YAG ceramics calcined at different temperatures were calculaied by X-ray diffraction (XRD, model D/maxrA, using nickel-filtered Cu-Ka radiation) patterns from the Scherrer s equation. Microstructures of the fractured and the thermal etched mirror-polished surfaces of Nd YAG specimens were observed by scanning electron microscopy (SEM, Model S-4800). Densities of the samples were measured by the Archimedes draining method. 

XRD analysis of Lu20j Eu powders calcinated at different temperatures for 2 hours are presented in Fig. 3. According to the XRD data, the precursor and products of its calcination at T<550 °C were amorphous. The formation of cubic lutetium oxide starts at T=550 °C. Temperature increase from 550 to 1000 °C leads to decrease of half-width of diffraction peaks and to increase of their intensity. This testifies an improvement of Lu203 Eu crystallinity. The average crystallite size estimated by Scherrer s equation also increases from 10.5 to 30.5 A (fig. 4). All the diffraction peaks on the XRD patterns were attributed to cubic Lu20j Eu. ... 

The high intensity and well defined lines located namely (200), (222), (321), (410), (421), (520), (530), (611), (541), (444), (650) and (741) lines. Theo value has been calculated using mean value of the cell parameters of the crystalline phase P-Si02 of sample 3 iso=l-3402nm close to the standard value a-1.3405nm ( 0.0034). The mean grain sizes evaluated by Scherrer s equation were lOnm around. 

Thin films of Ti02 are of interest as photocatalyzed depolluting layers. In such applications particle size and porosity are important material characteristics to be measured. In one such study (125) the crystallite sizes of Ti02 samples with different thermal treatments were measured. A sample treated at 600°C yielded a 20 = 1.2°, a full width at half maximum of the (101) diffraction peak at 20 = 25°, and a sample treated at 700°C yielded a 20 = 2.4° for the same (101) peak. For spherical particles (k = 1 and A = 0.154 nm) use Scherrer s equation to calculate the particle size at each treatment temperature. In the same study, the pore diameter of the 600°C and 700°C samples were found to be 5.5 nm and 8.5 nm, respectively. Are these pore sizes reasonable given the particle size results ... 

In 1910, Kolbe was the first to prove that dichroic nanocomposite samples based on gold contained the metal indeed in its zero-valence state. Such affirmation was confirmed a few years later by X-ray scattering. In particular, it was shown that zero-valence silver and gold were present in the respective nanocomposites made with oriented ramie fibers, and the ring-like interference patterns of the metal crystallites showed that the individual primary crystallites were not oriented (32). Based on Scherrer s equation, which was developed just in this period, the average particle diameter of silver and gold crystallites was determined in fibers of ramie, hemp, bamboo, silk, wool, viscose, and cellulose acetate to be between 5 and 14 nm (33). 

The data from Fig. 1.8 can also be used to determine the average crystallite size for each material present in the mixture. As indicated before, the positimi and angular width of each peak can be determined by peak fit. If the instrument resolution is properly taken into account in the determination of the peaks FWHM, then the resulting peak widths contain information about the material properties. A simple (and common) approach is to neglect micro-strain and defect contribution to the peak width, F (in radians), and assume that the only contribution to the line broadening is from the crystallite size, L, using Scherrer s equation ... 

Once the the instramental broadening and wavelength dispersion portions of the peak width are modeled, the remainder of the peak width can be described by Ae specimen function, which relates to the sample itself. More specifically, the peak broadening due to crystalhte size and microstiain can be described mathematically. Scherrer s equation expresses peak broadening, p, as k... 

Phase stability studies of the all the heat treated powders were then carried out by using X-ray diffractometer (Rigaku Geiger-Flex Difractometer, Japan). Subsequently the powder particle size was then obtained based on the FWHM theory, whereby the peak width was calculated according to the Scherrer s equation. The critical particle size was then obtained by comparing both XRD traces and calculated particle size. 

On the successful formation of forsterite, it was hypothesized that prolonged ball-milling hours could have led to the shrinkage of particle size, thus assisting the conversion of the starting precursors into pure forsterite. To affirm this, a FWHM study was conducted on the as-received, before heat treatment powders to obtain the particle size of the milled powders. According to Scherrer s equation (Equation 1), it is observed that the peak width is inversely proportional to the crystallite size. Therefore in order to obtain the powder particle size, further XRD examination as shown in Figure 3 on the ball-milled, untreated powders is carried out. 

By coupling the width of the most prominent peaks (= 28.6°) in Figure 3 with Scherrer s equation (Equation 1), calculated results indicated that particle size decreased with increasing ball-milling time as indicated in Figure 4. With the obtained results in Figure 4, this therefore validates the earlier drawn hypothesis on the refinement of powder particle size and its significance towards the formation of pure forsterite. In the current work, it has been observed that the successful formation of nano-crystalline forsterite occurred at a critical particle size of approximately 41 nm. 

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